hypothesis testing with critical value and p value

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7.5: Critical values, p-values, and significance level

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Foster et al.
University of Missouri-St. Louis, Rice University, & University of Houston, Downtown Campus via University of Missouri’s Affordable and Open Access Educational Resources Initiative

A low probability value casts doubt on the null hypothesis. How low must the probability value be in order to conclude that the null hypothesis is false? Although there is clearly no right or wrong answer to this question, it is conventional to conclude the null hypothesis is false if the probability value is less than 0.05. More conservative researchers conclude the null hypothesis is false only if the probability value is less than 0.01. When a researcher concludes that the null hypothesis is false, the researcher is said to have rejected the null hypothesis. The probability value below which the null hypothesis is rejected is called the α level or simply \(α\) (“alpha”). It is also called the significance level. If α is not explicitly specified, assume that \(α\) = 0.05.

The significance level is a threshold we set before collecting data in order to determine whether or not we should reject the null hypothesis. We set this value beforehand to avoid biasing ourselves by viewing our results and then determining what criteria we should use. If our data produce values that meet or exceed this threshold, then we have sufficient evidence to reject the null hypothesis; if not, we fail to reject the null (we never “accept” the null).

There are two criteria we use to assess whether our data meet the thresholds established by our chosen significance level, and they both have to do with our discussions of probability and distributions. Recall that probability refers to the likelihood of an event, given some situation or set of conditions. In hypothesis testing, that situation is the assumption that the null hypothesis value is the correct value, or that there is no effect. The value laid out in H0 is our condition under which we interpret our results. To reject this assumption, and thereby reject the null hypothesis, we need results that would be very unlikely if the null was true. Now recall that values of z which fall in the tails of the standard normal distribution represent unlikely values. That is, the proportion of the area under the curve as or more extreme than \(z\) is very small as we get into the tails of the distribution. Our significance level corresponds to the area under the tail that is exactly equal to α: if we use our normal criterion of \(α\) = .05, then 5% of the area under the curve becomes what we call the rejection region (also called the critical region) of the distribution. This is illustrated in Figure \(\PageIndex{1}\).

The shaded rejection region takes us 5% of the area under the curve. Any result which falls in that region is sufficient evidence to reject the null hypothesis.

The rejection region is bounded by a specific \(z\)-value, as is any area under the curve. In hypothesis testing, the value corresponding to a specific rejection region is called the critical value, \(z_{crit}\) (“\(z\)-crit”) or \(z*\) (hence the other name “critical region”). Finding the critical value works exactly the same as finding the z-score corresponding to any area under the curve like we did in Unit 1. If we go to the normal table, we will find that the z-score corresponding to 5% of the area under the curve is equal to 1.645 (\(z\) = 1.64 corresponds to 0.0405 and \(z\) = 1.65 corresponds to 0.0495, so .05 is exactly in between them) if we go to the right and -1.645 if we go to the left. The direction must be determined by your alternative hypothesis, and drawing then shading the distribution is helpful for keeping directionality straight.

Suppose, however, that we want to do a non-directional test. We need to put the critical region in both tails, but we don’t want to increase the overall size of the rejection region (for reasons we will see later). To do this, we simply split it in half so that an equal proportion of the area under the curve falls in each tail’s rejection region. For \(α\) = .05, this means 2.5% of the area is in each tail, which, based on the z-table, corresponds to critical values of \(z*\) = ±1.96. This is shown in Figure \(\PageIndex{2}\).

Thus, any \(z\)-score falling outside ±1.96 (greater than 1.96 in absolute value) falls in the rejection region. When we use \(z\)-scores in this way, the obtained value of \(z\) (sometimes called \(z\)-obtained) is something known as a test statistic, which is simply an inferential statistic used to test a null hypothesis. The formula for our \(z\)-statistic has not changed:

\[z=\dfrac{\overline{\mathrm{X}}-\mu}{\bar{\sigma} / \sqrt{\mathrm{n}}} \]

To formally test our hypothesis, we compare our obtained \(z\)-statistic to our critical \(z\)-value. If \(\mathrm{Z}_{\mathrm{obt}}>\mathrm{Z}_{\mathrm{crit}}\), that means it falls in the rejection region (to see why, draw a line for \(z\) = 2.5 on Figure \(\PageIndex{1}\) or Figure \(\PageIndex{2}\)) and so we reject \(H_0\). If \(\mathrm{Z}_{\mathrm{obt}}<\mathrm{Z}_{\mathrm{crit}}\), we fail to reject. Remember that as \(z\) gets larger, the corresponding area under the curve beyond \(z\) gets smaller. Thus, the proportion, or \(p\)-value, will be smaller than the area for \(α\), and if the area is smaller, the probability gets smaller. Specifically, the probability of obtaining that result, or a more extreme result, under the condition that the null hypothesis is true gets smaller.

The \(z\)-statistic is very useful when we are doing our calculations by hand. However, when we use computer software, it will report to us a \(p\)-value, which is simply the proportion of the area under the curve in the tails beyond our obtained \(z\)-statistic. We can directly compare this \(p\)-value to \(α\) to test our null hypothesis: if \(p < α\), we reject \(H_0\), but if \(p > α\), we fail to reject. Note also that the reverse is always true: if we use critical values to test our hypothesis, we will always know if \(p\) is greater than or less than \(α\). If we reject, we know that \(p < α\) because the obtained \(z\)-statistic falls farther out into the tail than the critical \(z\)-value that corresponds to \(α\), so the proportion (\(p\)-value) for that \(z\)-statistic will be smaller. Conversely, if we fail to reject, we know that the proportion will be larger than \(α\) because the \(z\)-statistic will not be as far into the tail. This is illustrated for a one-tailed test in Figure \(\PageIndex{3}\).

When the null hypothesis is rejected, the effect is said to be statistically significant. For example, in the Physicians Reactions case study, the probability value is 0.0057. Therefore, the effect of obesity is statistically significant and the null hypothesis that obesity makes no difference is rejected. It is very important to keep in mind that statistical significance means only that the null hypothesis of exactly no effect is rejected; it does not mean that the effect is important, which is what “significant” usually means. When an effect is significant, you can have confidence the effect is not exactly zero. Finding that an effect is significant does not tell you about how large or important the effect is. Do not confuse statistical significance with practical significance. A small effect can be highly significant if the sample size is large enough. Why does the word “significant” in the phrase “statistically significant” mean something so different from other uses of the word? Interestingly, this is because the meaning of “significant” in everyday language has changed. It turns out that when the procedures for hypothesis testing were developed, something was “significant” if it signified something. Thus, finding that an effect is statistically significant signifies that the effect is real and not due to chance. Over the years, the meaning of “significant” changed, leading to the potential misinterpretation.

P-Value vs. Critical Value: A Friendly Guide for Beginners

In the world of statistics, you may have come across the terms p-value and critical value . These concepts are essential in hypothesis testing, a process that helps you make informed decisions based on data. As you embark on your journey to understand the significance and applications of these values, don’t worry; you’re not alone. Many professionals and students alike grapple with these concepts, but once you get the hang of what they mean, they become powerful tools at your fingertips.

The main difference between p-value and critical value is that the p-value quantifies the strength of evidence against a null hypothesis, while the critical value sets a threshold for assessing the significance of a test statistic. Simply put, if your p-value is below the critical value, you reject the null hypothesis.

As you read on, you can expect to dive deeper into the definitions, applications, and interpretations of these often misunderstood statistical concepts. The remainder of the article will guide you through how p-values and critical values work in real-world scenarios, tips on interpreting their results, and potential pitfalls to avoid. By the end, you’ll have a clear understanding of their role in hypothesis testing, helping you become a more effective researcher or analyst.

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Table of Contents

Understanding P-Value and Critical Value

When you dive into the world of statistics, it’s essential to grasp the concepts of P-value and critical value . These two values play a crucial role in hypothesis testing, helping you make informed decisions based on data. In this section, we will focus on the concept of hypothesis testing and how P-value and critical value relate to it.

hypothesis testing with critical value and p value

Concept of Hypothesis Testing

Hypothesis testing is a statistical technique used to analyze data and draw conclusions. You start by creating a null hypothesis (H0) and an alternative hypothesis (H1). The null hypothesis represents the idea that there is no significant effect or relationship between the variables being tested, while the alternative hypothesis claims that there is a significant effect or relationship.

To conduct a hypothesis test, follow these steps:

Formulate your null and alternative hypotheses.
Choose an appropriate statistical test and significance level (α).
Collect and analyze your data.
Calculate the test statistic and P-value.
Compare the P-value to the critical value.

Now, let’s discuss how P-value and critical value come into play during hypothesis testing.

The P-value is the probability of observing a test statistic as extreme (or more extreme) than the one calculated if the null hypothesis were true. In simpler terms, it’s the likelihood of getting your observed results by chance alone. The lower the P-value, the more evidence you have against the null hypothesis.

Here’s what you need to know about P-values:

A low P-value (typically ≤ 0.05) indicates that the null hypothesis is unlikely to be true.
A high P-value (typically > 0.05) suggests that the observed results align with the null hypothesis.

Critical Value

The critical value is a threshold that defines whether the test statistic is extreme enough to reject the null hypothesis. It depends on the chosen significance level (α) and the specific statistical test being used. If the test statistic exceeds the critical value, you reject the null hypothesis in favor of the alternative.

To summarize:

If the P-value ≤ critical value, reject the null hypothesis.
If the P-value > critical value, fail to reject the null hypothesis (do not conclude that the alternative is true).

In conclusion, understanding P-value and critical value is crucial for hypothesis testing. They help you determine the significance of your findings and make data-driven decisions. By grasping these concepts, you’ll be well-equipped to analyze data and draw meaningful conclusions in a variety of contexts.

P-Value Essentials

Calculating and interpreting p-values is essential to understanding statistical significance in research. In this section, we’ll cover the basics of p-values and how they relate to critical values.

Calculating P-Values

A p-value represents the probability of obtaining a result at least as extreme as the observed data, assuming the null hypothesis is correct. To calculate a p-value, follow these steps:

Define your null and alternative hypotheses.
Determine the test statistic and its distribution.
Calculate the observed test statistic based on your sample data.
Find the probability of obtaining a test statistic at least as extreme as the observed value.

Let’s dive deeper into these steps:

Step 1: Formulate the null hypothesis (H₀) and alternative hypothesis (H₁). The null hypothesis typically states that there is no effect or relationship between variables, while the alternative hypothesis suggests otherwise.
Step 2: Determine your test statistic and its distribution. The choice of test statistic depends on your data and hypotheses. Some common test statistics include the t -test, z -test, or chi-square test.
Step 3: Using your sample data, compute the test statistic. This value quantifies the difference between your sample data and the null hypothesis.
Step 4: Find the probability of obtaining a test statistic at least as extreme as the observed value, under the assumption that the null hypothesis is true. This probability is the p-value .

Interpreting P-Values

Once you’ve calculated the p-value, it’s time to interpret your results. The interpretation depends on the pre-specified significance level (α) you’ve chosen. Here’s a simplified guideline:

If p-value ≤ α , you can reject the null hypothesis.
If p-value > α , you cannot reject the null hypothesis.

Keep in mind that:

A lower p-value indicates stronger evidence against the null hypothesis.
A higher p-value implies weaker evidence against the null hypothesis.

Remember that statistical significance (p-value ≤ α) does not guarantee practical or scientific significance. It’s essential not to take the p-value as the sole metric for decision-making, but rather as a tool to help gauge your research outcomes.

In summary, p-values are crucial in understanding and interpreting statistical research results. By calculating and appropriately interpreting p-values, you can deepen your knowledge of your data and make informed decisions based on statistical evidence.

Critical Value Essentials

In this section, we’ll discuss two important aspects of critical values: Significance Level and Rejection Region . Knowing these concepts helps you better understand hypothesis testing and make informed decisions about the statistical significance of your results.

Significance Level

The significance level , often denoted as α or alpha, is an essential part of hypothesis testing. You can think of it as the threshold for deciding whether your results are statistically significant or not. In general, a common significance level is 0.05 or 5% , which means that there is a 5% chance of rejecting a true null hypothesis.

To help you understand better, here are a few key points:

The lower the significance level, the more stringent the test.
Higher α-levels may increase the risk of Type I errors (incorrectly rejecting the null hypothesis).
Lower α-levels may increase the risk of Type II errors (failing to reject a false null hypothesis).

Rejection Region

The rejection region is the range of values that, if your test statistic falls within, leads to the rejection of the null hypothesis. This area depends on the critical value and the significance level. The critical value is a specific point that separates the rejection region from the rest of the distribution. Test statistics that fall in the rejection region provide evidence that the null hypothesis might not be true and should be rejected.

Here are essential points to consider when using the rejection region:

Z-score : The z-score is a measure of how many standard deviations away from the mean a given value is. If your test statistic lies in the rejection region, it means that the z-score is significant.
Rejection regions are tailored for both one-tailed and two-tailed tests.
In a one-tailed test, the rejection region is either on the left or right side of the distribution.
In a two-tailed test, there are two rejection regions, one on each side of the distribution.

By understanding and considering the significance level and rejection region, you can more effectively interpret your statistical results and avoid making false assumptions or claims. Remember that critical values are crucial in determining whether to reject or accept the null hypothesis.

Statistical Tests and Decision Making

When you’re comparing the means of two samples, a t-test is often used. This test helps you determine whether there is a significant difference between the means. Here’s how you can conduct a t-test:

Calculate the t-statistic for your samples
Determine the degrees of freedom
Compare the t-statistic to a critical value from a t-distribution table

If the t-statistic is greater than the critical value, you can reject the null hypothesis and conclude that there is a significant difference between the sample means. Some key points about t-test:

Test statistic : In a t-test, the t-statistic is the key value that you calculate
Sample : For a t-test, you’ll need two independent samples to compare

The Analysis of Variance (ANOVA) is another statistical test, often used when you want to compare the means of three or more treatment groups. With this method, you analyze the differences between group means and make decisions on whether the total variation in the dataset can be accounted for by the variance within the groups or the variance between the groups. Here are the main steps in conducting an ANOVA test:

Calculate the F statistic
Determine the degrees of freedom for between-groups and within-groups
Compare the F statistic to a critical value from an F-distribution table

When the F statistic is larger than the critical value, you can reject the null hypothesis and conclude that there is a significant difference among the treatment groups. Keep these points in mind for ANOVA tests:

Treatment Groups : ANOVA tests require three or more groups to compare
Observations : You need multiple observations within each treatment group

Confidence Intervals

Confidence intervals (CIs) are a way to estimate values within a certain range, with a specified level of confidence. They help to indicate the reliability of an estimated parameter, like the mean or difference between sample means. Here’s what you need to know about calculating confidence intervals:

Determine the point estimate (e.g., sample mean or difference in means)
Calculate the standard error
Multiply the standard error by the appropriate critical value

The result gives you a range within which the true population parameter is likely to fall, with a certain level of confidence (e.g., 95%). Remember these insights when working with confidence intervals:

Confidence Level : The confidence level is the probability that the true population parameter falls within the calculated interval
Critical Value : Based on the specified confidence level, you’ll determine a critical value from a table (e.g., t-distribution)

Remember, using appropriate statistical tests, test statistics, and critical values will help you make informed decisions in your data analysis.

Comparing P-Values and Critical Values

Differences and Similarities

When analyzing data, you may come across two important concepts – p-values and critical values . While they both help determine the significance of a data set, they have some differences and similarities.

P-values are probabilities, ranging from 0 to 1, indicating how likely it is a particular result could be observed if the null hypothesis is true. Lower p-values suggest the null hypothesis should be rejected, meaning the observed data is not due to chance alone.
On the other hand, critical values are preset thresholds that decide whether the null hypothesis should be rejected or not. Results that surpass the critical value support adopting the alternative hypothesis.

The main similarity between p-values and critical values is their role in hypothesis testing. Both are used to determine if observed data provides enough evidence to reject the null hypothesis in favor of the alternative hypothesis.

Applications in Geospatial Data Analysis

In the field of geospatial data analysis, p-values and critical values play essential roles in making data-driven decisions. Researchers like Hartmann, Krois, and Waske from the Department of Earth Sciences at Freie Universitaet Berlin often use these concepts in their e-Learning project SOGA.

To better understand the applications, let’s look at three main aspects:

Spatial autocorrelation : With geospatial data, points might be related not only by their values but also by their locations. P-values can help assess spatial autocorrelation and recognize underlying spatial patterns.
Geostatistical analysis : Techniques like kriging or semivariogram estimation depend on critical values and p-values to decide the suitability of a model. By finding the best fit model, geospatial data can be better represented, ensuring accurate and precise predictions.
Comparing geospatial data groups : When comparing two subsets of data (e.g., mineral concentrations, soil types), p-values can be used in permutation tests or t-tests to verify if the observed differences are significant or due to chance.

In summary, when working with geospatial data analysis, p-values and critical values are crucial tools that enable you to make informed decisions about your data and its implications. By understanding the differences and similarities between the two concepts, you can apply them effectively in your geospatial data analysis journey.

Standard Distributions and Scores

In this section, we will discuss the Standard Normal Distribution and its associated scores, namely Z-Score and T-Statistic . These concepts are crucial in understanding the differences between p-values and critical values.

Standard Normal Distribution

The Standard Normal Distribution is a probability distribution that has a mean of 0 and a standard deviation of 1. This distribution is crucial for hypothesis testing, as it helps you make inferences about your data based on standard deviations from the mean. Some characteristics of this distribution include:

68% of the data falls within ±1 standard deviation from the mean
95% of the data falls within ±2 standard deviations from the mean
99.7% of the data falls within ±3 standard deviations from the mean

The Z-Score is a measure of how many standard deviations away a data point is from the mean of the distribution. It is used to compare data points across different distributions with different means and standard deviations. To calculate the Z-Score, use the formula:

Key features of the Z-Score include:

Positive Z-Scores indicate values above the mean
Negative Z-Scores indicate values below the mean
A Z-Score of 0 is equal to the mean

T-Statistic

The T-Statistic , also known as the Student’s t-distribution , is another way to assess how far away a data point is from the mean. It comes in handy when:

You have a small sample size (generally less than 30)
Population variance is not known
Population is assumed to be normally distributed

The T-Statistic shares similarities with the Z-Score but adjusts for sample size, making it more appropriate for smaller samples. The formula for calculating the T-Statistic is:

In conclusion, understanding the Standard Normal Distribution , Z-Score , and T-Statistic will help you better differentiate between p-values and critical values, ultimately aiding in accurate statistical analysis and hypothesis testing.

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Frequently Asked Questions

What is the relationship between p-value and critical value.

The p-value represents the probability of observing the test statistic under the null hypothesis, while the critical value is a predetermined threshold for declaring significance. If the p-value is less than the critical value, you reject the null hypothesis.

How do you interpret p-value in comparison to critical value?

When the p-value is smaller than the critical value , there is strong evidence against the null hypothesis, which means you reject it. In contrast, if the p-value is larger, you fail to reject the null hypothesis and cannot conclude a significant effect.

What does it mean when the p-value is greater than the critical value?

If the p-value is greater than the critical value , it indicates that the observed data are consistent with the null hypothesis, and you do not have enough evidence to reject it. In other words, the finding is not statistically significant.

How are critical values used to determine significance?

Critical values are used as a threshold to determine if a test statistic is considered significant. When the test statistic is more extreme than the critical value, you reject the null hypothesis, indicating that the observed effect is unlikely due to chance alone.

Why is it important to know both p-value and critical value in hypothesis testing?

Knowing both p-value and critical value helps you to:

Understand the strength of evidence against the null hypothesis
Decide whether to reject or fail to reject the null hypothesis
Assess the statistical significance of your findings
Avoid misinterpretations and false conclusions

How do you calculate critical values and compare them to p-values?

To calculate critical values, you:

Choose a significance level (α)
Determine the appropriate test statistic distribution
Find the value that corresponds to α in the distribution

Then, you compare the calculated critical value with the p-value to determine if the result is statistically significant or not. If the p-value is less than the critical value, you reject the null hypothesis.

BEFORE YOU GO: Don’t forget to check out my latest article – 6 Proven Steps To Becoming a Data Scientist [Complete Guide] . We interviewed numerous data science professionals (data scientists, hiring managers, recruiters – you name it) and created this comprehensive guide to help you land that perfect data science job.

Affiliate Disclosure: We participate in several affiliate programs and may be compensated if you make a purchase using our referral link, at no additional cost to you. You can, however, trust the integrity of our recommendation. Affiliate programs exist even for products that we are not recommending. We only choose to recommend you the products that we actually believe in.

Daisy is the founder of DataScienceNerd.com. Passionate for the field of Data Science, she shares her learnings and experiences in this domain, with the hope to help other Data Science enthusiasts in their path down this incredible discipline.

StatPearls [Internet].

Hypothesis testing, p values, confidence intervals, and significance.

Jacob Shreffler ; Martin R. Huecker .

Affiliations

Last Update: March 13, 2023 .

Definition/Introduction

Medical providers often rely on evidence-based medicine to guide decision-making in practice. Often a research hypothesis is tested with results provided, typically with p values, confidence intervals, or both. Additionally, statistical or research significance is estimated or determined by the investigators. Unfortunately, healthcare providers may have different comfort levels in interpreting these findings, which may affect the adequate application of the data.

Issues of Concern

Without a foundational understanding of hypothesis testing, p values, confidence intervals, and the difference between statistical and clinical significance, it may affect healthcare providers' ability to make clinical decisions without relying purely on the research investigators deemed level of significance. Therefore, an overview of these concepts is provided to allow medical professionals to use their expertise to determine if results are reported sufficiently and if the study outcomes are clinically appropriate to be applied in healthcare practice.

Hypothesis Testing

Investigators conducting studies need research questions and hypotheses to guide analyses. Starting with broad research questions (RQs), investigators then identify a gap in current clinical practice or research. Any research problem or statement is grounded in a better understanding of relationships between two or more variables. For this article, we will use the following research question example:

Research Question: Is Drug 23 an effective treatment for Disease A?

Research questions do not directly imply specific guesses or predictions; we must formulate research hypotheses. A hypothesis is a predetermined declaration regarding the research question in which the investigator(s) makes a precise, educated guess about a study outcome. This is sometimes called the alternative hypothesis and ultimately allows the researcher to take a stance based on experience or insight from medical literature. An example of a hypothesis is below.

Research Hypothesis: Drug 23 will significantly reduce symptoms associated with Disease A compared to Drug 22.

The null hypothesis states that there is no statistical difference between groups based on the stated research hypothesis.

Researchers should be aware of journal recommendations when considering how to report p values, and manuscripts should remain internally consistent.

Regarding p values, as the number of individuals enrolled in a study (the sample size) increases, the likelihood of finding a statistically significant effect increases. With very large sample sizes, the p-value can be very low significant differences in the reduction of symptoms for Disease A between Drug 23 and Drug 22. The null hypothesis is deemed true until a study presents significant data to support rejecting the null hypothesis. Based on the results, the investigators will either reject the null hypothesis (if they found significant differences or associations) or fail to reject the null hypothesis (they could not provide proof that there were significant differences or associations).

To test a hypothesis, researchers obtain data on a representative sample to determine whether to reject or fail to reject a null hypothesis. In most research studies, it is not feasible to obtain data for an entire population. Using a sampling procedure allows for statistical inference, though this involves a certain possibility of error. [1] When determining whether to reject or fail to reject the null hypothesis, mistakes can be made: Type I and Type II errors. Though it is impossible to ensure that these errors have not occurred, researchers should limit the possibilities of these faults. [2]

Significance

Significance is a term to describe the substantive importance of medical research. Statistical significance is the likelihood of results due to chance. [3] Healthcare providers should always delineate statistical significance from clinical significance, a common error when reviewing biomedical research. [4] When conceptualizing findings reported as either significant or not significant, healthcare providers should not simply accept researchers' results or conclusions without considering the clinical significance. Healthcare professionals should consider the clinical importance of findings and understand both p values and confidence intervals so they do not have to rely on the researchers to determine the level of significance. [5] One criterion often used to determine statistical significance is the utilization of p values.

P values are used in research to determine whether the sample estimate is significantly different from a hypothesized value. The p-value is the probability that the observed effect within the study would have occurred by chance if, in reality, there was no true effect. Conventionally, data yielding a p<0.05 or p<0.01 is considered statistically significant. While some have debated that the 0.05 level should be lowered, it is still universally practiced. [6] Hypothesis testing allows us to determine the size of the effect.

An example of findings reported with p values are below:

Statement: Drug 23 reduced patients' symptoms compared to Drug 22. Patients who received Drug 23 (n=100) were 2.1 times less likely than patients who received Drug 22 (n = 100) to experience symptoms of Disease A, p<0.05.

Statement:Individuals who were prescribed Drug 23 experienced fewer symptoms (M = 1.3, SD = 0.7) compared to individuals who were prescribed Drug 22 (M = 5.3, SD = 1.9). This finding was statistically significant, p= 0.02.

For either statement, if the threshold had been set at 0.05, the null hypothesis (that there was no relationship) should be rejected, and we should conclude significant differences. Noticeably, as can be seen in the two statements above, some researchers will report findings with < or > and others will provide an exact p-value (0.000001) but never zero [6] . When examining research, readers should understand how p values are reported. The best practice is to report all p values for all variables within a study design, rather than only providing p values for variables with significant findings. [7] The inclusion of all p values provides evidence for study validity and limits suspicion for selective reporting/data mining.

While researchers have historically used p values, experts who find p values problematic encourage the use of confidence intervals. [8] . P-values alone do not allow us to understand the size or the extent of the differences or associations. [3] In March 2016, the American Statistical Association (ASA) released a statement on p values, noting that scientific decision-making and conclusions should not be based on a fixed p-value threshold (e.g., 0.05). They recommend focusing on the significance of results in the context of study design, quality of measurements, and validity of data. Ultimately, the ASA statement noted that in isolation, a p-value does not provide strong evidence. [9]

When conceptualizing clinical work, healthcare professionals should consider p values with a concurrent appraisal study design validity. For example, a p-value from a double-blinded randomized clinical trial (designed to minimize bias) should be weighted higher than one from a retrospective observational study [7] . The p-value debate has smoldered since the 1950s [10] , and replacement with confidence intervals has been suggested since the 1980s. [11]

Confidence Intervals

A confidence interval provides a range of values within given confidence (e.g., 95%), including the accurate value of the statistical constraint within a targeted population. [12] Most research uses a 95% CI, but investigators can set any level (e.g., 90% CI, 99% CI). [13] A CI provides a range with the lower bound and upper bound limits of a difference or association that would be plausible for a population. [14] Therefore, a CI of 95% indicates that if a study were to be carried out 100 times, the range would contain the true value in 95, [15] confidence intervals provide more evidence regarding the precision of an estimate compared to p-values. [6]

In consideration of the similar research example provided above, one could make the following statement with 95% CI:

Statement: Individuals who were prescribed Drug 23 had no symptoms after three days, which was significantly faster than those prescribed Drug 22; there was a mean difference between the two groups of days to the recovery of 4.2 days (95% CI: 1.9 – 7.8).

It is important to note that the width of the CI is affected by the standard error and the sample size; reducing a study sample number will result in less precision of the CI (increase the width). [14] A larger width indicates a smaller sample size or a larger variability. [16] A researcher would want to increase the precision of the CI. For example, a 95% CI of 1.43 – 1.47 is much more precise than the one provided in the example above. In research and clinical practice, CIs provide valuable information on whether the interval includes or excludes any clinically significant values. [14]

Null values are sometimes used for differences with CI (zero for differential comparisons and 1 for ratios). However, CIs provide more information than that. [15] Consider this example: A hospital implements a new protocol that reduced wait time for patients in the emergency department by an average of 25 minutes (95% CI: -2.5 – 41 minutes). Because the range crosses zero, implementing this protocol in different populations could result in longer wait times; however, the range is much higher on the positive side. Thus, while the p-value used to detect statistical significance for this may result in "not significant" findings, individuals should examine this range, consider the study design, and weigh whether or not it is still worth piloting in their workplace.

Similarly to p-values, 95% CIs cannot control for researchers' errors (e.g., study bias or improper data analysis). [14] In consideration of whether to report p-values or CIs, researchers should examine journal preferences. When in doubt, reporting both may be beneficial. [13] An example is below:

Reporting both: Individuals who were prescribed Drug 23 had no symptoms after three days, which was significantly faster than those prescribed Drug 22, p = 0.009. There was a mean difference between the two groups of days to the recovery of 4.2 days (95% CI: 1.9 – 7.8).

Clinical Significance

Recall that clinical significance and statistical significance are two different concepts. Healthcare providers should remember that a study with statistically significant differences and large sample size may be of no interest to clinicians, whereas a study with smaller sample size and statistically non-significant results could impact clinical practice. [14] Additionally, as previously mentioned, a non-significant finding may reflect the study design itself rather than relationships between variables.

Healthcare providers using evidence-based medicine to inform practice should use clinical judgment to determine the practical importance of studies through careful evaluation of the design, sample size, power, likelihood of type I and type II errors, data analysis, and reporting of statistical findings (p values, 95% CI or both). [4] Interestingly, some experts have called for "statistically significant" or "not significant" to be excluded from work as statistical significance never has and will never be equivalent to clinical significance. [17]

The decision on what is clinically significant can be challenging, depending on the providers' experience and especially the severity of the disease. Providers should use their knowledge and experiences to determine the meaningfulness of study results and make inferences based not only on significant or insignificant results by researchers but through their understanding of study limitations and practical implications.

Nursing, Allied Health, and Interprofessional Team Interventions

All physicians, nurses, pharmacists, and other healthcare professionals should strive to understand the concepts in this chapter. These individuals should maintain the ability to review and incorporate new literature for evidence-based and safe care.

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Disclosure: Jacob Shreffler declares no relevant financial relationships with ineligible companies.

Disclosure: Martin Huecker declares no relevant financial relationships with ineligible companies.

This book is distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ), which permits others to distribute the work, provided that the article is not altered or used commercially. You are not required to obtain permission to distribute this article, provided that you credit the author and journal.

Cite this Page Shreffler J, Huecker MR. Hypothesis Testing, P Values, Confidence Intervals, and Significance. [Updated 2023 Mar 13]. In: StatPearls [Internet]. Treasure Island (FL): StatPearls Publishing; 2024 Jan-.

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Understanding P values | Definition and Examples

Understanding P-values | Definition and Examples

Published on July 16, 2020 by Rebecca Bevans . Revised on June 22, 2023.

The p value is a number, calculated from a statistical test, that describes how likely you are to have found a particular set of observations if the null hypothesis were true.

P values are used in hypothesis testing to help decide whether to reject the null hypothesis. The smaller the p value, the more likely you are to reject the null hypothesis.

What is a null hypothesis, what exactly is a p value, how do you calculate the p value, p values and statistical significance, reporting p values, caution when using p values, other interesting articles, frequently asked questions about p-values.

All statistical tests have a null hypothesis. For most tests, the null hypothesis is that there is no relationship between your variables of interest or that there is no difference among groups.

For example, in a two-tailed t test , the null hypothesis is that the difference between two groups is zero.

Null hypothesis ( H 0 ): there is no difference in longevity between the two groups.
Alternative hypothesis ( H A or H 1 ): there is a difference in longevity between the two groups.

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The p value , or probability value, tells you how likely it is that your data could have occurred under the null hypothesis. It does this by calculating the likelihood of your test statistic , which is the number calculated by a statistical test using your data.

The p value tells you how often you would expect to see a test statistic as extreme or more extreme than the one calculated by your statistical test if the null hypothesis of that test was true. The p value gets smaller as the test statistic calculated from your data gets further away from the range of test statistics predicted by the null hypothesis.

The p value is a proportion: if your p value is 0.05, that means that 5% of the time you would see a test statistic at least as extreme as the one you found if the null hypothesis was true.

P values are usually automatically calculated by your statistical program (R, SPSS, etc.).

You can also find tables for estimating the p value of your test statistic online. These tables show, based on the test statistic and degrees of freedom (number of observations minus number of independent variables) of your test, how frequently you would expect to see that test statistic under the null hypothesis.

The calculation of the p value depends on the statistical test you are using to test your hypothesis :

Different statistical tests have different assumptions and generate different test statistics. You should choose the statistical test that best fits your data and matches the effect or relationship you want to test.
The number of independent variables you include in your test changes how large or small the test statistic needs to be to generate the same p value.

No matter what test you use, the p value always describes the same thing: how often you can expect to see a test statistic as extreme or more extreme than the one calculated from your test.

P values are most often used by researchers to say whether a certain pattern they have measured is statistically significant.

Statistical significance is another way of saying that the p value of a statistical test is small enough to reject the null hypothesis of the test.

How small is small enough? The most common threshold is p < 0.05; that is, when you would expect to find a test statistic as extreme as the one calculated by your test only 5% of the time. But the threshold depends on your field of study – some fields prefer thresholds of 0.01, or even 0.001.

The threshold value for determining statistical significance is also known as the alpha value.

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P values of statistical tests are usually reported in the results section of a research paper , along with the key information needed for readers to put the p values in context – for example, correlation coefficient in a linear regression , or the average difference between treatment groups in a t -test.

P values are often interpreted as your risk of rejecting the null hypothesis of your test when the null hypothesis is actually true.

In reality, the risk of rejecting the null hypothesis is often higher than the p value, especially when looking at a single study or when using small sample sizes. This is because the smaller your frame of reference, the greater the chance that you stumble across a statistically significant pattern completely by accident.

P values are also often interpreted as supporting or refuting the alternative hypothesis. This is not the case. The p value can only tell you whether or not the null hypothesis is supported. It cannot tell you whether your alternative hypothesis is true, or why.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

Normal distribution
Descriptive statistics
Measures of central tendency
Correlation coefficient
Null hypothesis

Methodology

Cluster sampling
Stratified sampling
Types of interviews
Cohort study
Thematic analysis

Research bias

Implicit bias
Cognitive bias
Survivorship bias
Availability heuristic
Nonresponse bias
Regression to the mean

A p -value , or probability value, is a number describing how likely it is that your data would have occurred under the null hypothesis of your statistical test .

P -values are usually automatically calculated by the program you use to perform your statistical test. They can also be estimated using p -value tables for the relevant test statistic .

P -values are calculated from the null distribution of the test statistic. They tell you how often a test statistic is expected to occur under the null hypothesis of the statistical test, based on where it falls in the null distribution.

If the test statistic is far from the mean of the null distribution, then the p -value will be small, showing that the test statistic is not likely to have occurred under the null hypothesis.

Statistical significance is a term used by researchers to state that it is unlikely their observations could have occurred under the null hypothesis of a statistical test . Significance is usually denoted by a p -value , or probability value.

Statistical significance is arbitrary – it depends on the threshold, or alpha value, chosen by the researcher. The most common threshold is p < 0.05, which means that the data is likely to occur less than 5% of the time under the null hypothesis .

When the p -value falls below the chosen alpha value, then we say the result of the test is statistically significant.

No. The p -value only tells you how likely the data you have observed is to have occurred under the null hypothesis .

If the p -value is below your threshold of significance (typically p < 0.05), then you can reject the null hypothesis, but this does not necessarily mean that your alternative hypothesis is true.

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S.3.1 hypothesis testing (critical value approach).

The critical value approach involves determining "likely" or "unlikely" by determining whether or not the observed test statistic is more extreme than would be expected if the null hypothesis were true. That is, it entails comparing the observed test statistic to some cutoff value, called the " critical value ." If the test statistic is more extreme than the critical value, then the null hypothesis is rejected in favor of the alternative hypothesis. If the test statistic is not as extreme as the critical value, then the null hypothesis is not rejected.

Specifically, the four steps involved in using the critical value approach to conducting any hypothesis test are:

Specify the null and alternative hypotheses.
Using the sample data and assuming the null hypothesis is true, calculate the value of the test statistic. To conduct the hypothesis test for the population mean μ , we use the t -statistic \(t^*=\frac{\bar{x}-\mu}{s/\sqrt{n}}\) which follows a t -distribution with n - 1 degrees of freedom.
Determine the critical value by finding the value of the known distribution of the test statistic such that the probability of making a Type I error — which is denoted \(\alpha\) (greek letter "alpha") and is called the " significance level of the test " — is small (typically 0.01, 0.05, or 0.10).
Compare the test statistic to the critical value. If the test statistic is more extreme in the direction of the alternative than the critical value, reject the null hypothesis in favor of the alternative hypothesis. If the test statistic is less extreme than the critical value, do not reject the null hypothesis.

Example S.3.1.1

Mean gpa section .

In our example concerning the mean grade point average, suppose we take a random sample of n = 15 students majoring in mathematics. Since n = 15, our test statistic t * has n - 1 = 14 degrees of freedom. Also, suppose we set our significance level α at 0.05 so that we have only a 5% chance of making a Type I error.

Right-Tailed

The critical value for conducting the right-tailed test H 0 : μ = 3 versus H A : μ > 3 is the t -value, denoted t \(\alpha\) , n - 1 , such that the probability to the right of it is \(\alpha\). It can be shown using either statistical software or a t -table that the critical value t 0.05,14 is 1.7613. That is, we would reject the null hypothesis H 0 : μ = 3 in favor of the alternative hypothesis H A : μ > 3 if the test statistic t * is greater than 1.7613. Visually, the rejection region is shaded red in the graph.

t distribution graph for a t value of 1.76131

Left-Tailed

The critical value for conducting the left-tailed test H 0 : μ = 3 versus H A : μ < 3 is the t -value, denoted -t ( \(\alpha\) , n - 1) , such that the probability to the left of it is \(\alpha\). It can be shown using either statistical software or a t -table that the critical value -t 0.05,14 is -1.7613. That is, we would reject the null hypothesis H 0 : μ = 3 in favor of the alternative hypothesis H A : μ < 3 if the test statistic t * is less than -1.7613. Visually, the rejection region is shaded red in the graph.

There are two critical values for the two-tailed test H 0 : μ = 3 versus H A : μ ≠ 3 — one for the left-tail denoted -t ( \(\alpha\) / 2, n - 1) and one for the right-tail denoted t ( \(\alpha\) / 2, n - 1) . The value - t ( \(\alpha\) /2, n - 1) is the t -value such that the probability to the left of it is \(\alpha\)/2, and the value t ( \(\alpha\) /2, n - 1) is the t -value such that the probability to the right of it is \(\alpha\)/2. It can be shown using either statistical software or a t -table that the critical value -t 0.025,14 is -2.1448 and the critical value t 0.025,14 is 2.1448. That is, we would reject the null hypothesis H 0 : μ = 3 in favor of the alternative hypothesis H A : μ ≠ 3 if the test statistic t * is less than -2.1448 or greater than 2.1448. Visually, the rejection region is shaded red in the graph.

t distribution graph for a two tailed test of 0.05 level of significance

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7.5: Critical values, p-values, and significance level

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Understanding the Difference Between P-Value and Critical Value in Hypothesis Testing

Key Takeaways

In statistical hypothesis testing, the p-value and critical value are two important concepts that help determine the significance of a statistical test. The p-value represents the probability of obtaining a test statistic as extreme as the one observed, assuming the null hypothesis is true. On the other hand, the critical value is a threshold value that is compared to the test statistic to determine whether to reject or fail to reject the null hypothesis. Understanding the difference between p-value and critical value is crucial for interpreting statistical results accurately.

Introduction

Statistical hypothesis testing is a fundamental tool in data analysis and research. It allows researchers to make inferences about population parameters based on sample data. In this process, the p-value and critical value play significant roles in determining the validity of the statistical test. This article aims to provide a comprehensive understanding of the differences between p-value and critical value, their significance, and how they are used in hypothesis testing.

P-Value: A Measure of Evidence Against the Null Hypothesis

The p-value is a statistical measure that quantifies the evidence against the null hypothesis. The null hypothesis assumes that there is no significant difference or relationship between variables, while the alternative hypothesis suggests otherwise. The p-value represents the probability of obtaining a test statistic as extreme as the one observed, assuming the null hypothesis is true.

When conducting a hypothesis test, the researcher calculates the test statistic based on the sample data and compares it to the p-value. If the p-value is small (typically less than a predetermined significance level, such as 0.05), it suggests that the observed data is unlikely to occur under the null hypothesis. In such cases, the researcher rejects the null hypothesis in favor of the alternative hypothesis, indicating that there is sufficient evidence to support the alternative hypothesis.

It is important to note that the p-value does not provide information about the magnitude or practical significance of the observed effect. It only indicates the strength of evidence against the null hypothesis. Therefore, researchers should consider the context and practical implications of the results when interpreting the p-value.

Critical Value: A Threshold for Decision Making

The critical value, also known as the cutoff value or rejection region, is a predetermined threshold that is compared to the test statistic to make a decision about the null hypothesis. It is determined based on the desired significance level, which represents the maximum probability of rejecting the null hypothesis when it is true.

When conducting a hypothesis test, the researcher calculates the test statistic and compares it to the critical value. If the test statistic exceeds the critical value, it falls into the rejection region, leading to the rejection of the null hypothesis. On the other hand, if the test statistic is less than or equal to the critical value, it falls into the non-rejection region, indicating that there is insufficient evidence to reject the null hypothesis.

The critical value is determined based on the desired significance level and the distribution of the test statistic. Different statistical tests have different critical values, depending on the assumptions and characteristics of the data. For example, in a t-test, the critical value is determined based on the degrees of freedom and the desired significance level.

Relationship Between P-Value and Critical Value

The p-value and critical value are closely related but serve different purposes in hypothesis testing. While the p-value quantifies the evidence against the null hypothesis, the critical value provides a threshold for decision making.

If the p-value is less than the predetermined significance level (e.g., 0.05), it suggests that the observed data is unlikely to occur under the null hypothesis. In this case, the researcher rejects the null hypothesis. On the other hand, if the p-value is greater than or equal to the significance level, the researcher fails to reject the null hypothesis.

The critical value, on the other hand, is compared directly to the test statistic. If the test statistic exceeds the critical value, the null hypothesis is rejected. If the test statistic is less than or equal to the critical value, the null hypothesis is not rejected.

It is important to note that the p-value and critical value are not interchangeable. The p-value provides a continuous measure of evidence against the null hypothesis, while the critical value provides a binary decision threshold. Both measures are essential for accurate hypothesis testing and should be interpreted together.

In statistical hypothesis testing, the p-value and critical value are crucial concepts that help determine the significance of a statistical test. The p-value quantifies the evidence against the null hypothesis, while the critical value provides a threshold for decision making. Understanding the differences between these two concepts is essential for interpreting statistical results accurately and making informed conclusions based on data analysis. By considering both the p-value and critical value, researchers can make sound judgments about the validity of their hypotheses and draw meaningful insights from their data.

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Basics of Statistics
Hypothesis Tests
Introduction to Hypothesis Testing
Critical Value and the p-Value

The Critical Value and the p-Value Approach to Hypothesis Testing

Published: 18 March 2002

Statistics review 3: Hypothesis testing and P values

Elise Whitley 1 &
Jonathan Ball 2

Critical Care volume 6 , Article number: 222 ( 2002 ) Cite this article

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An Erratum to this article was published on 13 December 2002

The present review introduces the general philosophy behind hypothesis (significance) testing and calculation of P values. Guidelines for the interpretation of P values are also provided in the context of a published example, along with some of the common pitfalls. Examples of specific statistical tests will be covered in future reviews.

Introduction

The previous review in this series described how to use confidence intervals to draw inferences about a population from a representative sample. A common next step in data analysis is calculation of P values, also known as hypothesis testing. Hypothesis testing is generally used when some comparison is to be made. This comparison may be a single observed value versus some hypothesized quantity (e.g. the number of babies born in a single delivery to mothers undergoing fertility treatment as compared with typical singleton birth), or it may be a comparison of two or more groups (e.g. mortality rates in intensive care unit patients who require renal replacement therapy versus those who do not). The choice of which statistical test to use depends on the format of the data and the study design. Examples of some of the more common techniques will be covered in subsequent reviews. However, the philosophy behind these statistical tests and the interpretation of the resulting P values are always the same, and it is these ideas that are covered in the present review.

The null hypothesis

A typical research question is most easily expressed in terms of there being some difference between groups. For example, 'In patients with acute myocardial infarction (AMI), does the administration of intravenous nitrate (as compared with none) reduce mortality?' To answer this question, the most appropriate study design would be a randomized controlled trial comparing AMI patients who receive intravenous nitrate with control patients. The challenge then is to interpret the results of that study. Even if there is no real effect of intravenous nitrate on mortality, sampling variation means that it is extremely unlikely that exactly the same proportion of patients in each group will die. Thus, any observed difference between the two groups may be due to the treatment or it may simply be a coincidence, in other words due to chance. The aim of hypothesis testing is to establish which of these explanations is most likely. Note that statistical analyses can never prove the truth of a hypothesis, but rather merely provide evidence to support or refute it.

To do this, the research question is more formally expressed in terms of there being no difference. This is known as the null hypothesis. In the current example the null hypothesis would be expressed as, 'The administration of intravenous nitrate has no effect on mortality in AMI patients.'

In hypothesis testing any observed differences between two (or more) groups are interpreted within the context of this null hypothesis. More formally, hypothesis testing explores how likely it is that the observed difference would be seen by chance alone if the null hypothesis were true.

What is a P value?

There is a wide range of statistical tests available, depending on the nature of the investigation. However, the end result of any statistical test is a P value. The ' P ' stands for probability, and measures how likely it is that any observed difference between groups is due to chance. In other words, the P value is the probability of seeing the observed difference, or greater, just by chance if the null hypothesis is true. Being a probability, P can take any value between 0 and 1. Values close to 0 indicate that the observed difference is unlikely to be due to chance, whereas a P value close to 1 suggests there is no difference between groups other than that due to random variation. The interpretation of a P value is not always straightforward and several important factors must be taken into account, as outlined below. Put simply, however, the P value measures the strength of evidence against the null hypothesis.

Note that the aim of hypothesis testing is not to 'accept' or 'reject' the null hypothesis. Rather, it is simply to gauge how likely it is that the observed difference is genuine if the null hypothesis is true.

Interpreting P values

Continuing with the previous example, a number of trials of intravenous nitrates in patients with AMI have been carried out. In 1988 an overview of those that had been conducted at that time was performed in order to synthesize all the available evidence [ 1 ]. The results from six trials of intravenous nitrate are given in Table 1 .

In the first trial (Chiche), 50 patients were randomly assigned to receive intravenous nitrate and 45 were randomly assigned to the control group. At the end of follow up, three of the 50 patients given intravenous nitrate had died versus eight in the control group. The calculation and interpretation of odds ratios will be covered in a future review. However, the interpretation in this context is that the odds ratio approximately represents the risk of dying in the nitrate group as compared with that in the control group. The odds ratio can take any positive value (above 0); in this context, values less than 1 indicate a protective effect of intravenous nitrate (a reduction in risk of death in patients administered intravenous nitrate), whereas an odds ratio greater than 1 points to a harmful effect (i.e. an increase in risk of death in patients administered intravenous nitrate). An odds ratio close to 1 is consistent with no effect of intravenous nitrate (i.e. no difference between the two groups). Interpretation of the confidence intervals is just as described in Statistics review 2, with the first confidence interval (Chiche) indicating that the true odds ratio in the population from which the trial subjects were drawn is likely to be between 0.09 and 1.13.

Initially ignoring the confidence intervals, five of the six trials summarized in Table 1 have odds ratios that are consistent with a protective effect of intravenous nitrate (odds ratio <1). These range from a risk reduction of 17% (Flaherty) to one of 76% (Bussman). In other words, in the Bussman trial the risk of dying in the nitrate group is about one-quarter of that in the control group. The remaining trial (Jaffe) has an odds ratio of 2.04, suggesting that the effect of intravenous nitrate might be harmful, with a doubling of risk in patients given this treatment as compared with those in the control group.

The P values shown in the final column of Table 1 give an indication of how likely it is that these differences are simply due to chance. The P value for the first trial (Chiche) indicates that the probability of observing an odds ratio of 0.33 or more extreme, if the null hypothesis is true, is 0.08. In other words, if there is genuinely no effect of intravenous nitrate on the mortality of patients with AMI, then 8 out of 100 such trials would show a risk reduction of 66% or more just by chance. Equivalently, 2 out of 25 would show such a chance effect. The question of whether this is sufficiently unlikely to suggest that there is a real effect is highly subjective. However, it is unlikely that the management of critically ill patients would be altered on the basis of this evidence alone, and an isolated result such as this would probably be interpreted as being consistent with no effect. Similarly the P value for the Bussman trial indicates that 1 in 100 trials would have an odds ratio of 0.24 or more extreme by chance alone; this is a smaller probability than in the previous trial but, in isolation, perhaps still not sufficiently unlikely to alter clinical care in practice. The P value of 0.70 in the Flaherty trial suggests that the observed odds ratio of 0.83 is very likely to be a chance finding.

Comparing the P values across different trials there are two main features of interest. The first is that the size of the P value is related, to some extent, to the size of the trial (and, in this context, the proportion of deaths). For example, the odds ratios in the Lis and Jugdutt trials are reasonably similar, both of which are consistent with an approximate halving of risk in patients given intravenous nitrate, but the P value for the larger Jugdutt trial is substantially smaller than that for the Lis trial. This pattern tends to be apparent in general, with larger studies giving rise to smaller P values. The second feature relates to how the P values change with the size of the observed effect. The Chiche and Flaherty trials have broadly similar numbers of patients (in fact, the numbers are somewhat higher in the Flaherty trial) but the smaller P value occurs in the Chiche study, which suggests that the effect of intravenous nitrate is much larger than that in the Flaherty study (67% versus 17% reduction in mortality). Again, this pattern will tend to hold in general, with more extreme effects corresponding to smaller P values. Both of these properties are discussed in considerably more detail in the next review, on sample size/power calculations.

There are two additional points to note when interpreting P values. It was common in the past for researchers to classify results as statistically 'significant' or 'non-significant', based on whether the P value was smaller than some prespecified cut point, commonly 0.05. This practice is now becoming increasingly obsolete, and the use of exact P values is much preferred. This is partly for practical reasons, because the increasing use of statistical software renders calculation of exact P values increasingly simple as compared with the past when tabulated values were used. However, there is also a more pragmatic reason for this shift. The use of a cut-off for statistical significance based on a purely arbitrary value such as 0.05 tends to lead to a misleading conclusion of accepting or rejecting the null hypothesis, in other words of concluding that a 'statistically significant' result is real in some sense. Recall that a P value of 0.05 means that one out of 20 studies would result in a difference at least as big as that observed just by chance. Thus, a researcher who accepts a 'significant' result as real will be wrong 5% of the time (this is sometimes known as a type I error). Similarly, dismissing an apparently 'non-significant' finding as a null result may also be incorrect (sometimes known as a type II error), particularly in a small study, in which the lack of statistical significance may simply be due to the small sample size rather than to any real lack of clinical effect (see the next review for details). Both of these scenarios have serious implications in terms of practical identification of risk factors and treatment of disease. The presentation of exact P values allows the researcher to make an educated judgement as to whether the observed effect is likely to be due to chance and this, taken in the context of other available evidence, will result in a far more informed conclusion being reached.

Finally, P values give no indication as to the clinical importance of an observed effect. For example, suppose a new drug for lowering blood pressure is tested against standard treatment, and the resulting P value is extremely small. This indicates that the difference is unlikely to be due to chance, but decisions on whether to prescribe the new drug will depend on many other factors, including the cost of the new treatment, any potential contraindications or side effects, and so on. In particular, just as a small study may fail to detect a genuine effect, a very large study may result in a very small P value based on a small difference of effect that is unlikely to be important when translated into clinical practice.

P values and confidence intervals

Although P values provide a measure of the strength of an association, there is a great deal of additional information to be obtained from confidence intervals. Recall that a confidence interval gives a range of values within which it is likely that the true population value lies. Consider the confidence intervals shown in Table 1 . The odds ratio for the Chiche study is 0.33, suggesting that the effect of intravenous nitrate is to reduce mortality by two thirds. However, the confidence interval indicates that the true effect is likely to be somewhere between a reduction of 91% and an increase of 13%. The results from that study show that there may be a substantial reduction in mortality due to intravenous nitrate, but equally it is not possible to rule out an important increase in mortality. Clearly, if the latter were the case then it would be extremely dangerous to administer intravenous nitrate to patients with AMI.

The confidence interval for the Bussman study (0.08, 0.74) provides a rather more positive picture. It indicates that, although the reduction in mortality may be as little as 26%, there is little evidence to suggest that the effect of intravenous nitrate may be harmful. Administration of intravenous nitrate therefore appears more reasonable based on the results of that study, although the P value indicates a 1 in 100 probability that this may be a chance finding and so the result in isolation might not be sufficient evidence to change clinical practice.

The overview of those trials was carried out because the results did not appear to be consistent, largely because the individual trials were generally too small to provide reliable estimates of effect. A pooled analysis of the data from all of the nitrate trials shown in Table 1 (and including one other trial with no deaths) was therefore conducted to obtain a more robust estimate of effect (for details of the methods used, see Yusuf et al . [ 1 ]). The odds ratios and 95% confidence intervals for the individual trials in Table 1 are shown in Fig. 1 . The odds ratio for each trial is represented by a box, the size of which is proportional to the amount of statistical information available for that estimate, and the 95% confidence interval is indicated by a horizontal line. The solid vertical line indicates an odds ratio of 1.0; in other words it shows the line of 'no effect'. The combined odds ratio from all six trials is indicated by the dashed vertical line, and its associated 95% confidence interval by the diamond at the bottom.

Individual and combined odds ratios and 95% confidence intervals for six intravenous nitrate trials.

This pooled analysis resulted in an estimated overall odds ratio of 0.53 with a 95% confidence interval of (0.36, 0.75), suggesting a true reduction in mortality of somewhere between one-quarter and two-thirds. Examination of the confidence intervals from individual studies shows a high degree of overlap with the pooled confidence interval, and so all of the evidence appears to be consistent with this pooled estimate; this includes the evidence from the Jaffe study, which, at first glance, appears to suggest a harmful effect. The P value for the pooled analysis was 0.0002, which indicates that the result is extremely unlikely to have been due to chance.

Note that, since that meta-analysis was reported, treatment of AMI patients has changed dramatically with the introduction of thrombolysis. In addition, the Fourth International Study of Infarct Survival (ISIS-4) [ 2 ], which randomized over 58,000 patients with suspected AMI, found no evidence to suggest that mortality was reduced in those given oral nitrates. Thus, in practice the indications for intravenous nitrates in patients with AMI are restricted to symptom and blood pressure control.

Specific methods for comparing two or more means or proportions will be introduced in subsequent reviews. In general, these will tend to focus on the calculation of P values. However, there is still much to be learned from examination of confidence intervals in this context. For example, when comparing the risk for developing secondary infection following trauma in patients with or without a history of chronic alcohol abuse, it may be enlightening to compare the confidence intervals for the two groups and to examine the extent to which they do or do not overlap. Alternatively, it is possible to calculate a confidence interval for the difference in two means or the difference or ratio of proportions directly. This can also give a useful indication of the likely effect of chronic alcohol abuse, in particular by exploring the extent to which the range of likely values includes or excludes 0 or 1, the respective expected values of a difference or ratio if there is no effect of chronic alcohol abuse, or in other words under the null hypothesis.

Although P values provide a measure of the strength of an association, an estimate of the size of any effect along with an associated confidence interval is always required for meaningful interpretation of results. P values and confidence intervals are frequently calculated using similar quantities (see subsequent reviews for details), and so it is not surprising that the two are closely related. In particular, larger studies will in general result in narrower confidence intervals and smaller P values, and this should be taken into account when interpreting the results from statistical analyses. Both P values and confidence intervals have an important role to play in understanding data analyses, and both should be presented wherever possible.

Key messages

A P value is the probability that an observed effect is simply due to chance; it therefore provides a measure of the strength of an association. A P value does not provide any measure of the size of an effect, and cannot be used in isolation to inform clinical judgement.

P values are affected both by the magnitude of the effect and by the size of the study from which they are derived, and should therefore be interpreted with caution. In particular, a large P value does not always indicate that there is no association and, similarly, a small P value does not necessarily signify an important clinical effect.

Subdividing P values into 'significant' and 'non-significant' is poor statistical practice and should be avoided. Exact P values should always be presented, along with estimates of effect and associated confidence intervals.

Abbreviations

AMI=acute myocardial infarction.

Yusuf S, Collins R, MacMahon S, Peto R: Effect of intravenous nitrates on mortality in acute myocardial infarction: an overview of the randomised trials. Lancet 1988, 1: 1088-1092. 10.1016/S0140-6736(88)91906-X

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Anonymous: ISIS-4: a randomised factorial trial assessing early oral captopril, oral mononitrate, and intravenous magnesium sulphate in 58,050 patients with suspected acute myocardial infarction. Lancet 2002, 345: 669-685.

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Whitley E, Ball J: Statistics review 1: Presenting and summarising data. Crit Care 2002, 6: 66-71. 10.1186/cc1455

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Whitley E, Ball J: Statistics review 2: Samples and populations. Crit Care 2002., 6 (2):

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Lecturer in Medical Statistics, University of Bristol, Bristol, UK

Elise Whitley

Lecturer in Intensive Care Medicine, St George's Hospital Medical School, London, UK

Jonathan Ball

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This article is the third in an ongoing, educational review series on medical statistics in critical care. Previous articles have covered 'presenting and summarising data' [ 3 ] and 'samples and populations' [ 4 ]. Future topics to be covered include power calculations, comparison of means, comparison of proportions, and analysis of survival data to name but a few. If there is a medical statistics topic you would like explained contact us on [email protected].

An erratum to this article is available at http://dx.doi.org/10.1186/cc1868 .

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Whitley, E., Ball, J. Statistics review 3: Hypothesis testing and P values. Crit Care 6 , 222 (2002). https://doi.org/10.1186/cc1493

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Published : 18 March 2002

DOI : https://doi.org/10.1186/cc1493

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What is the P-value from the t-distribution table?

Table of Contents

The P-value from the t-distribution table is a statistical measure used to determine the probability of obtaining a test statistic as extreme as the one observed in a t-test, assuming the null hypothesis is true. It is calculated by comparing the observed t-value to the critical t-value from the t-distribution table. A lower P-value indicates stronger evidence against the null hypothesis, while a higher P-value suggests that the observed data is likely to have occurred by chance. The t-distribution table is used to find the critical t-value at a given significance level, which is compared to the calculated t-value to determine the P-value. Overall, the P-value from the t-distribution table is a crucial tool in determining the statistical significance of results in t-tests.

Here is Find the P-Value from the t-Distribution Table

The t distribution table is a table that shows the critical values of the t distribution. To use the t distribution table, you only need three values:

A significance level (common choices are 0.01, 0.05, and 0.10)
The degrees of freedom
The type of test (one-tailed or two-tailed)

The t distribution table is commonly used in the following hypothesis tests:

A hypothesis test for a mean
A hypothesis test for a difference in means
A hypothesis test for a difference in paired means

When you conduct each of these tests, you’ll end up with a test statistic t . To find out if this test statistic is statistically significant at some alpha level, you have two options:

Compare the test statistic t to a critical value from the t distribution table.
Compare the p-value of the test statistic t to a chosen alpha level.

Let’s walk through an example of how to use each of these approaches.

Suppose we conduct a two-sided hypothesis test at alpha level 0.05 to find out if mean weight loss differs between two diets. Suppose our test statistic t is 1.34 and our degrees of freedom is 22 . We would like to know if these results are statistically significant.

Compare the test statistic t to a critical value

The first approach we can use to determine if our results are statistically significant is to compare the test statistic t of 1.34 to the critical value in the t distribution table. The critical value is the value in the table that aligns with a two-tailed value of 0.05 and a degrees of freedom of 22 . This number turns out to be 2.074 :

Since out test statistic t ( 1.34 ) is smaller than the critical value ( 2.074 ), we fail to reject the null hypothesis of our test. We do not have sufficient evidence to say that the mean weight loss between the two diets is statistically significant at alpha level 0.05.

Compare the p-value to a chosen alpha level

The second approach we can use to determine if our results are statistically significant is to find the p-value for the test statistic t of 1.34 . In order to find this p-value, we can’t use the t distribution table because it only provides us with critical values, not p-values .

S o, in order to find this p-value we need to use a with the following inputs:

The p-value for a test statistic t of 1.34 for a two-tailed test with 22 degrees of freedom is 0.19392 . Since this number is greater than our alpha level of 0.05 , we fail to reject the null hypothesis of our test. We do not have sufficient evidence to say that the mean weight loss between the two diets is statistically significant at alpha level 0.05.

When to Use the t Distribution Table

If you are interested in finding the t critical value for a given significance level, degrees of freedom, and type of test (one-tailed or two-tailed), then you should use the .

Instead, if you have a given test statistic t and you simply want to know the p-value of that test statistic, then you would need to use a to do so.

Related terms:

How can I find the P-value from the F-distribution table?
How do you find the P-Value from the Chi-Square Distribution Table?
What is the Chi-square Distribution Table?
What is the purpose of a distribution table and how is it used in statistical analysis?
What is the table for the t-Distribution?
What is the Binomial Distribution Table?
What information can be found in the F-Distribution Table?
What’s the difference between the Poisson distribution and the Normal Distribution?
Difference between Normal Distribution and Standard Normal Distribution?
How can I create a frequency table in SPSS and what is an example of a frequency table?

Do you get more food when you order in-person at Chipotle?

April 7, 2024

Inspired by this Reddit post , we will conduct a hypothesis test to determine if there is a difference in the weight of Chipotle orders between in-person and online orders. The data was originally collected by Zackary Smigel , and a cleaned copy be found in data/chipotle.csv .

Throughout the application exercise we will use the infer package which is part of tidymodels to conduct our permutation tests.

Variable type: character

Variable type: Date

Variable type: numeric

The variable we will use in this analysis is weight which records the total weight of the meal in grams.

We wish to test the claim that the difference in weight between in-person and online orders must be due to something other than chance.

Your turn: Write out the correct null and alternative hypothesis in terms of the difference in means between in-person and online orders. Do this in both words and in proper notation.

Null hypothesis: TODO

\[H_0: \mu_{\text{online}} - \mu_{\text{in-person}} = TODO\]

Alternative hypothesis: The difference in means between in-person and online Chipotle orders is not \(0\) .

\[H_A: \mu_{\text{online}} - \mu_{\text{in-person}} TODO\]

Observed data

Our goal is to use the collected data and calculate the probability of a sample statistic at least as extreme as the one observed in our data if in fact the null hypothesis is true.

Demo: Calculate and report the sample statistic below using proper notation.

The null distribution

Let’s use permutation-based methods to conduct the hypothesis test specified above.

We’ll start by generating the null distribution.

Demo: Generate the null distribution.
Your turn: Take a look at null_dist . What does each element in this distribution represent?

Add response here.

Question: Before you visualize the distribution of null_dist – at what value would you expect this distribution to be centered? Why?

Demo: Create an appropriate visualization for your null distribution. Does the center of the distribution match what you guessed in the previous question?

Demo: Now, add a vertical red line on your null distribution that represents your sample statistic.

Question: Based on the position of this line, does your observed sample difference in means appear to be an unusual observation under the assumption of the null hypothesis?

Above, we eyeballed how likely/unlikely our observed mean is. Now, let’s actually quantify it using a p-value.

Question: What is a p-value?

Guesstimate the p-value

Demo: Visualize the p-value.

Your turn: What is you guesstimate of the p-value?

Calculate the p-value

Your turn: What is the conclusion of the hypothesis test based on the p-value you calculated? Make sure to frame it in context of the data and the research question. Use a significance level of 5% to make your conclusion.

Demo: Interpret the p-value in context of the data and the research question.

Reframe as a linear regression model

While we originally evaluated the null/alternative hypotheses as a difference in means, we could also frame this as a regression problem where the outcome of interest (weight of the order) is a continuous variable. Framing it this way allows us to include additional explanatory variables in our model which may account for some of the variation in weight.

Single explanatory variable

Demo: Let’s reevaluate the original hypotheses using a linear regression model. Notice the similarities and differences in the code compared to a difference in means, and that the obtained p-value should be nearly identical to the results from the difference in means test.

Multiple explanatory variables

Demo: Now let’s also account for additional variables that likely influence the weight of the order.

Protein type ( meat )
Type of meal ( meal_type ) - burrito or bowl
Store ( store ) - at which Chipotle location the order was placed

Your turn: Interpret the p-value for the order in context of the data and the research question.

Compare to CLT-based method

Demo: Let’s compare the p-value obtained from the permutation test to the p-value obtained from that derived using the Central Limit Theorem (CLT).

Your turn: What is the p-value obtained from the CLT-based method? How does it compare to the p-value obtained from the permutation test?

IMAGES

What is P-value in hypothesis testing
Hypothesis testing tutorial using p value method
P-Value Method For Hypothesis Testing
Critical Value
Hypothesis testing tutorial using p value method
p-Value in Hypothesis Testing

VIDEO

Hypothesis Testing for Proportion: p-value is more than the level of significance (Degree Example)
Hypothesis Testing for Mean: p-value is more than the level of significance (Hat Size Example)
FA II STATISTICS/ Chapter no 7 / Testing of hypothesis/ Z distribution / Example 7.8
Hypothesis Testing: critical value(s)
Hypothesis Test
P-Value, Confidence Interval and Significance explained

COMMENTS

7.5: Critical values, p-values, and significance level
When we use z z -scores in this way, the obtained value of z z (sometimes called z z -obtained) is something known as a test statistic, which is simply an inferential statistic used to test a null hypothesis. The formula for our z z -statistic has not changed: z = X¯¯¯¯ − μ σ¯/ n−−√ (7.5.1) (7.5.1) z = X ¯ − μ σ ¯ / n.
P-Value vs. Critical Value: A Friendly Guide for Beginners
Daisy. The main difference between p-value and critical value is that the p-value quantifies the strength of evidence against a null hypothesis, while the critical value sets a threshold for assessing the significance of a test statistic. Simply put, if your p-value is below the critical value, you reject the null hypothesis.
S.3.2 Hypothesis Testing (P-Value Approach)
The P -value is, therefore, the area under a tn - 1 = t14 curve to the left of -2.5 and to the right of 2.5. It can be shown using statistical software that the P -value is 0.0127 + 0.0127, or 0.0254. The graph depicts this visually. Note that the P -value for a two-tailed test is always two times the P -value for either of the one-tailed tests.
Hypothesis Testing, P Values, Confidence Intervals, and Significance
Definition/Introduction. Medical providers often rely on evidence-based medicine to guide decision-making in practice. Often a research hypothesis is tested with results provided, typically with p values, confidence intervals, or both. Additionally, statistical or research significance is estimated or determined by the investigators.
How Hypothesis Tests Work: Significance Levels (Alpha) and P values
Using P values and Significance Levels Together. If your P value is less than or equal to your alpha level, reject the null hypothesis. The P value results are consistent with our graphical representation. The P value of 0.03112 is significant at the alpha level of 0.05 but not 0.01.
Critical Value: Definition, Finding & Calculator
A critical value defines regions in the sampling distribution of a test statistic. These values play a role in both hypothesis tests and confidence intervals. In hypothesis tests, critical values determine whether the results are statistically significant. For confidence intervals, they help calculate the upper and lower limits.
Understanding P-values
The p value gets smaller as the test statistic calculated from your data gets further away from the range of test statistics predicted by the null hypothesis. The p value is a proportion: if your p value is 0.05, that means that 5% of the time you would see a test statistic at least as extreme as the one you found if the null hypothesis was true.
Interpreting P values
Here is the technical definition of P values: P values are the probability of observing a sample statistic that is at least as extreme as your sample statistic when you assume that the null hypothesis is true. Let's go back to our hypothetical medication study. Suppose the hypothesis test generates a P value of 0.03.
How to Calculate Critical Values for Statistical Hypothesis Testing
Many statistical hypothesis tests return a p-value that is used to interpret the outcome of the test. Some tests do not return a p-value, requiring an alternative method for interpreting the calculated test statistic directly. ... Test Statistic <= Critical Value: Fail to reject the null hypothesis of the statistical test.
Hypothesis testing and p-values (video)
In this video there was no critical value set for this experiment. In the last seconds of the video, Sal briefly mentions a p-value of 5% (0.05), which would have a critical of value of z = (+/-) 1.96. Since the experiment produced a z-score of 3, which is more extreme than 1.96, we reject the null hypothesis.
7.1.3.1. Critical values and p values
Critical values and. p. values. Determination of critical values. Critical values for a test of hypothesis depend upon a test statistic, which is specific to the type of test, and the significance level, α , which defines the sensitivity of the test. A value of α = 0.05 implies that the null hypothesis is rejected 5 % of the time when it is ...
S.3.1 Hypothesis Testing (Critical Value Approach)
The critical value for conducting the right-tailed test H0 : μ = 3 versus HA : μ > 3 is the t -value, denoted t\ (\alpha\), n - 1, such that the probability to the right of it is \ (\alpha\). It can be shown using either statistical software or a t -table that the critical value t 0.05,14 is 1.7613. That is, we would reject the null ...
7.5: Critical values, p-values, and significance level
7: Introduction to Hypothesis Testing 7.5: Critical values, p-values, and significance level Expand/collapse global location
Understanding the Difference Between P-Value and Critical Value in
In statistical hypothesis testing, the p-value and critical value are two important concepts that help determine the significance of a statistical test. The p-value represents the probability of obtaining a test statistic as extreme as the one observed, assuming the null hypothesis is true. On the other hand, the critical value is a threshold ...
How to Find the P value: Process and Calculations
To find the p value for your sample, do the following: Identify the correct test statistic. Calculate the test statistic using the relevant properties of your sample. Specify the characteristics of the test statistic's sampling distribution. Place your test statistic in the sampling distribution to find the p value.
P-Value in Statistical Hypothesis Tests: What is it?
The p value is the evidence against a null hypothesis. The smaller the p-value, the stronger the evidence that you should reject the null hypothesis. P values are expressed as decimals although it may be easier to understand what they are if you convert them to a percentage. For example, a p value of 0.0254 is 2.54%.
Critical Value and the p-Value • SOGA-R • Department of Earth Sciences
The p-value approach. For the p-value approach the likelihood (p-value) of the numerical value of the test statistic is compared to the specified significance level (α) of the hypothesis test.. The p-value corresponds to the probability of observing sample data at least as extreme as the actually obtained test statistic.Small p-values provide evidence against the null hypothesis.
Hypothesis Testing: Critical Value Approach versus P-Value ...
In this short video, we consider the critical value approach to hypothesis testing decisiuon making and compare this approach to the p-value approach to hypo...
Statistics review 3: Hypothesis testing and P values
The ' P ' stands for probability, and measures how likely it is that any observed difference between groups is due to chance. In other words, the P value is the probability of seeing the observed difference, or greater, just by chance if the null hypothesis is true. Being a probability, P can take any value between 0 and 1.
How to Find P Value from a Test Statistic
Hypothesis tests are used to test the validity of a claim that is made about a population. This claim that's on trial, in essence, is called the null hypothesis (H 0).The alternative hypothesis (H a) is the one you would believe if the null hypothesis is concluded to be untrue.Learning how to find the p-value in statistics is a fundamental skill in testing, helping you weigh the evidence ...
What Is The P-value From The T-distribution Table?
The first approach we can use to determine if our results are statistically significant is to compare the test statistic t of 1.34 to the critical value in the t distribution table.The critical value is the value in the table that aligns with a two-tailed value of 0.05 and a degrees of freedom of 22.This number turns out to be 2.074: Since out test statistic t (1.34) is smaller than the ...
Do you get more food when you order in-person at Chipotle?
Your turn: What is the conclusion of the hypothesis test based on the p-value you calculated? Make sure to frame it in context of the data and the research question. Use a significance level of 5% to make your conclusion. Add response here. Your turn: Interpret the p-value for the order in context of the data and the research question. Add ...

7.5: Critical values, p-values, and significance level

P-Value vs. Critical Value: A Friendly Guide for Beginners

Understanding P-Value and Critical Value

Concept of Hypothesis Testing

Critical Value

P-Value Essentials

Calculating P-Values

Interpreting P-Values

Critical Value Essentials

Significance Level

Rejection Region

Statistical Tests and Decision Making

Confidence Intervals

Comparing P-Values and Critical Values

Differences and Similarities

Applications in Geospatial Data Analysis

Standard Distributions and Scores

Standard Normal Distribution

T-Statistic

Author’s Recommendations: Top Data Science Resources To Consider

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