the hypothesis should be rejected

Statistics Made Easy

When Do You Reject the Null Hypothesis? (3 Examples)

A hypothesis test is a formal statistical test we use to reject or fail to reject a statistical hypothesis.

We always use the following steps to perform a hypothesis test:

Step 1: State the null and alternative hypotheses.

The null hypothesis , denoted as H 0 , is the hypothesis that the sample data occurs purely from chance.

The alternative hypothesis , denoted as H A , is the hypothesis that the sample data is influenced by some non-random cause.

2. Determine a significance level to use.

Decide on a significance level. Common choices are .01, .05, and .1.

3. Calculate the test statistic and p-value.

Use the sample data to calculate a test statistic and a corresponding p-value .

4. Reject or fail to reject the null hypothesis.

If the p-value is less than the significance level, then you reject the null hypothesis.

If the p-value is not less than the significance level, then you fail to reject the null hypothesis.

You can use the following clever line to remember this rule:

“If the p is low, the null must go.”

In other words, if the p-value is low enough then we must reject the null hypothesis.

The following examples show when to reject (or fail to reject) the null hypothesis for the most common types of hypothesis tests.

Example 1: One Sample t-test

A one sample t-test is used to test whether or not the mean of a population is equal to some value.

For example, suppose we want to know whether or not the mean weight of a certain species of turtle is equal to 310 pounds.

We go out and collect a simple random sample of 40 turtles with the following information:

Sample size n = 40
Sample mean weight x = 300
Sample standard deviation s = 18.5

We can use the following steps to perform a one sample t-test:

Step 1: State the Null and Alternative Hypotheses

We will perform the one sample t-test with the following hypotheses:

H 0 : μ = 310 (population mean is equal to 310 pounds)
H A : μ ≠ 310 (population mean is not equal to 310 pounds)

We will choose to use a significance level of 0.05 .

We can plug in the numbers for the sample size, sample mean, and sample standard deviation into this One Sample t-test Calculator to calculate the test statistic and p-value:

t test statistic: -3.4187
two-tailed p-value: 0.0015

Since the p-value (0.0015) is less than the significance level (0.05) we reject the null hypothesis .

We conclude that there is sufficient evidence to say that the mean weight of turtles in this population is not equal to 310 pounds.

Example 2: Two Sample t-test

A two sample t-test is used to test whether or not two population means are equal.

For example, suppose we want to know whether or not the mean weight between two different species of turtles is equal.

We go out and collect a simple random sample from each population with the following information:

Sample size n 1 = 40
Sample mean weight x 1 = 300
Sample standard deviation s 1 = 18.5
Sample size n 2 = 38
Sample mean weight x 2 = 305
Sample standard deviation s 2 = 16.7

We can use the following steps to perform a two sample t-test:

We will perform the two sample t-test with the following hypotheses:

H 0 : μ 1 = μ 2 (the two population means are equal)
H 1 : μ 1 ≠ μ 2 (the two population means are not equal)

We will choose to use a significance level of 0.10 .

We can plug in the numbers for the sample sizes, sample means, and sample standard deviations into this Two Sample t-test Calculator to calculate the test statistic and p-value:

t test statistic: -1.2508
two-tailed p-value: 0.2149

Since the p-value (0.2149) is not less than the significance level (0.10) we fail to reject the null hypothesis .

We do not have sufficient evidence to say that the mean weight of turtles between these two populations is different.

Example 3: Paired Samples t-test

A paired samples t-test is used to compare the means of two samples when each observation in one sample can be paired with an observation in the other sample.

For example, suppose we want to know whether or not a certain training program is able to increase the max vertical jump of college basketball players.

To test this, we may recruit a simple random sample of 20 college basketball players and measure each of their max vertical jumps. Then, we may have each player use the training program for one month and then measure their max vertical jump again at the end of the month:

We can use the following steps to perform a paired samples t-test:

We will perform the paired samples t-test with the following hypotheses:

H 0 : μ before = μ after (the two population means are equal)
H 1 : μ before ≠ μ after (the two population means are not equal)

We will choose to use a significance level of 0.01 .

We can plug in the raw data for each sample into this Paired Samples t-test Calculator to calculate the test statistic and p-value:

t test statistic: -3.226
two-tailed p-value: 0.0045

Since the p-value (0.0045) is less than the significance level (0.01) we reject the null hypothesis .

We have sufficient evidence to say that the mean vertical jump before and after participating in the training program is not equal.

Bonus: Decision Rule Calculator

You can use this decision rule calculator to automatically determine whether you should reject or fail to reject a null hypothesis for a hypothesis test based on the value of the test statistic.

Published by Zach

Hypothesis Testing (cont...)

Hypothesis testing, the null and alternative hypothesis.

In order to undertake hypothesis testing you need to express your research hypothesis as a null and alternative hypothesis. The null hypothesis and alternative hypothesis are statements regarding the differences or effects that occur in the population. You will use your sample to test which statement (i.e., the null hypothesis or alternative hypothesis) is most likely (although technically, you test the evidence against the null hypothesis). So, with respect to our teaching example, the null and alternative hypothesis will reflect statements about all statistics students on graduate management courses.

The null hypothesis is essentially the "devil's advocate" position. That is, it assumes that whatever you are trying to prove did not happen ( hint: it usually states that something equals zero). For example, the two different teaching methods did not result in different exam performances (i.e., zero difference). Another example might be that there is no relationship between anxiety and athletic performance (i.e., the slope is zero). The alternative hypothesis states the opposite and is usually the hypothesis you are trying to prove (e.g., the two different teaching methods did result in different exam performances). Initially, you can state these hypotheses in more general terms (e.g., using terms like "effect", "relationship", etc.), as shown below for the teaching methods example:

Depending on how you want to "summarize" the exam performances will determine how you might want to write a more specific null and alternative hypothesis. For example, you could compare the mean exam performance of each group (i.e., the "seminar" group and the "lectures-only" group). This is what we will demonstrate here, but other options include comparing the distributions , medians , amongst other things. As such, we can state:

Now that you have identified the null and alternative hypotheses, you need to find evidence and develop a strategy for declaring your "support" for either the null or alternative hypothesis. We can do this using some statistical theory and some arbitrary cut-off points. Both these issues are dealt with next.

Significance levels

The level of statistical significance is often expressed as the so-called p -value . Depending on the statistical test you have chosen, you will calculate a probability (i.e., the p -value) of observing your sample results (or more extreme) given that the null hypothesis is true . Another way of phrasing this is to consider the probability that a difference in a mean score (or other statistic) could have arisen based on the assumption that there really is no difference. Let us consider this statement with respect to our example where we are interested in the difference in mean exam performance between two different teaching methods. If there really is no difference between the two teaching methods in the population (i.e., given that the null hypothesis is true), how likely would it be to see a difference in the mean exam performance between the two teaching methods as large as (or larger than) that which has been observed in your sample?

So, you might get a p -value such as 0.03 (i.e., p = .03). This means that there is a 3% chance of finding a difference as large as (or larger than) the one in your study given that the null hypothesis is true. However, you want to know whether this is "statistically significant". Typically, if there was a 5% or less chance (5 times in 100 or less) that the difference in the mean exam performance between the two teaching methods (or whatever statistic you are using) is as different as observed given the null hypothesis is true, you would reject the null hypothesis and accept the alternative hypothesis. Alternately, if the chance was greater than 5% (5 times in 100 or more), you would fail to reject the null hypothesis and would not accept the alternative hypothesis. As such, in this example where p = .03, we would reject the null hypothesis and accept the alternative hypothesis. We reject it because at a significance level of 0.03 (i.e., less than a 5% chance), the result we obtained could happen too frequently for us to be confident that it was the two teaching methods that had an effect on exam performance.

Whilst there is relatively little justification why a significance level of 0.05 is used rather than 0.01 or 0.10, for example, it is widely used in academic research. However, if you want to be particularly confident in your results, you can set a more stringent level of 0.01 (a 1% chance or less; 1 in 100 chance or less).

One- and two-tailed predictions

When considering whether we reject the null hypothesis and accept the alternative hypothesis, we need to consider the direction of the alternative hypothesis statement. For example, the alternative hypothesis that was stated earlier is:

The alternative hypothesis tells us two things. First, what predictions did we make about the effect of the independent variable(s) on the dependent variable(s)? Second, what was the predicted direction of this effect? Let's use our example to highlight these two points.

Sarah predicted that her teaching method (independent variable: teaching method), whereby she not only required her students to attend lectures, but also seminars, would have a positive effect (that is, increased) students' performance (dependent variable: exam marks). If an alternative hypothesis has a direction (and this is how you want to test it), the hypothesis is one-tailed. That is, it predicts direction of the effect. If the alternative hypothesis has stated that the effect was expected to be negative, this is also a one-tailed hypothesis.

Alternatively, a two-tailed prediction means that we do not make a choice over the direction that the effect of the experiment takes. Rather, it simply implies that the effect could be negative or positive. If Sarah had made a two-tailed prediction, the alternative hypothesis might have been:

In other words, we simply take out the word "positive", which implies the direction of our effect. In our example, making a two-tailed prediction may seem strange. After all, it would be logical to expect that "extra" tuition (going to seminar classes as well as lectures) would either have a positive effect on students' performance or no effect at all, but certainly not a negative effect. However, this is just our opinion (and hope) and certainly does not mean that we will get the effect we expect. Generally speaking, making a one-tail prediction (i.e., and testing for it this way) is frowned upon as it usually reflects the hope of a researcher rather than any certainty that it will happen. Notable exceptions to this rule are when there is only one possible way in which a change could occur. This can happen, for example, when biological activity/presence in measured. That is, a protein might be "dormant" and the stimulus you are using can only possibly "wake it up" (i.e., it cannot possibly reduce the activity of a "dormant" protein). In addition, for some statistical tests, one-tailed tests are not possible.

Rejecting or failing to reject the null hypothesis

Let's return finally to the question of whether we reject or fail to reject the null hypothesis.

If our statistical analysis shows that the significance level is below the cut-off value we have set (e.g., either 0.05 or 0.01), we reject the null hypothesis and accept the alternative hypothesis. Alternatively, if the significance level is above the cut-off value, we fail to reject the null hypothesis and cannot accept the alternative hypothesis. You should note that you cannot accept the null hypothesis, but only find evidence against it.

Chapter 2: Summarizing and Visualizing Data

Chapter 3: measure of central tendency, chapter 4: measures of variation, chapter 5: measures of relative standing, chapter 6: probability distributions, chapter 7: estimates, chapter 8: distributions, chapter 9: hypothesis testing, chapter 10: analysis of variance, chapter 11: correlation and regression, chapter 12: statistics in practice.

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In an experiment, a farm with infected plants is subjected to a widely applicable insecticide.

This insecticide is expected to increase the number of healthy plants after its application. However, at the end of the experiment, the proportion of healthy and infected plants remained the same.

Here, the null hypothesis that the insecticide has no effect seems to hold, but should one accept the hypothesis or fail to reject it?

Accepting this hypothesis would mean that the insecticide is ineffective and cannot improve the plants' health.

This decision actually overlooks the other plausible explanations for the observed results.

In this case, using an unprescribed amount or concentration of insecticide might have resulted in no effect.

There is a possibility of plants being infected by something that the insecticide cannot target.

Failing to reject a null hypothesis means there is no sufficient evidence for the expected or the observed effect.

Today, if scientists had accepted null hypotheses, the discovery of plant viruses or the rediscovery of many extinct species would not have been possible.

9.8: Hypothesis: Accept or Fail to Reject?

The outcome of any hypothesis testing leads to rejecting or not rejecting the null hypothesis. This decision is taken based on the analysis of the data, an appropriate test statistic, an appropriate confidence level, the critical values, and P -values. However, when the evidence suggests that the null hypothesis cannot be rejected, is it right to say, 'Accept' the null hypothesis?

There are two ways to indicate that the null hypothesis is not rejected. 'Accept' the null hypothesis and 'fail to reject' the null hypothesis. Superficially, both these phrases mean the same, but in statistics, the meanings are somewhat different. The phrase 'accept the null hypothesis' implies that the null hypothesis is by nature true, and it is proved. But a hypothesis test simply provides information that there is no sufficient evidence in support of the alternative hypothesis, and therefore the null hypothesis cannot be rejected. The null hypothesis cannot be proven, although the hypothesis test begins with an assumption that the hypothesis is true, and the final result indicates the failure of the rejection of the null hypothesis. Thus, it is always advisable to state 'fail to reject the null hypothesis' instead of 'accept the null hypothesis.'

'Accepting' a hypothesis may also imply that the given hypothesis is now proven, so there is no need to study it further. Nevertheless, that is never the case, as newer scientific evidence often challenges the existing studies. Discovery of viruses and fossils, rediscovery of presumed extinct species, criminal trials, and novel drug tests follow the same principles of testing hypotheses. In those cases, 'accepting' a hypothesis may lead to severe consequences.

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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

How to accept or reject a hypothesis?

A hypothesis is a proposed statement to explore a possible theory. Many studies in the fields of social sciences, sciences, and mathematics make use of hypothesis testing to prove a theory. Assumptions in a hypothesis help in making predictions. It is presented in the form of null and alternate hypotheses. When a hypothesis is presented negatively (for example, TV advertisements do not affect consumer behavior), it is called a null hypothesis. This article explains the conditions to accept or reject a hypothesis.

Why is it important to reject the null hypothesis?

A null hypothesis is a statement that describes that there is no difference in the assumed characteristics of the population. For example, in a study wherein the impact of the level of education on the efficiency of the employee need to be determined, null (Ho) and alternate (HA) hypothesis would be:

In the above-stated null hypothesis, there is very little chance of a relationship between both the variables (education and employee’s efficiency). When a null hypothesis is accepted, it shows that the study has a lack of evidence in showing any significant connection between the variables. This could be due to problems with the data such as:

high variability,
small sample size,
inappropriate sample and,
wrong data testing method.

Hence, for efficient, appropriate, and reliable results, it is suggested to reject the null hypothesis.

Conditions for rejecting a null hypothesis

Rejection of the null hypothesis provides sufficient evidence for supporting the perception of the researcher. Thus, a statistician always prefers to reject the null hypothesis. However, there are certain conditions which need to be fulfilled for the required results i.e.

Condition 1: Sample data should be reasonably random

A random sample is the one every person in the sample universe has an equal possibility of being selected for the analysis. Random sampling is necessary for deriving accurate results and rejecting the null hypothesis. This is because when a sample is randomly selected, characteristic traits of each participant in the study are the same, so there is no error in decision making. For example, in the sample hypothesis, instead of collecting data from all employees, the data was collected from only the board members of the company. This hypothesis testing would not provide good results as the sample does not represent all the employees of the company.

Condition 2: Distribution of the sample should be known

A dataset can be of two types: normally distributed or skewed. Normally distributed datasets require application of parametric tests i.e. Z-test, T-test, χ2-test, and F-distribution. On the other hand, skewed dataset uses non-parametric test i.e. Wilcoxon rank sum test, Wilcoxon signed rank test, and Kruskal Wallis test. For reliable hypothesis test result, it is essential that the distribution of the sample be tested.

Condition 3: Value of test statistic should not fall in the rejection region

Test statistic value is compared with critical value when the null hypothesis is true (critical value). If the test statistic is more extreme as compared to the critical value, then the null hypothesis would be rejected.

For example, in the sample hypothesis if the sample size is 50 and the significance level of the study is 5% then the critical value for the given two-tailed test would be 1.960. Hence, null hypothesis would be rejected if,

Condition 4: P-value should be less than the significance of the study

P-value represents the probability that the null hypothesis true. In order to reject the null hypothesis, it is essential that the p-value should be less that the significance or the precision level considered for the study. Hence,

Reject null hypothesis (H0) if ‘p’ value < statistical significance (0.01/0.05/0.10)
Accept null hypothesis (H0) if ‘p’ value > statistical significance (0.01/0.05/0.10)

For example, in the sample hypothesis if the considered statistical significance level is 5% and the p-value of the model is 0.12. Hence, the hypothesis of having no significant impact would not be rejected as 0.12 > 0.05.

Important points to note

While making the final decision of the hypothesis, these points should be noted i.e.

A large sample size i.e. at least greater than 30 should be considered. As per the Central Limit Theorem (CLT) large sample size i.e. at least greater than 30 is considered to be approximately normally distributed.
For deriving the results either p-value approach or rejection approach could be used. However, the p-value is a more preferable approach.
Statistical significance should be maintained at a minimum level.
The choice of the rejection region should be appropriately made by verifying the direction of the alternative hypothesis.
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11.8: Significance Testing and Confidence Intervals

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Learning Objectives

Explain why a confidence interval makes clear that one should not accept the null hypothesis

There is a close relationship between confidence intervals and significance tests. Specifically, if a statistic is significantly different from $0$ at the $0.05$ level, then the $95\%$ confidence interval will not contain $0$. All values in the confidence interval are plausible values for the parameter, whereas values outside the interval are rejected as plausible values for the parameter. In the Physicians' Reactions case study, the $95\%$ confidence interval for the difference between means extends from $2.00$ to $11.26$. Therefore, any value lower than $2.00$ or higher than $11.26$ is rejected as a plausible value for the population difference between means. Since zero is lower than $2.00$, it is rejected as a plausible value and a test of the null hypothesis that there is no difference between means is significant. It turns out that the $p$ value is $0.0057$. There is a similar relationship between the $99\%$ confidence interval and significance at the $0.01$ level.

Whenever an effect is significant, all values in the confidence interval will be on the same side of zero (either all positive or all negative). Therefore, a significant finding allows the researcher to specify the direction of the effect. There are many situations in which it is very unlikely two conditions will have exactly the same population means. For example, it is practically impossible that aspirin and acetaminophen provide exactly the same degree of pain relief. Therefore, even before an experiment comparing their effectiveness is conducted, the researcher knows that the null hypothesis of exactly no difference is false. However, the researcher does not know which drug offers more relief. If a test of the difference is significant, then the direction of the difference is established because the values in the confidence interval are either all positive or all negative.

If the $95\%$ confidence interval contains zero (more precisely, the parameter value specified in the null hypothesis), then the effect will not be significant at the $0.05$ level. Looking at non-significant effects in terms of confidence intervals makes clear why the null hypothesis should not be accepted when it is not rejected: Every value in the confidence interval is a plausible value of the parameter. Since zero is in the interval, it cannot be rejected. However, there is an infinite number of other values in the interval (assuming continuous measurement), and none of them can be rejected either.

What 'Fail to Reject' Means in a Hypothesis Test

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In statistics , scientists can perform a number of different significance tests to determine if there is a relationship between two phenomena. One of the first they usually perform is a null hypothesis test. In short, the null hypothesis states that there is no meaningful relationship between two measured phenomena. After a performing a test, scientists can:

Reject the null hypothesis (meaning there is a definite, consequential relationship between the two phenomena), or
Fail to reject the null hypothesis (meaning the test has not identified a consequential relationship between the two phenomena)

Key Takeaways: The Null Hypothesis

• In a test of significance, the null hypothesis states that there is no meaningful relationship between two measured phenomena.

• By comparing the null hypothesis to an alternative hypothesis, scientists can either reject or fail to reject the null hypothesis.

• The null hypothesis cannot be positively proven. Rather, all that scientists can determine from a test of significance is that the evidence collected does or does not disprove the null hypothesis.

It is important to note that a failure to reject does not mean that the null hypothesis is true—only that the test did not prove it to be false. In some cases, depending on the experiment, a relationship may exist between two phenomena that is not identified by the experiment. In such cases, new experiments must be designed to rule out alternative hypotheses.

Null vs. Alternative Hypothesis

The null hypothesis is considered the default in a scientific experiment . In contrast, an alternative hypothesis is one that claims that there is a meaningful relationship between two phenomena. These two competing hypotheses can be compared by performing a statistical hypothesis test, which determines whether there is a statistically significant relationship between the data.

For example, scientists studying the water quality of a stream may wish to determine whether a certain chemical affects the acidity of the water. The null hypothesis—that the chemical has no effect on the water quality—can be tested by measuring the pH level of two water samples, one of which contains some of the chemical and one of which has been left untouched. If the sample with the added chemical is measurably more or less acidic—as determined through statistical analysis—it is a reason to reject the null hypothesis. If the sample's acidity is unchanged, it is a reason to not reject the null hypothesis.

When scientists design experiments, they attempt to find evidence for the alternative hypothesis. They do not try to prove that the null hypothesis is true. The null hypothesis is assumed to be an accurate statement until contrary evidence proves otherwise. As a result, a test of significance does not produce any evidence pertaining to the truth of the null hypothesis.

Failing to Reject vs. Accept

In an experiment, the null hypothesis and the alternative hypothesis should be carefully formulated such that one and only one of these statements is true. If the collected data supports the alternative hypothesis, then the null hypothesis can be rejected as false. However, if the data does not support the alternative hypothesis, this does not mean that the null hypothesis is true. All it means is that the null hypothesis has not been disproven—hence the term "failure to reject." A "failure to reject" a hypothesis should not be confused with acceptance.

In mathematics, negations are typically formed by simply placing the word “not” in the correct place. Using this convention, tests of significance allow scientists to either reject or not reject the null hypothesis. It sometimes takes a moment to realize that “not rejecting” is not the same as "accepting."

Null Hypothesis Example

In many ways, the philosophy behind a test of significance is similar to that of a trial. At the beginning of the proceedings, when the defendant enters a plea of “not guilty,” it is analogous to the statement of the null hypothesis. While the defendant may indeed be innocent, there is no plea of “innocent” to be formally made in court. The alternative hypothesis of “guilty” is what the prosecutor attempts to demonstrate.

The presumption at the outset of the trial is that the defendant is innocent. In theory, there is no need for the defendant to prove that he or she is innocent. The burden of proof is on the prosecuting attorney, who must marshal enough evidence to convince the jury that the defendant is guilty beyond a reasonable doubt. Likewise, in a test of significance, a scientist can only reject the null hypothesis by providing evidence for the alternative hypothesis.

If there is not enough evidence in a trial to demonstrate guilt, then the defendant is declared “not guilty.” This claim has nothing to do with innocence; it merely reflects the fact that the prosecution failed to provide enough evidence of guilt. In a similar way, a failure to reject the null hypothesis in a significance test does not mean that the null hypothesis is true. It only means that the scientist was unable to provide enough evidence for the alternative hypothesis.

For example, scientists testing the effects of a certain pesticide on crop yields might design an experiment in which some crops are left untreated and others are treated with varying amounts of pesticide. Any result in which the crop yields varied based on pesticide exposure—assuming all other variables are equal—would provide strong evidence for the alternative hypothesis (that the pesticide does affect crop yields). As a result, the scientists would have reason to reject the null hypothesis.

Hypothesis Test for the Difference of Two Population Proportions
Type I and Type II Errors in Statistics
Null Hypothesis and Alternative Hypothesis
Null Hypothesis Examples
How to Conduct a Hypothesis Test
An Example of a Hypothesis Test
What Is a P-Value?
The Difference Between Type I and Type II Errors in Hypothesis Testing
What Is a Hypothesis? (Science)
Null Hypothesis Definition and Examples
Hypothesis Test Example
The Runs Test for Random Sequences
How to Do Hypothesis Tests With the Z.TEST Function in Excel
Scientific Method Vocabulary Terms
What Is the Difference Between Alpha and P-Values?
Chi-Square Goodness of Fit Test

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Knowledge Base
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.

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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The p-value and rejecting the null (for one- and two-tail tests)

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What is the p-value?

The ???p??? -value (or the observed level of significance) is the smallest level of significance at which you can reject the null hypothesis, assuming the null hypothesis is true.

You can also think about the ???p???-value as the total area of the region of rejection. Remember that in a one-tailed test, the region of rejection is consolidated into one tail, whereas in a two-tailed test, the rejection region is split between two tails.

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So, as you might expect, calculating the ???p???-value as the area of the rejection region will be slightly different depending on whether we’re using a two-tailed test or a one-tailed test, and whether the one-tailed test is an upper-tail test or lower-tail test.

Calculating the ???p???-value

For a one-tailed, lower-tail test

For a one-tailed test, first calculate your ???z???-test statistic. For a lower-tail test, ???z??? will be negative. Look up the ???z???-value in a ???z???-table, and the value you find in the body of the table represents the area under the probability distribution curve to the left of your negative ???z???-value.

For instance, assume you found ???z=-1.46???. In a ???z???-table, you find

So ???0.0721??? is the area under the curve to the left of ???z=-1.46???, and this is the ???p???-value also. So ???p=0.0721???.

For a one-tailed, upper-tail test

For a one-tailed test, first calculate your ???z???-test statistic. For an upper-tail test, ???z??? will be positive. Look up the ???z???-value in a ???z???-table, and the value you find in the body of the table represents the area under the probability distribution curve to the left of your positive ???z???-value.

For instance, assume you found ???z=1.46???. In a ???z???-table, you find

But in an upper-tail test, you’re interested in the area to the right of the ???z???-value, not the area to the left. To find the area to the right, you need to subtract the value in the ???z???-table from ???1???.

???1-0.9279=0.0721???

So ???0.0721??? is the area under the curve to the right of ???z=1.46???, and this is the ???p???-value also. So ???p=0.0721???.

For a two-tailed test

For a two-tailed test, first calculate your ???z???-test statistic. For an two-tail test, ???z??? could be either positive or negative. Look up the ???z???-value in a -table, and the value you find in the body of the table represents the area under the probability distribution curve to the left of your ???z???-value.

For instance, assume you found ???z=1.23???. In a ???z???-table, you find

But for a positive ???z???-value, you’re interested in the area to the right of the ???z???-value, not the area to the left. To find the area to the right, you need to subtract the value in the ???z???-table from ???1???.

???1-0.8907=0.1093???

So ???0.1093??? is the area under the curve to the right of ???z=1.23???. Because this is a two-tail test, the region of rejection is not only the ???10.93\%??? of area under the upper tail, but also the symmetrical ???10.93\%??? of area under the lower tail. So we’ll double ???0.1093??? to get ???2(0.1093)=0.2186???, and this is the ???p???-value also. So ???p=0.2186???.

How to reject the null hypothesis

The reason we’ve gone through all this work to understand the ???p???-value is because using a ???p???-value is a really quick way to decide whether or not to reject the null hypothesis.

Whether or not you should reject ???H_0??? can be determined by the relationship between the ???\alpha??? level and the ???p???-value.

If ???p\leq \alpha???, reject the null hypothesis

If ???p>\alpha???, do not reject the null hypothesis

In our earlier examples, we found

???p=0.0721??? for the lower-tail one-tailed test

???p=0.0721??? for the upper-tail one-tailed test

???p=0.2186??? for the two-tailed test

With these in mind, let’s say for instance you set the confidence level of your hypothesis test at ???90\%???, which is the same as setting the ???\alpha??? level at ???\alpha=0.10???. In that case,

???p=0.0721\leq\alpha=0.10???

???p=0.2186>\alpha=0.10???

So we would have rejected the null hypothesis for both one-tailed tests, but we would have failed to reject the null in the two-tailed test. If, however, we’d picked a more rigorous ???\alpha=0.05??? or ???\alpha=0.01???, we would have failed to reject the null hypothesis every time.

Significance

The significance (or statistical significance ) of a test is the probability of obtaining your result by chance. The less likely it is that we obtained a result by chance, the more significant our results.

Hopefully by now it’s not too surprising by now that all of these are equivalent statements:

The finding is significant at the ???0.01??? level

The confidence level is ???99\%???

The Type I error rate is ???0.01???

The alpha level is ???0.01???, ???\alpha=0.01???

The area of the rejection region is ???0.01???

The ???p???-value is ???0.01???, ???p=0.01???

There’s a ???1??? in ???100??? chance of getting a result as, or more, extreme as this one

The smaller the ???p???-value, or the smaller the alpha value, or the lower the Type I error rate, and the smaller the region of rejection, the higher the confidence level, and the less likely it is that you got your result by chance.

In other words, an alpha level of ???0.10??? (or a ???p???-value of ???0.10???, or a confidence level of ???90\%???) is a lower bar to clear. At that significance level, there’s a ???1??? in ???10??? chance that the result we got was just by chance. And therefore there’s a ???1??? in ???10??? chance that we’ll reject the null hypothesis when we really shouldn’t have, thinking that we provided support for the alternative hypothesis when we shouldn’t have.

But a stricter alpha level of ???0.01??? (or a ???p???-value of ???0.01???, or a confidence level of ???99\%???) is a higher bar to clear. At that significance level, there’s only a ???1??? in ???100??? chance that the result we got was just by chance. And therefore there’s only a ???1??? in ???100??? chance that we’ll reject the null hypothesis when we really shouldn’t have, thinking that we provided support for the alternative hypothesis when we shouldn’t have.

If we find a result that clears the bar we’ve set for ourselves, then we reject the null hypothesis and we say that the finding is significant at the ???p???-value that we find. Otherwise, we fail to reject the null.

How to use the p-value to determine whether or not you can reject the null hypothesis

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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.

Important Sidenote: We interviewed numerous data science professionals (data scientists, hiring managers, recruiters – you name it) and identified 6 proven steps to follow for becoming a data scientist. Read my article: ‘6 Proven Steps To Becoming a Data Scientist [Complete Guide] for in-depth findings and recommendations! – This is perhaps the most comprehensive article on the subject you will find on the internet!

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.

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.

1.1: The Scientific Method

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Adapted from http://www.biologycorner.com/

Introduction:

The scientific method is central to the study of biology: it is a process of acquiring and verifying information through experimentation. The general steps of the scientific method are depicted in the figure below. The hypothesis , or suggested explanation for the observation, is the basis for setting up experiments. A good experimental design is essential to the scientific method. A few keys to good experimental design include effective use of controls, reproducibility, a large sample size, and multiple trials.

In an experiment, in order to determine that any changes that occur are due to investigator manipulation only, there must be some basis for comparison. A control group is necessary to establish this basis of comparison. In the control group, everything is kept the same as the experimental group except for the independent variable .

The experimental group is actually being experimented upon. For example, in a drug trial, there will be a group that receives the drug (the experimental group) and a group that receives a placebo (the control group). The drug itself is considered the independent variable and any change(s) that occur because of the drug are considered the dependent variable .

In order to ensure that it is only the drug causing changes, all other variables must be tightly controlled (such as diet, exercise, smoking, etc.). These are referred to as controlled variables .

Part 1: The Strange Case of BeriBeri:

In 1887 a strange nerve disease attacked the people in the Dutch East Indies. The disease was beriberi. Symptoms of the disease included weakness and loss of appetite, victims often died of heart failure. Scientists thought the disease might be caused by bacteria. They injected chickens with bacteria from the blood of patients with beriberi. The injected chickens became sick. However, so did a group of chickens that were not injected with bacteria. One of the scientists, Dr. Eijkman, noticed something. Before the experiment, all the chickens had eaten whole-grain rice, but during the experiment, the chickens were fed polished rice. Dr. Eijkman researched this interesting case and found that polished rice lacked thiamine, a vitamin necessary for good health.

1. State the problem.

2. What was the hypothesis?

3. How was the hypothesis tested?

4. Do the results indicate that the hypothesis should be rejected?

5. What should be the new hypothesis and how would you test it?

Part 2: How Penicillin Was Discovered:

In 1928, Sir Alexander Fleming was studying Staphylococcus bacteria growing in culture dishes. He noticed that a mold called Penicillium was also growing in some of the dishes. A clear area existed around the mold because all the bacteria that had grown in this area had died. In the culture dishes without the mold, no clear areas were present. Fleming hypothesized that the mold must be producing a chemical that killed the bacteria. He decided to isolate this substance and test it to see if it would kill bacteria. Fleming transferred the mold to a nutrient broth solution. This solution contained all the materials the mold needed to grow. After the mold grew, he removed it from the nutrient broth. Fleming then added the nutrient broth in which the mold had grown to a culture of bacteria. He observed that the bacteria died which was later used to develop antibiotics used to treat a variety of diseases.

1. Identify the problem.

2. What was Fleming's hypothesis?

5. This experiment led to the development of what major medical advancement…?

Part 3: Identify the Controls and Variables

Smithers thinks that a special juice will increase the productivity of workers. He creates two groups of 50 workers each and assigns each group the same task (in this case, they're supposed to staple a set of papers). Group A is given the special juice to drink while they work. Group B is not given the special juice. After an hour, Smithers counts how many stacks of papers each group has made. Group A made 1,587 stacks; Group B made 2,113 stacks.

Identify the:

Control Group:
Independent Variable:
Dependent Variable:

What should Smithers' conclusion be?

How could this experiment be improved?

Homer notices that his shower is covered in a strange green slime. His friend Barney tells him that coconut juice will get rid of the green slime. Homer decides to check this out by spraying half of the shower with coconut juice. He sprays the other half of the shower with water. After 3 days of "treatment", there is no change in the appearance of the green slime on either side of the shower.

What was the initial observation?

What should Homer's conclusion be?

Bart believes that mice exposed to radio waves will become extra strong (maybe he's been reading too much Radioactive Man). He decides to perform this experiment by placing 10 mice near a radio for 5 hours. He compared these 10 mice to another 10 mice that had not been exposed. His test consisted of a heavy block of wood that blocked the mouse food. He found that 8 out of 10 of the exposed mice were able to push the block away, while 7 out of 10 of the other mice were able to do the same.

What should Bart's conclusion be?

How could Bart's experiment be improved?

Krusty was told that a certain itching powder was the newest best thing on the market: it even claims to cause 50% longer lasting itches. Interested in this product, he buys the itching powder and compares it to his usual product. One test subject (A) is sprinkled with the original itching powder, and another test subject (B) was sprinkled with the Experimental itching powder. Subject A reported having itches for 30 minutes. Subject B reported having itches for 45 minutes

Explain whether the data supports the advertisement's claims about its product.

Design Lisa's experiment.

Type 1 and Type 2 Errors in Statistics

Saul Mcleod, PhD

Editor-in-Chief for Simply Psychology

BSc (Hons) Psychology, MRes, PhD, University of Manchester

Saul Mcleod, PhD., is a qualified psychology teacher with over 18 years of experience in further and higher education. He has been published in peer-reviewed journals, including the Journal of Clinical Psychology.

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A statistically significant result cannot prove that a research hypothesis is correct (which implies 100% certainty). Because a p -value is based on probabilities, there is always a chance of making an incorrect conclusion regarding accepting or rejecting the null hypothesis ( H 0 ).

Anytime we make a decision using statistics, there are four possible outcomes, with two representing correct decisions and two representing errors.

The chances of committing these two types of errors are inversely proportional: that is, decreasing type I error rate increases type II error rate and vice versa.

As the significance level (α) increases, it becomes easier to reject the null hypothesis, decreasing the chance of missing a real effect (Type II error, β). If the significance level (α) goes down, it becomes harder to reject the null hypothesis , increasing the chance of missing an effect while reducing the risk of falsely finding one (Type I error).

Type I error

A type 1 error is also known as a false positive and occurs when a researcher incorrectly rejects a true null hypothesis. Simply put, it’s a false alarm.

This means that you report that your findings are significant when they have occurred by chance.

The probability of making a type 1 error is represented by your alpha level (α), the p- value below which you reject the null hypothesis.

A p -value of 0.05 indicates that you are willing to accept a 5% chance of getting the observed data (or something more extreme) when the null hypothesis is true.

You can reduce your risk of committing a type 1 error by setting a lower alpha level (like α = 0.01). For example, a p-value of 0.01 would mean there is a 1% chance of committing a Type I error.

However, using a lower value for alpha means that you will be less likely to detect a true difference if one really exists (thus risking a type II error).

Scenario: Drug Efficacy Study

Imagine a pharmaceutical company is testing a new drug, named “MediCure”, to determine if it’s more effective than a placebo at reducing fever. They experimented with two groups: one receives MediCure, and the other received a placebo.

Null Hypothesis (H0) : MediCure is no more effective at reducing fever than the placebo.
Alternative Hypothesis (H1) : MediCure is more effective at reducing fever than the placebo.

After conducting the study and analyzing the results, the researchers found a p-value of 0.04.

If they use an alpha (α) level of 0.05, this p-value is considered statistically significant, leading them to reject the null hypothesis and conclude that MediCure is more effective than the placebo.

However, MediCure has no actual effect, and the observed difference was due to random variation or some other confounding factor. In this case, the researchers have incorrectly rejected a true null hypothesis.

Error : The researchers have made a Type 1 error by concluding that MediCure is more effective when it isn’t.

Implications

Resource Allocation : Making a Type I error can lead to wastage of resources. If a business believes a new strategy is effective when it’s not (based on a Type I error), they might allocate significant financial and human resources toward that ineffective strategy.

Unnecessary Interventions : In medical trials, a Type I error might lead to the belief that a new treatment is effective when it isn’t. As a result, patients might undergo unnecessary treatments, risking potential side effects without any benefit.

Reputation and Credibility : For researchers, making repeated Type I errors can harm their professional reputation. If they frequently claim groundbreaking results that are later refuted, their credibility in the scientific community might diminish.

Type II error

A type 2 error (or false negative) happens when you accept the null hypothesis when it should actually be rejected.

Here, a researcher concludes there is not a significant effect when actually there really is.

The probability of making a type II error is called Beta (β), which is related to the power of the statistical test (power = 1- β). You can decrease your risk of committing a type II error by ensuring your test has enough power.

You can do this by ensuring your sample size is large enough to detect a practical difference when one truly exists.

Scenario: Efficacy of a New Teaching Method

Educational psychologists are investigating the potential benefits of a new interactive teaching method, named “EduInteract”, which utilizes virtual reality (VR) technology to teach history to middle school students.

They hypothesize that this method will lead to better retention and understanding compared to the traditional textbook-based approach.

Null Hypothesis (H0) : The EduInteract VR teaching method does not result in significantly better retention and understanding of history content than the traditional textbook method.
Alternative Hypothesis (H1) : The EduInteract VR teaching method results in significantly better retention and understanding of history content than the traditional textbook method.

The researchers designed an experiment where one group of students learns a history module using the EduInteract VR method, while a control group learns the same module using a traditional textbook.

After a week, the student’s retention and understanding are tested using a standardized assessment.

Upon analyzing the results, the psychologists found a p-value of 0.06. Using an alpha (α) level of 0.05, this p-value isn’t statistically significant.

Therefore, they fail to reject the null hypothesis and conclude that the EduInteract VR method isn’t more effective than the traditional textbook approach.

However, let’s assume that in the real world, the EduInteract VR truly enhances retention and understanding, but the study failed to detect this benefit due to reasons like small sample size, variability in students’ prior knowledge, or perhaps the assessment wasn’t sensitive enough to detect the nuances of VR-based learning.

Error : By concluding that the EduInteract VR method isn’t more effective than the traditional method when it is, the researchers have made a Type 2 error.

This could prevent schools from adopting a potentially superior teaching method that might benefit students’ learning experiences.

Missed Opportunities : A Type II error can lead to missed opportunities for improvement or innovation. For example, in education, if a more effective teaching method is overlooked because of a Type II error, students might miss out on a better learning experience.

Potential Risks : In healthcare, a Type II error might mean overlooking a harmful side effect of a medication because the research didn’t detect its harmful impacts. As a result, patients might continue using a harmful treatment.

Stagnation : In the business world, making a Type II error can result in continued investment in outdated or less efficient methods. This can lead to stagnation and the inability to compete effectively in the marketplace.

How do Type I and Type II errors relate to psychological research and experiments?

Type I errors are like false alarms, while Type II errors are like missed opportunities. Both errors can impact the validity and reliability of psychological findings, so researchers strive to minimize them to draw accurate conclusions from their studies.

How does sample size influence the likelihood of Type I and Type II errors in psychological research?

Sample size in psychological research influences the likelihood of Type I and Type II errors. A larger sample size reduces the chances of Type I errors, which means researchers are less likely to mistakenly find a significant effect when there isn’t one.

A larger sample size also increases the chances of detecting true effects, reducing the likelihood of Type II errors.

Are there any ethical implications associated with Type I and Type II errors in psychological research?

Yes, there are ethical implications associated with Type I and Type II errors in psychological research.

Type I errors may lead to false positive findings, resulting in misleading conclusions and potentially wasting resources on ineffective interventions. This can harm individuals who are falsely diagnosed or receive unnecessary treatments.

Type II errors, on the other hand, may result in missed opportunities to identify important effects or relationships, leading to a lack of appropriate interventions or support. This can also have negative consequences for individuals who genuinely require assistance.

Therefore, minimizing these errors is crucial for ethical research and ensuring the well-being of participants.

Further Information

Publication manual of the American Psychological Association
Statistics for Psychology Book Download

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COMMENTS

When Do You Reject the Null Hypothesis? (3 Examples)
two-tailed p-value: 0.0045. 4. Reject or fail to reject the null hypothesis. Since the p-value (0.0045) is less than the significance level (0.01) we reject the null hypothesis. We have sufficient evidence to say that the mean vertical jump before and after participating in the training program is not equal.
What Is The Null Hypothesis & When To Reject It
The observed value is statistically significant (p ≤ 0.05), so the null hypothesis (N0) is rejected, and the alternative hypothesis (Ha) is accepted. Usually, a researcher uses a confidence level of 95% or 99% (p-value of 0.05 or 0.01) as general guidelines to decide if you should reject or keep the null.
Null Hypothesis: Definition, Rejecting & Examples
When your sample contains sufficient evidence, you can reject the null and conclude that the effect is statistically significant. Statisticians often denote the null hypothesis as H 0 or H A.. Null Hypothesis H 0: No effect exists in the population.; Alternative Hypothesis H A: The effect exists in the population.; In every study or experiment, researchers assess an effect or relationship.
Hypothesis Testing
Let's return finally to the question of whether we reject or fail to reject the null hypothesis. If our statistical analysis shows that the significance level is below the cut-off value we have set (e.g., either 0.05 or 0.01), we reject the null hypothesis and accept the alternative hypothesis. Alternatively, if the significance level is above ...
Support or Reject Null Hypothesis in Easy Steps
Use the P-Value method to support or reject null hypothesis. Step 1: State the null hypothesis and the alternate hypothesis ("the claim"). H o :p ≤ 0.23; H 1 :p > 0.23 (claim) Step 2: Compute by dividing the number of positive respondents from the number in the random sample: 63 / 210 = 0.3.
9.1: Null and Alternative Hypotheses
Review. In a hypothesis test, sample data is evaluated in order to arrive at a decision about some type of claim.If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we: Evaluate the null hypothesis, typically denoted with $H_{0}$.The null is not rejected unless the hypothesis test shows otherwise.
Hypothesis Testing
There are 5 main steps in hypothesis testing: State your research hypothesis as a null hypothesis and alternate hypothesis (H o) and (H a or H 1 ). Collect data in a way designed to test the hypothesis. Perform an appropriate statistical test. Decide whether to reject or fail to reject your null hypothesis. Present the findings in your results ...
4.4: Hypothesis Testing
Now if we obtain any observation with a Z score greater than 1.65, we would reject H 0. If the null hypothesis is true, we incorrectly reject the null hypothesis about 5% of the time when the sample mean is above the null value, as shown in Figure 4.19. Suppose the sample mean was smaller than the null value.
Hypothesis: Accept or Fail to Reject?
The outcome of any hypothesis testing leads to rejecting or not rejecting the null hypothesis. This decision is taken based on the analysis of the data, an appropriate test statistic, an appropriate confidence level, the critical values, and P-values.However, when the evidence suggests that the null hypothesis cannot be rejected, is it right to say, 'Accept' the null hypothesis?
6a.1
The first step in hypothesis testing is to set up two competing hypotheses. The hypotheses are the most important aspect. If the hypotheses are incorrect, your conclusion will also be incorrect. The two hypotheses are named the null hypothesis and the alternative hypothesis. The null hypothesis is typically denoted as H 0.
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 ...
How to accept or reject a hypothesis?
P-value represents the probability that the null hypothesis true. In order to reject the null hypothesis, it is essential that the p-value should be less that the significance or the precision level considered for the study. Hence, Reject null hypothesis (H0) if 'p' value < statistical significance (0.01/0.05/0.10)
Failing to Reject the Null Hypothesis
There is something I am confused about. If our significance level is .05 and our resulting p-value is .02 (thus the strength of our evidence is strong enough to reject the null hypothesis), do we state that we reject the null hypothesis with 95% confidence or 98% confidence? My guess is our confidence level is 95% since or alpha was .05.
11.8: Significance Testing and Confidence Intervals
Since zero is lower than $2.00$, it is rejected as a plausible value and a test of the null hypothesis that there is no difference between means is significant. ... (0.05\) level. Looking at non-significant effects in terms of confidence intervals makes clear why the null hypothesis should not be accepted when it is not rejected: Every value ...
What 'Fail to Reject' Means in a Hypothesis Test
A "failure to reject" a hypothesis should not be confused with acceptance. In mathematics, negations are typically formed by simply placing the word "not" in the correct place. Using this convention, tests of significance allow scientists to either reject or not reject the null hypothesis.
Understanding P-values
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.
The p-value and rejecting the null (for one- and two-tail tests)
The p-value (or the observed level of significance) is the smallest level of significance at which you can reject the null hypothesis, assuming the null hypothesis is true. You can also think about the p-value as the total area of the region of rejection. Remember that in a one-tailed test, the region of rejection is consolidated into one tail ...
P-Value vs. Critical Value: A Friendly Guide for Beginners
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.
1.1: The Scientific Method
The hypothesis, or suggested explanation for the observation, is the basis for setting up experiments. A good experimental design is essential to the scientific method. ... Do the results indicate that the hypothesis should be rejected? 5. What should be the new hypothesis and how would you test it? Part 2: How Penicillin Was Discovered:
Understanding P-Values and Statistical Significance
In statistical hypothesis testing, you reject the null hypothesis when the p-value is less than or equal to the significance level (α) you set before conducting your test. The significance level is the probability of rejecting the null hypothesis when it is true. Commonly used significance levels are 0.01, 0.05, and 0.10.
Type 1 and Type 2 Errors in Statistics
A statistically significant result cannot prove that a research hypothesis is correct (which implies 100% certainty). Because a p-value is based on probabilities, there is always a chance of making an incorrect conclusion regarding accepting or rejecting the null hypothesis (H 0).
Bio Chap 13, 14, 15 AP Classroom Questions Flashcards
D The null hypothesis should be rejected because the critical value is greater than the calculated value. and more. Study with Quizlet and memorize flashcards containing terms like In dogs, one pair of alleles determines coat color (dark and albino). Another pair of alleles determines hair length (short and long).
unit 5 bio Flashcards
B The null hypothesis should not be rejected because the critical value is less than the calculated value. C The null hypothesis should not be rejected because the critical value is greater than the calculated value. D The null hypothesis should be rejected because the critical value is greater than the calculated value. and more.