two tailed hypothesis test example

Upper-tailed, Lower-tailed, Two-tailed Tests

The research or alternative hypothesis can take one of three forms. An investigator might believe that the parameter has increased, decreased or changed. For example, an investigator might hypothesize:  

The exact form of the research hypothesis depends on the investigator's belief about the parameter of interest and whether it has possibly increased, decreased or is different from the null value. The research hypothesis is set up by the investigator before any data are collected.

Rejection Region for Upper-Tailed Z Test (H : μ > μ ) with α=0.05

The decision rule is: Reject H if Z 1.645.

Rejection Region for Lower-Tailed Z Test (H 1 : μ < μ 0 ) with α =0.05

The decision rule is: Reject H 0 if Z < 1.645.

Rejection Region for Two-Tailed Z Test (H 1 : μ ≠ μ 0 ) with α =0.05

The decision rule is: Reject H 0 if Z < -1.960 or if Z > 1.960.

The complete table of critical values of Z for upper, lower and two-tailed tests can be found in the table of Z values to the right in "Other Resources."

Critical values of t for upper, lower and two-tailed tests can be found in the table of t values in "Other Resources."

• Step 4. Compute the test statistic.  

Here we compute the test statistic by substituting the observed sample data into the test statistic identified in Step 2.

• Step 5. Conclusion.  

The final conclusion is made by comparing the test statistic (which is a summary of the information observed in the sample) to the decision rule. The final conclusion will be either to reject the null hypothesis (because the sample data are very unlikely if the null hypothesis is true) or not to reject the null hypothesis (because the sample data are not very unlikely).  

If the null hypothesis is rejected, then an exact significance level is computed to describe the likelihood of observing the sample data assuming that the null hypothesis is true. The exact level of significance is called the p-value and it will be less than the chosen level of significance if we reject H 0 .

Statistical computing packages provide exact p-values as part of their standard output for hypothesis tests. In fact, when using a statistical computing package, the steps outlined about can be abbreviated. The hypotheses (step 1) should always be set up in advance of any analysis and the significance criterion should also be determined (e.g., α =0.05). Statistical computing packages will produce the test statistic (usually reporting the test statistic as t) and a p-value. The investigator can then determine statistical significance using the following: If p < α then reject H 0 .  

• Step 1. Set up hypotheses and determine level of significance

H 0 : μ = 191 H 1 : μ > 191                 α =0.05

The research hypothesis is that weights have increased, and therefore an upper tailed test is used.

• Step 2. Select the appropriate test statistic.

Because the sample size is large (n > 30) the appropriate test statistic is

• Step 3. Set up decision rule.  

In this example, we are performing an upper tailed test (H 1 : μ> 191), with a Z test statistic and selected α =0.05.   Reject H 0 if Z > 1.645.

We now substitute the sample data into the formula for the test statistic identified in Step 2.  

We reject H 0 because 2.38 > 1.645. We have statistically significant evidence at a =0.05, to show that the mean weight in men in 2006 is more than 191 pounds. Because we rejected the null hypothesis, we now approximate the p-value which is the likelihood of observing the sample data if the null hypothesis is true. An alternative definition of the p-value is the smallest level of significance where we can still reject H 0 . In this example, we observed Z=2.38 and for α=0.05, the critical value was 1.645. Because 2.38 exceeded 1.645 we rejected H 0 . In our conclusion we reported a statistically significant increase in mean weight at a 5% level of significance. Using the table of critical values for upper tailed tests, we can approximate the p-value. If we select α=0.025, the critical value is 1.96, and we still reject H 0 because 2.38 > 1.960. If we select α=0.010 the critical value is 2.326, and we still reject H 0 because 2.38 > 2.326. However, if we select α=0.005, the critical value is 2.576, and we cannot reject H 0 because 2.38 < 2.576. Therefore, the smallest α where we still reject H 0 is 0.010. This is the p-value. A statistical computing package would produce a more precise p-value which would be in between 0.005 and 0.010. Here we are approximating the p-value and would report p < 0.010.                  

In all tests of hypothesis, there are two types of errors that can be committed. The first is called a Type I error and refers to the situation where we incorrectly reject H 0 when in fact it is true. This is also called a false positive result (as we incorrectly conclude that the research hypothesis is true when in fact it is not). When we run a test of hypothesis and decide to reject H 0 (e.g., because the test statistic exceeds the critical value in an upper tailed test) then either we make a correct decision because the research hypothesis is true or we commit a Type I error. The different conclusions are summarized in the table below. Note that we will never know whether the null hypothesis is really true or false (i.e., we will never know which row of the following table reflects reality).

Table - Conclusions in Test of Hypothesis

In the first step of the hypothesis test, we select a level of significance, α, and α= P(Type I error). Because we purposely select a small value for α, we control the probability of committing a Type I error. For example, if we select α=0.05, and our test tells us to reject H 0 , then there is a 5% probability that we commit a Type I error. Most investigators are very comfortable with this and are confident when rejecting H 0 that the research hypothesis is true (as it is the more likely scenario when we reject H 0 ).

When we run a test of hypothesis and decide not to reject H 0 (e.g., because the test statistic is below the critical value in an upper tailed test) then either we make a correct decision because the null hypothesis is true or we commit a Type II error. Beta (β) represents the probability of a Type II error and is defined as follows: β=P(Type II error) = P(Do not Reject H 0 | H 0 is false). Unfortunately, we cannot choose β to be small (e.g., 0.05) to control the probability of committing a Type II error because β depends on several factors including the sample size, α, and the research hypothesis. When we do not reject H 0 , it may be very likely that we are committing a Type II error (i.e., failing to reject H 0 when in fact it is false). Therefore, when tests are run and the null hypothesis is not rejected we often make a weak concluding statement allowing for the possibility that we might be committing a Type II error. If we do not reject H 0 , we conclude that we do not have significant evidence to show that H 1 is true. We do not conclude that H 0 is true.

 The most common reason for a Type II error is a small sample size.

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Content ©2017. All Rights Reserved. Date last modified: November 6, 2017. Wayne W. LaMorte, MD, PhD, MPH

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4.2 Two-tailed tests

Hypotheses that have an equal (=) or not equal (≠) supposition (sign) in the statement are called non-directional hypotheses . In non-directional hypotheses, the researcher is interested in whether there is a statistically significant difference or relationship between two or more variables, but does not have any specific expectation about which group or variable will be higher or lower. For example, a non-directional hypothesis might be: ‘There is a difference in the preference for brand X between male and female consumers.’ In this hypothesis, the researcher is interested in whether there is a statistically significant difference in the preference for brand X between male and female consumers, but does not have a specific prediction about which gender will have a higher preference. The researcher may conduct a survey or experiment to collect data on the brand preference of male and female consumers and then use statistical analysis to determine whether there is a significant difference between the two groups.

Non-directional hypotheses are also known as two-tailed hypotheses. The term ‘two-tailed’ comes from the fact that the statistical test used to evaluate the hypothesis is based on the assumption that the difference or relationship could occur in either direction, resulting in two ‘tails’ in the probability distribution. Using the coffee foam example (from Activity 1), you have the following set of hypotheses:

H 0 : µ = 1cm foam

H a : µ ≠ 1cm foam

In this case, the researcher can reject the null hypothesis for the mean value that is either ‘much higher’ or ‘much lower’ than 1 cm foam. This is called a two-tailed test because the rejection region includes outcomes from both the upper and lower tails of the sample distribution when determining a decision rule. To give an illustration, if you set alpha level (α) equal to 0.05, that would give you a 95% confidence level. Then, you would reject the null hypothesis for obtained values of z 1.96 (you will look at how to calculate z-scores later in the course).

This can be plotted on a graph as shown in Figure 7.

A symmetrical graph reminiscent of a bell. The x-axis is labelled ‘z-score’ and the y-axis is labelled ‘probability density’. The x-axis increases in increments of 1 from -2 to 2.

The top of the bell-shaped curve is labelled ‘Foam height = 1cm’. The graph circles the rejection regions of the null hypothesis on both sides of the bell curve. Within these circles are two areas shaded orange: beneath the curve from -2 downwards which is labelled z 1.96 and α = 0.025.

In a two-tailed hypothesis test, the null hypothesis assumes that there is no significant difference or relationship between the two groups or variables, and the alternative hypothesis suggests that there is a significant difference or relationship, but does not specify the direction of the difference or relationship.

When performing a two-tailed test, you need to determine the level of significance, which is denoted by alpha (α). The value of alpha, in this case, is 0.05. To perform a two-tailed test at a significance level of 0.05, you need to divide alpha by 2, giving a significance level of 0.025 for each distribution tail (0.05/2 = 0.025). This is done because the two-tailed test is looking for significance in either tail of the distribution. If the calculated test statistic falls in the rejection region of either tail of the distribution, then the null hypothesis is rejected and the alternative hypothesis is accepted. In this case, the researcher can conclude that there is a significant difference or relationship between the two groups or variables.

Assuming that the population follows a normal distribution, the tail located below the critical value of z = –1.96 (in a later section, you will discuss how this value was determined) and the tail above the critical value of z = +1.96 each represent a proportion of 0.025. These tails are referred to as the lower and upper tails, respectively, and they correspond to the extreme values of the distribution that are far from the central part of the bell curve. These critical values are used in a two-tailed hypothesis test to determine whether to reject or fail to reject the null hypothesis. The null hypothesis represents the default assumption that there is no significant difference between the observed data and what would be expected under a specific condition.

If the calculated test statistic falls within the critical values, then the null hypothesis cannot be rejected at the 0.05 level of significance. However, if the calculated test statistic falls outside the critical values (orange-coloured areas in Figure 7), then the null hypothesis can be rejected in favour of the alternative hypothesis, suggesting that there is evidence of a significant difference between the observed data and what would be expected under the specified condition.

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Hypothesis Testing | A Step-by-Step Guide with Easy Examples

Published on November 8, 2019 by Rebecca Bevans . Revised on June 22, 2023.

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics . It is most often used by scientists to test specific predictions, called hypotheses, that arise from theories.

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 and discussion section.

Though the specific details might vary, the procedure you will use when testing a hypothesis will always follow some version of these steps.

Table of contents

Step 1: state your null and alternate hypothesis, step 2: collect data, step 3: perform a statistical test, step 4: decide whether to reject or fail to reject your null hypothesis, step 5: present your findings, other interesting articles, frequently asked questions about hypothesis testing.

After developing your initial research hypothesis (the prediction that you want to investigate), it is important to restate it as a null (H o ) and alternate (H a ) hypothesis so that you can test it mathematically.

The alternate hypothesis is usually your initial hypothesis that predicts a relationship between variables. The null hypothesis is a prediction of no relationship between the variables you are interested in.

• H 0 : Men are, on average, not taller than women. H a : Men are, on average, taller than women.

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For a statistical test to be valid , it is important to perform sampling and collect data in a way that is designed to test your hypothesis. If your data are not representative, then you cannot make statistical inferences about the population you are interested in.

There are a variety of statistical tests available, but they are all based on the comparison of within-group variance (how spread out the data is within a category) versus between-group variance (how different the categories are from one another).

If the between-group variance is large enough that there is little or no overlap between groups, then your statistical test will reflect that by showing a low p -value . This means it is unlikely that the differences between these groups came about by chance.

Alternatively, if there is high within-group variance and low between-group variance, then your statistical test will reflect that with a high p -value. This means it is likely that any difference you measure between groups is due to chance.

Your choice of statistical test will be based on the type of variables and the level of measurement of your collected data .

• an estimate of the difference in average height between the two groups.
• a p -value showing how likely you are to see this difference if the null hypothesis of no difference is true.

Based on the outcome of your statistical test, you will have to decide whether to reject or fail to reject your null hypothesis.

In most cases you will use the p -value generated by your statistical test to guide your decision. And in most cases, your predetermined level of significance for rejecting the null hypothesis will be 0.05 – that is, when there is a less than 5% chance that you would see these results if the null hypothesis were true.

In some cases, researchers choose a more conservative level of significance, such as 0.01 (1%). This minimizes the risk of incorrectly rejecting the null hypothesis ( Type I error ).

The results of hypothesis testing will be presented in the results and discussion sections of your research paper , dissertation or thesis .

In the results section you should give a brief summary of the data and a summary of the results of your statistical test (for example, the estimated difference between group means and associated p -value). In the discussion , you can discuss whether your initial hypothesis was supported by your results or not.

In the formal language of hypothesis testing, we talk about rejecting or failing to reject the null hypothesis. You will probably be asked to do this in your statistics assignments.

However, when presenting research results in academic papers we rarely talk this way. Instead, we go back to our alternate hypothesis (in this case, the hypothesis that men are on average taller than women) and state whether the result of our test did or did not support the alternate hypothesis.

If your null hypothesis was rejected, this result is interpreted as “supported the alternate hypothesis.”

These are superficial differences; you can see that they mean the same thing.

You might notice that we don’t say that we reject or fail to reject the alternate hypothesis . This is because hypothesis testing is not designed to prove or disprove anything. It is only designed to test whether a pattern we measure could have arisen spuriously, or by chance.

If we reject the null hypothesis based on our research (i.e., we find that it is unlikely that the pattern arose by chance), then we can say our test lends support to our hypothesis . But if the pattern does not pass our decision rule, meaning that it could have arisen by chance, then we say the test is inconsistent with our hypothesis .

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

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

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

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Explaining two-tailed tests

I am looking for various ways of explaining to my students (in an elementary statistics course) what is a two tailed test, and how its P value is calculated.

How do you explain to your students the two- vs one- tailed test?

• hypothesis-testing

2 Answers 2

This is a great question and I'm looking forward to everyones version of explaining the p-value and the two-tailed v.s. one-tailed test. I've been teaching fellow orthopaedic surgeons statistics and therefore I tried to keep it as basic as possible since most of them haven't done any advanced math for 10-30 years.

My way of explaining calculating p-values & the tails

I start with a explaining that if we believe that we have a fair coin we know it should end up tails 50 % of the flips on average ($=H_0$). Now if you wonder what the probability of getting only 2 tails out of 10 flips with this fair coin you can calculate that probability as I've done in the bar graph. From the graph you can see that the probability of getting 8 out of 10 flips with a fair coin is about about $\approx 4.4\%$.

Since we would question the fairness of the coin if we got 9 or 10 tails we have to include these possibilities, the tail of the test. By adding the values we get that the probability now is a little more than $\approx 5.5\%$ of getting 2 tails or less.

Now if we would get only 2 heads, ie 8 heads (the other tail), we would probably be just as willing to question the fairness of the coin. This means that you end up with a probability of $5.4...\%+5.4...\% \approx 10.9\%$ for a two-tailed test .

Since we in medicine usually are interested in studying failures we need to include the opposite side of the probability even if our intent is to do good and to introduce a beneficial treatment.

Reflections slightly out of topic

This simple example also shows how dependent we are on the null hypothesis to calculate the p-value. I also like to point out the resemblance between the binomial curve and the bell curve. When changing into 200 flips you get a natural way of explaining why the probability of getting exactly 100 flips starts to lack relevance. The defining intervals of interest is a natural transition to probability density/mass function functions and their cumulative counterparts.

In my class I recommend them the Khan academy statistics videos and I also use some of his explanations for certain concepts. They also get to flip coins where we look into the randomness of the coin flipping - the thing that I try to show is that randomness is more random than what we usually believe inspired by this Radiolab episode .

I usually have one graph/slide, the R-code that I used to create the graph:

• $\begingroup$ Great answer Max - and thank you for recognizing the non-triviality of my question :) $\endgroup$ –  Tal Galili Commented Dec 1, 2011 at 12:31
• $\begingroup$ +1 nice answer, very thorough. Forgive me, but I'm going to nitpick on two things. 1) the p-value is understood as the probability of data being as extreme or more extreme as yours under the null, thus your answer is right. However, when using discrete data like your coin flips, this is inappropriately conservative. It's best to use what's called the "mid p-value", i.e. 1/2 the probability of data as extreme as yours + the probability of data being more extreme. An easy discussion of these issues can be found in Agresti (2007) 2.6.3. (cont.) $\endgroup$ –  gung - Reinstate Monica Commented Dec 2, 2011 at 5:48
• $\begingroup$ 2) You state that randomness is more random than we believe. I can guess what you might mean by that (I haven't had a chance to listen to the Radiolab episode you link, but I will). Curiously enough, I've always told students that randomness is less random than you believe. I'm referring here to the perception of streaks (e.g., in gambling). People believe that random events should alternate much more than random events actually do, and as a result believe they see streaks. See Falk (1997) Making sense of randomness Psych Rev 104,2. Again, you're not wrong--just food for thought. $\endgroup$ –  gung - Reinstate Monica Commented Dec 2, 2011 at 6:00
• $\begingroup$ Thank you @gung for your input. I've actually not heard of the mid-pvalue - it makes sense though. I'm not sure about if it's something I would mention when teaching basic statistics since it may give a feeling of loosing the hands-on feeling that I try to give. Concerning randomness we mean exactly the same - when seeing a truly random number we are fooled to think there's a pattern to it. I think I heard on the Freakonomics podcast folly of prediction that... $\endgroup$ –  Max Gordon Commented Dec 2, 2011 at 16:43
• $\begingroup$ ... the human mind has over the years learned that failing to detect a predator is costlier than thinking it's probably nothing. I like that analogy and I try to tell my colleagues that one of the primary reasons for using statistics is to help us with this defect that we're all born with. $\endgroup$ –  Max Gordon Commented Dec 2, 2011 at 16:47

Suppose that you want to test the hypothesis that the average height of men is "5 ft 7 inches". You select a random sample of men, measure their heights and calculate the sample mean. Your hypothesis then is:

$H_0: \mu = 5\ \text{ft} \ 7 \ \text{inches}$

$H_A: \mu \ne 5\ \text{ft} \ 7 \ \text{inches}$

In the above situation you do a two-tailed test as you would reject your null if the sample average is either too low or too high.

In this case, the p-value represents the probability of realizing a sample mean that is at least as extreme as the one we actually obtained assuming that the null is in fact true. Thus, if observe the sample mean to be "5 ft 8 inches" then the p-value will represent the probability that we will observe heights greater than "5 ft 8 inches" or heights less than "5 ft 6 inches" provided the null is true.

If on the other hand your alternative was framed like so:

$H_A: \mu > 5\ \text{ft} \ 7 \ \text{inches}$

In the above situation you would a one-tailed test on the right side. The reason is that you would prefer to reject the null in favor of the alternative only if the sample mean is extremely high.

The interpretation of the p-value stays the same with the slight nuance that we are now talking about the probability of realizing a sample mean that is greater than the one we actually obtained. Thus, if observe the sample mean to be "5 ft 8 inches" then the p-value will represent the probability that we will observe heights greater than "5 ft 8 inches" provided the null is true.

• 2 $\begingroup$ Formerly, for your second $H_A$ the null should read $H_0:\, \mu\le 5\ \text{ft}\ 7\ \text{inches}$, not $H_0:\, \mu = 5\ \text{ft}\ 7\ \text{inches}$. See one of @whuber's comments to this question, Do null and alternative hypotheses have be to exhaustive or not? . $\endgroup$ –  chl Commented Dec 1, 2011 at 10:23
• 2 $\begingroup$ @chl I agree. However, for a person who is just being introduced to statistical ideas, re-writing the null for a one-tailed test may be a distraction when the focus is on how and why things change with respect to interpretation of the p-value. $\endgroup$ –  varty Commented Dec 1, 2011 at 14:55
• 1 $\begingroup$ Fair enough. That's worth mentioning though, even for teaching purpose. $\endgroup$ –  chl Commented Dec 2, 2011 at 13:19

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Two Sample t-test: Definition, Formula, and Example

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

This tutorial explains the following:

• The motivation for performing a two sample t-test.
• The formula to perform a two sample t-test.
• The assumptions that should be met to perform a two sample t-test.
• An example of how to perform a two sample t-test.

Two Sample t-test: Motivation

Suppose we want to know whether or not the mean weight between two different species of turtles is equal. Since there are thousands of turtles in each population, it would be too time-consuming and costly to go around and weigh each individual turtle.

Instead, we might take a simple random sample of 15 turtles from each population and use the mean weight in each sample to determine if the mean weight is equal between the two populations:

However, it’s virtually guaranteed that the mean weight between the two samples will be at least a little different. The question is whether or not this difference is statistically significant . Fortunately, a two sample t-test allows us to answer this question.

Two Sample t-test: Formula

A two-sample t-test always uses the following null hypothesis:

• H 0 : μ 1  = μ 2 (the two population means are equal)

The alternative hypothesis can be either two-tailed, left-tailed, or right-tailed:

• H 1 (two-tailed): μ 1  ≠ μ 2 (the two population means are not equal)
• H 1 (left-tailed): μ 1  < μ 2  (population 1 mean is less than population 2 mean)
• H 1 (right-tailed):  μ 1 > μ 2  (population 1 mean is greater than population 2 mean)

We use the following formula to calculate the test statistic t:

Test statistic:  ( x 1  –  x 2 )  /  s p (√ 1/n 1  + 1/n 2 )

where  x 1  and  x 2 are the sample means, n 1 and n 2  are the sample sizes, and where s p is calculated as:

s p = √  (n 1 -1)s 1 2  +  (n 2 -1)s 2 2  /  (n 1 +n 2 -2)

where s 1 2  and s 2 2  are the sample variances.

If the p-value that corresponds to the test statistic t with (n 1 +n 2 -1) degrees of freedom is less than your chosen significance level (common choices are 0.10, 0.05, and 0.01) then you can reject the null hypothesis.

Two Sample t-test: Assumptions

For the results of a two sample t-test to be valid, the following assumptions should be met:

• The observations in one sample should be independent of the observations in the other sample.
• The data should be approximately normally distributed.
• The two samples should have approximately the same variance. If this assumption is not met, you should instead perform Welch’s t-test .
• The data in both samples was obtained using a random sampling method .

Two Sample t-test : Example

Suppose we want to know whether or not the mean weight between two different species of turtles is equal. To test this, will perform a two sample t-test at significance level α = 0.05 using the following steps:

Step 1: Gather the sample data.

Suppose we collect a random sample of turtles 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

Step 2: Define the hypotheses.

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)

Step 3: Calculate the test statistic  t .

First, we will calculate the pooled standard deviation s p :

s p = √  (n 1 -1)s 1 2  +  (n 2 -1)s 2 2  /  (n 1 +n 2 -2)  = √  (40-1)18.5 2  +  (38-1)16.7 2  /  (40+38-2)  = 17.647

Next, we will calculate the test statistic  t :

t = ( x 1  –  x 2 )  /  s p (√ 1/n 1  + 1/n 2 ) =  (300-305) / 17.647(√ 1/40 + 1/38 ) =  -1.2508

Step 4: Calculate the p-value of the test statistic  t .

According to the T Score to P Value Calculator , the p-value associated with t = -1.2508 and degrees of freedom = n 1 +n 2 -2 = 40+38-2 = 76 is  0.21484 .

Step 5: Draw a conclusion.

Since this p-value is not less than our significance level α = 0.05, 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.

Note:  You can also perform this entire two sample t-test by simply using the Two Sample t-test Calculator .

Additional Resources

The following tutorials explain how to perform a two-sample t-test using different statistical programs:

How to Perform a Two Sample t-test in Excel How to Perform a Two Sample t-test in SPSS How to Perform a Two Sample t-test in Stata How to Perform a Two Sample t-test in R How to Perform a Two Sample t-test in Python How to Perform a Two Sample t-test on a TI-84 Calculator

Featured Posts

Hey there. My name is Zach Bobbitt. I have a Masters of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike.  My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.

2 Replies to “Two Sample t-test: Definition, Formula, and Example”

I like the detailed information and simplified in the way I can understand and relate easily. Thank you

It seems a couple of parenthesis is missed at the pooled standard deviation formula. Under square root you have (n1-1)s12 + (n2-1)s22 / (n1+n2-2) but it should be [(n1-1)s12 + (n2-1)s22] / (n1+n2-2) I used square bracket

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Using R for Hypothesis Testing

This tutorial demonstrates step-by step how to use R and Jupyter Notebook to conduct a two-tailed or two sided hypothesis test. An eight step approach that begins with the formulation of Null and Alternative Hypothesis and ends by stating what the results of the test mean in plain English.

What is hypothesis testing?

Hypothesis testing is one of the cornerstones of inferential statistics. It is generally used to test whether some phenomenon observed in a sample is likely the result of random change or is "real", that is statistically significant.

What are the steps of hypothesis testing?

Hypothesis testing is statistics is usually challenging at first, but a step-by-step approach can help keep everything straight. We always start with a hypothesis we would like to prove, i.e. Human activity is contributing to global warming. This is what we need to prove, and is called the alternative hypothesis . Its opposite, humans are not contributing to global warming, is the null hypothesis . To prove the alternative hypothesis we would need to collect data and if the collected sample data deviates enough from what we can attribute to normal variation we will have proved the alternative hypothesis is much more likely than not. The global warming question is devilishly hard to test, and as you may have heard the jury is still out. But it does serve as a nice vignette.

When presented with the sorts of questions found in introductory statistics, you can use the following steps:

• State the null and alternative hypotheses - the null is the status quo, and requires sufficient evidence to disprove. It is often easier to start wit hthe alternative hypothesis and then present its opposite as the null
• Choose the level of significance at which you would reject the Null - so at what point would you be reasonably sure that the difference between the sample mean and status quo mean is not the result of random chance? The level of significance is usually 0.05, but not always
• Choose a sample size to test your hypothesis. The textbook problem usually does this for you, but you will need to use the sample size to claculate your test statistic.
• Determine the appropriate statistical technique to use, i.e. Z-test or t-test
• Determine the critical value(s) that separate the rejection region from the non-rejection region. You will always need to either reject or fail to reject he null hyopthesis after you calculate your test statistic. This is the line or lines in thesand that allow you to do that.
• Collect data and calculate the test statistic. Again, this stepo is usually done for you in textbook problems.
• Compare the test statistic to the critical value(s). Is the test statistic beyond your line in the sand?
• State the statistical decision - if the test statistic is beyond a boundary established by a critical value, then you reject the null hypothesis, oitherwise you do not reject the null.
• State your decision in terms of the original question, i.e the results either do or do nbot support the alternative hypothese.

The video tutorial presented here is an example of a so called two-tailed hypothesis test, and walks through a problem using these steps.

You can download the Jupyter Notebook used in the video here

Statistics Tutorial

Descriptive statistics, inferential statistics, stat reference, statistics - hypothesis testing a proportion (two tailed).

A population proportion is the share of a population that belongs to a particular category .

Hypothesis tests are used to check a claim about the size of that population proportion.

Hypothesis Testing a Proportion

The following steps are used for a hypothesis test:

• Check the conditions
• Define the claims
• Decide the significance level
• Calculate the test statistic

For example:

• Population : Nobel Prize winners
• Category : Women

And we want to check the claim:

"The share of Nobel Prize winners that are women is not 50%"

By taking a sample of 100 randomly selected Nobel Prize winners we could find that:

10 out of 100 Nobel Prize winners in the sample were women

The sample proportion is then: \(\displaystyle \frac{10}{100} = 0.1\), or 10%.

From this sample data we check the claim with the steps below.

1. Checking the Conditions

The conditions for calculating a confidence interval for a proportion are:

• The sample is randomly selected
• Being in the category
• Not being in the category
• 5 members in the category
• 5 members not in the category

In our example, we randomly selected 10 people that were women.

The rest were not women, so there are 90 in the other category.

The conditions are fulfilled in this case.

Note: It is possible to do a hypothesis test without having 5 of each category. But special adjustments need to be made.

2. Defining the Claims

We need to define a null hypothesis (\(H_{0}\)) and an alternative hypothesis (\(H_{1}\)) based on the claim we are checking.

The claim was:

In this case, the parameter is the proportion of Nobel Prize winners that are women (\(p\)).

The null and alternative hypothesis are then:

Null hypothesis : 50% of Nobel Prize winners were women.

Alternative hypothesis : The share of Nobel Prize winners that are women is not 50%

Which can be expressed with symbols as:

\(H_{0}\): \(p = 0.50 \)

\(H_{1}\): \(p \neq 0.50 \)

This is a ' two-tailed ' test, because the alternative hypothesis claims that the proportion is different (larger or smaller) than in the null hypothesis.

If the data supports the alternative hypothesis, we reject the null hypothesis and accept the alternative hypothesis.

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3. Deciding the Significance Level

The significance level (\(\alpha\)) is the uncertainty we accept when rejecting the null hypothesis in a hypothesis test.

The significance level is a percentage probability of accidentally making the wrong conclusion.

Typical significance levels are:

• \(\alpha = 0.1\) (10%)
• \(\alpha = 0.05\) (5%)
• \(\alpha = 0.01\) (1%)

A lower significance level means that the evidence in the data needs to be stronger to reject the null hypothesis.

There is no "correct" significance level - it only states the uncertainty of the conclusion.

Note: A 5% significance level means that when we reject a null hypothesis:

We expect to reject a true null hypothesis 5 out of 100 times.

4. Calculating the Test Statistic

The test statistic is used to decide the outcome of the hypothesis test.

The test statistic is a standardized value calculated from the sample.

The formula for the test statistic (TS) of a population proportion is:

\(\displaystyle \frac{\hat{p} - p}{\sqrt{p(1-p)}} \cdot \sqrt{n} \)

\(\hat{p}-p\) is the difference between the sample proportion (\(\hat{p}\)) and the claimed population proportion (\(p\)).

\(n\) is the sample size.

In our example:

The claimed (\(H_{0}\)) population proportion (\(p\)) was \( 0.50 \)

The sample size (\(n\)) was \(100\)

So the test statistic (TS) is then:

\(\displaystyle \frac{0.1-0.5}{\sqrt{0.5(1-0.5)}} \cdot \sqrt{100} = \frac{-0.4}{\sqrt{0.5(0.5)}} \cdot \sqrt{100} = \frac{-0.4}{\sqrt{0.25}} \cdot \sqrt{100} = \frac{-0.4}{0.5} \cdot 10 = \underline{-8}\)

You can also calculate the test statistic using programming language functions:

With Python use the scipy and math libraries to calculate the test statistic for a proportion.

With R use the built-in math functions to calculate the test statistic for a proportion.

5. Concluding

There are two main approaches for making the conclusion of a hypothesis test:

• The critical value approach compares the test statistic with the critical value of the significance level.
• The P-value approach compares the P-value of the test statistic and with the significance level.

Note: The two approaches are only different in how they present the conclusion.

The Critical Value Approach

For the critical value approach we need to find the critical value (CV) of the significance level (\(\alpha\)).

For a population proportion test, the critical value (CV) is a Z-value from a standard normal distribution .

This critical Z-value (CV) defines the rejection region for the test.

The rejection region is an area of probability in the tails of the standard normal distribution.

Because the claim is that the population proportion is different from 50%, the rejection region is split into both the left and right tail:

Choosing a significance level (\(\alpha\)) of 0.01, or 1%, we can find the critical Z-value from a Z-table , or with a programming language function:

Note: Because this is a two-tailed test the tail area (\(\alpha\)) needs to be split in half (divided by 2).

With Python use the Scipy Stats library norm.ppf() function find the Z-value for an \(\alpha\)/2 = 0.005 in the left tail.

With R use the built-in qnorm() function to find the Z-value for an \(\alpha\) = 0.005 in the left tail.

Using either method we can find that the critical Z-value in the left tail is \(\approx \underline{-2.5758}\)

Since a normal distribution i symmetric, we know that the critical Z-value in the right tail will be the same number, only positive: \(\underline{2.5758}\)

For a two-tailed test we need to check if the test statistic (TS) is smaller than the negative critical value (-CV), or bigger than the positive critical value (CV).

If the test statistic is smaller than the negative critical value, the test statistic is in the rejection region .

If the test statistic is bigger than the positive critical value, the test statistic is in the rejection region .

When the test statistic is in the rejection region, we reject the null hypothesis (\(H_{0}\)).

Here, the test statistic (TS) was \(\approx \underline{-8}\) and the critical value was \(\approx \underline{-2.5758}\)

Here is an illustration of this test in a graph:

Since the test statistic was smaller than the negative critical value we reject the null hypothesis.

This means that the sample data supports the alternative hypothesis.

And we can summarize the conclusion stating:

The sample data supports the claim that "The share of Nobel Prize winners that are women is not 50%" at a 1% significance level .

The P-Value Approach

For the P-value approach we need to find the P-value of the test statistic (TS).

If the P-value is smaller than the significance level (\(\alpha\)), we reject the null hypothesis (\(H_{0}\)).

The test statistic was found to be \( \approx \underline{-8} \)

For a population proportion test, the test statistic is a Z-Value from a standard normal distribution .

Because this is a two-tailed test, we need to find the P-value of a Z-value smaller than -8 and multiply it by 2 .

We can find the P-value using a Z-table , or with a programming language function:

With Python use the Scipy Stats library norm.cdf() function find the P-value of a Z-value smaller than -8 for a two tailed test:

With R use the built-in pnorm() function find the P-value of a Z-value smaller than -8 for a two tailed test:

Using either method we can find that the P-value is \(\approx \underline{1.25 \cdot 10^{-15}}\) or \(0.00000000000000125\)

This tells us that the significance level (\(\alpha\)) would need to be bigger than 0.000000000000125%, to reject the null hypothesis.

This P-value is smaller than any of the common significance levels (10%, 5%, 1%).

So the null hypothesis is rejected at all of these significance levels.

The sample data supports the claim that "The share of Nobel Prize winners that are women is not 50%" at a 10%, 5%, and 1% significance level .

Calculating a P-Value for a Hypothesis Test with Programming

Many programming languages can calculate the P-value to decide outcome of a hypothesis test.

Using software and programming to calculate statistics is more common for bigger sets of data, as calculating manually becomes difficult.

The P-value calculated here will tell us the lowest possible significance level where the null-hypothesis can be rejected.

With Python use the scipy and math libraries to calculate the P-value for a two-tailed tailed hypothesis test for a proportion.

Here, the sample size is 100, the occurrences are 10, and the test is for a proportion different from than 0.50.

With R use the built-in prop.test() function find the P-value for a left tailed hypothesis test for a proportion.

Here, the sample size is 100, the occurrences are 10, and the test is for a proportion different from 0.50.

Note: The conf.level in the R code is the reverse of the significance level.

Here, the significance level is 0.01, or 1%, so the conf.level is 1-0.01 = 0.99, or 99%.

Left-Tailed and Two-Tailed Tests

This was an example of a two tailed test, where the alternative hypothesis claimed that parameter is different from the null hypothesis claim.

You can check out an equivalent step-by-step guide for other types here:

• Right-Tailed Test
• Left-Tailed Test

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Two Tailed Hypothesis

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In the vast realm of scientific inquiry, the two-tailed hypothesis holds a special place, serving as a compass for researchers exploring possibilities in two opposing directions. Instead of predicting a specific direction of the relationship between variables, it remains open to outcomes on both ends of the spectrum. Understanding how to craft such a hypothesis, enriched with insights and nuances, can elevate the robustness of one’s research. Delve into its world, discover thesis statement examples, learn the art of its formulation, and grasp tips to master its intricacies.

What is Two Tailed Hypothesis? – Definition

A two-tailed hypothesis, also known as a non-directional hypothesis , is a type of hypothesis used in statistical testing that predicts a relationship between variables without specifying the direction of the relationship. In other words, it tests for the possibility of the relationship in both directions. This approach is used when a researcher believes there might be a difference due to the experiment but doesn’t have enough preliminary evidence or basis to predict a specific direction of that difference.

What is an example of a Two Tailed hypothesis statement?

Let’s consider a study on the impact of a new teaching method on student performance:

Hypothesis Statement : The new teaching method will have an effect on student performance.

Notice that the hypothesis doesn’t specify whether the effect will be positive or negative (i.e., whether student performance will improve or decline). It’s open to both possibilities, making it a two-tailed hypothesis.

Two Tailed Hypothesis Statement Examples

The two-tailed hypothesis, an essential tool in research, doesn’t predict a specific directional outcome between variables. Instead, it posits that an effect exists, without specifying its nature. This approach offers flexibility, as it remains open to both positive and negative outcomes. Below are various examples from diverse fields to shed light on this versatile research method. You may also be interested to browse through our other  one-tailed hypothesis .

• Sleep and Cognitive Ability : Sleep duration affects cognitive performance in adults.
• Dietary Fiber and Digestion : Consumption of dietary fiber influences digestion rates.
• Exercise and Stress Levels : Engaging in physical activity impacts stress levels.
• Vitamin C and Immunity : Intake of Vitamin C has an effect on immunity strength.
• Noise Levels and Concentration : Ambient noise levels influence individual concentration ability.
• Artificial Sweeteners and Appetite : Consumption of artificial sweeteners affects appetite.
• UV Light and Skin Health : Exposure to UV light influences skin health.
• Coffee Intake and Sleep Quality : Consuming coffee has an effect on sleep quality.
• Air Pollution and Respiratory Issues : Levels of air pollution impact respiratory health.
• Meditation and Blood Pressure : Practicing meditation affects blood pressure readings.
• Pet Ownership and Loneliness : Having a pet influences feelings of loneliness.
• Green Spaces and Mental Wellbeing : Exposure to green spaces impacts mental health.
• Music Tempo and Heart Rate : Listening to music of varying tempos affects heart rate.
• Chocolate Consumption and Mood : Eating chocolate has an effect on mood.
• Social Media Usage and Self-Esteem : The frequency of social media usage influences self-esteem.
• E-reading and Eye Strain : Using e-readers affects eye strain levels.
• Vegan Diets and Energy Levels : Following a vegan diet influences daily energy levels.
• Carbonated Drinks and Tooth Decay : Consumption of carbonated drinks has an effect on tooth decay rates.
• Distance Learning and Student Engagement : Engaging in distance learning impacts student involvement.
• Organic Foods and Health Perceptions : Consuming organic foods influences perceptions of health.
• Urban Living and Stress Levels : Living in urban environments affects stress levels.
• Plant-Based Diets and Cholesterol : Adopting a plant-based diet impacts cholesterol levels.
• Virtual Reality Training and Skill Acquisition : Using virtual reality for training influences the rate of skill acquisition.
• Video Game Play and Hand-Eye Coordination : Playing video games has an effect on hand-eye coordination.
• Aromatherapy and Sleep Quality : Using aromatherapy impacts the quality of sleep.
• Bilingualism and Cognitive Flexibility : Being bilingual affects cognitive flexibility.
• Microplastics and Marine Health : The presence of microplastics in oceans influences marine organism health.
• Yoga Practice and Joint Health : Engaging in yoga has an effect on joint health.
• Processed Foods and Metabolism : Consuming processed foods impacts metabolic rates.
• Home Schooling and Social Skills : Being homeschooled influences the development of social skills.
• Smartphone Usage and Attention Span : Regular smartphone use affects attention spans.
• E-commerce and Consumer Trust : Engaging with e-commerce platforms influences levels of consumer trust.
• Work-from-Home and Productivity : The practice of working from home has an effect on productivity levels.
• Classical Music and Plant Growth : Exposing plants to classical music impacts their growth rate.
• Public Transport and Community Engagement : Using public transport influences community engagement levels.
• Digital Note-taking and Memory Retention : Taking notes digitally affects memory retention.
• Acoustic Music and Relaxation : Listening to acoustic music impacts feelings of relaxation.
• GMO Foods and Public Perception : Consuming GMO foods influences public perception of food safety.
• LED Lights and Eye Comfort : Using LED lights affects visual comfort.
• Fast Fashion and Consumer Satisfaction : Engaging with fast fashion brands influences consumer satisfaction levels.
• Diverse Teams and Innovation : Working in diverse teams impacts the level of innovation.
• Local Produce and Nutritional Value : Consuming local produce affects its nutritional value.
• Podcasts and Language Acquisition : Listening to podcasts influences the speed of language acquisition.
• Augmented Reality and Learning Efficiency : Using augmented reality in education has an effect on learning efficiency.
• Museums and Historical Interest : Visiting museums impacts interest in history.
• E-books vs. Physical Books and Reading Retention : The type of book, whether e-book or physical, affects memory retention from reading.
• Biophilic Design and Worker Well-being : Implementing biophilic designs in office spaces influences worker well-being.
• Recycled Products and Consumer Preference : Using recycled materials in products impacts consumer preferences.
• Interactive Learning and Critical Thinking : Engaging in interactive learning environments affects the development of critical thinking skills.
• High-Intensity Training and Muscle Growth : Participating in high-intensity training has an effect on muscle growth rate.
• Pet Therapy and Anxiety Levels : Engaging with therapy animals influences anxiety levels.
• 3D Printing and Manufacturing Efficiency : Implementing 3D printing in manufacturing affects production efficiency.
• Electric Cars and Public Adoption Rates : Introducing more electric cars impacts the rate of public adoption.
• Ancient Architectural Study and Modern Design Inspiration : Studying ancient architecture influences modern design inspirations.
• Natural Lighting and Productivity : The amount of natural lighting in a workspace affects worker productivity.
• Streaming Platforms and Traditional TV Viewing : The rise of streaming platforms has an effect on traditional TV viewing habits.
• Handwritten Notes and Conceptual Understanding : Taking notes by hand influences the depth of conceptual understanding.
• Urban Farming and Community Engagement : Implementing urban farming practices impacts levels of community engagement.
• Influencer Marketing and Brand Loyalty : Collaborating with influencers affects brand loyalty among consumers.
• Online Workshops and Skill Enhancement : Participating in online workshops influences skill enhancement.
• Virtual Reality and Empathy Development : Using virtual reality experiences influences the development of empathy.
• Gardening and Mental Well-being : Engaging in gardening activities affects overall mental well-being.
• Drones and Wildlife Observation : The use of drones impacts the accuracy of wildlife observations.
• Artificial Intelligence and Job Markets : The introduction of artificial intelligence in industries has an effect on job availability.
• Online Reviews and Purchase Decisions : Reading online reviews influences purchase decisions for consumers.
• Blockchain Technology and Financial Security : Implementing blockchain technology affects financial transaction security.
• Minimalism and Life Satisfaction : Adopting a minimalist lifestyle influences levels of life satisfaction.
• Microlearning and Long-term Retention : Engaging in microlearning practices impacts long-term information retention.
• Virtual Teams and Communication Efficiency : Operating in virtual teams has an effect on the efficiency of communication.
• Plant Music and Growth Rates : Exposing plants to specific music frequencies influences their growth rates.
• Green Building Practices and Energy Consumption : Implementing green building designs affects overall energy consumption.
• Fermented Foods and Gut Health : Consuming fermented foods impacts gut health.
• Digital Art Platforms and Creative Expression : Using digital art platforms influences levels of creative expression.
• Aquatic Therapy and Physical Rehabilitation : Engaging in aquatic therapy has an effect on the rate of physical rehabilitation.
• Solar Energy and Utility Bills : Adopting solar energy solutions influences monthly utility bills.
• Immersive Theatre and Audience Engagement : Experiencing immersive theatre performances affects audience engagement levels.
• Podcast Popularity and Radio Listening Habits : The rise in podcast popularity impacts traditional radio listening habits.
• Vertical Farming and Crop Yield : Implementing vertical farming techniques has an effect on crop yields.
• DIY Culture and Craftsmanship Appreciation : The rise of DIY culture influences public appreciation for craftsmanship.
• Crowdsourcing and Solution Innovation : Utilizing crowdsourcing methods affects the innovativeness of solutions derived.
• Urban Beekeeping and Local Biodiversity : Introducing urban beekeeping practices impacts local biodiversity levels.
• Digital Nomad Lifestyle and Work-Life Balance : Adopting a digital nomad lifestyle affects perceptions of work-life balance.
• Virtual Tours and Tourism Interest : Offering virtual tours of destinations influences interest in real-life visits.
• Neurofeedback Training and Cognitive Abilities : Engaging in neurofeedback training has an effect on various cognitive abilities.
• Sensory Gardens and Stress Reduction : Visiting sensory gardens impacts levels of stress reduction.
• Subscription Box Services and Consumer Spending : The popularity of subscription box services influences overall consumer spending patterns.
• Makerspaces and Community Collaboration : Introducing makerspaces in communities affects collaboration levels among members.
• Remote Work and Company Loyalty : Adopting long-term remote work policies impacts employee loyalty towards the company.
• Upcycling and Environmental Awareness : Engaging in upcycling activities influences levels of environmental awareness.
• Mixed Reality in Education and Engagement : Implementing mixed reality tools in education affects student engagement.
• Microtransactions in Gaming and Player Commitment : The presence of microtransactions in video games impacts player commitment and longevity.
• Floating Architecture and Sustainable Living : Adopting floating architectural solutions influences perceptions of sustainable living.
• Edible Packaging and Waste Reduction : Introducing edible packaging in markets has an effect on overall waste reduction.
• Space Tourism and Interest in Astronomy : The advent of space tourism influences the general public’s interest in astronomy.
• Urban Green Roofs and Air Quality : Implementing green roofs in urban settings impacts the local air quality.
• Smart Mirrors and Fitness Consistency : Using smart mirrors for workouts affects consistency in fitness routines.
• Open Source Software and Technological Innovation : Promoting open-source software has an effect on the rate of technological innovation.
• Microgreens and Nutrient Intake : Consuming microgreens influences nutrient intake.
• Aquaponics and Sustainable Farming : Implementing aquaponic systems impacts perceptions of sustainable farming.
• Esports Popularity and Physical Sport Engagement : The rise of esports affects engagement in traditional physical sports.

Two Tailed Hypothesis Statement Examples in Research

In academic research, a two-tailed hypothesis is versatile, not pointing to a specific direction of effect but remaining open to outcomes on both ends of the spectrum. Such hypothesis aim to determine if a particular variable affects another, without specifying how. Here are examples tailored to research scenarios.

• Interdisciplinary Collaboration and Innovation : Engaging in interdisciplinary collaborations impacts the degree of innovation in research findings.
• Open Access Journals and Citation Rates : Publishing in open-access journals influences the citation rates of the papers.
• Research Grants and Publication Quality : Receiving larger research grants affects the quality of resulting publications.
• Laboratory Environment and Data Accuracy : The physical conditions of a research laboratory impact the accuracy of experimental data.
• Peer Review Process and Research Integrity : The stringency of the peer review process influences the overall integrity of published research.
• Researcher Mobility and Knowledge Transfer : The mobility of researchers between institutions affects the rate of knowledge transfer.
• Interdisciplinary Conferences and Networking Opportunities : Attending interdisciplinary conferences impacts the depth and breadth of networking opportunities.
• Qualitative Methods and Research Depth : Incorporating qualitative methods in research affects the depth of findings.
• Data Visualization Tools and Research Comprehension : Utilizing advanced data visualization tools influences the comprehension of complex research data.
• Collaborative Tools and Research Efficiency : The adoption of modern collaborative tools impacts research efficiency and productivity.

Two Tailed Testing Hypothesis Statement Examples

In hypothesis testing , a two-tailed test examines the possibility of a relationship in both directions. Unlike one-tailed tests, it doesn’t anticipate a specific direction of the relationship. The following are examples that encapsulate this approach within varied testing scenarios.

• Load Testing and Website Speed : Conducting load testing on a website influences its loading speed.
• A/B Testing and Conversion Rates : Implementing A/B testing affects the conversion rates of a webpage.
• Drug Efficacy Testing and Patient Recovery : Testing a new drug’s efficacy impacts patient recovery rates.
• Usability Testing and User Engagement : Conducting usability testing on an app influences user engagement metrics.
• Genetic Testing and Disease Prediction : Utilizing genetic testing affects the accuracy of disease prediction.
• Water Quality Testing and Contaminant Levels : Performing water quality tests influences our understanding of contaminant levels.
• Battery Life Testing and Device Longevity : Conducting battery life tests impacts claims about device longevity.
• Product Safety Testing and Recall Rates : Implementing rigorous product safety tests affects the rate of product recalls.
• Emissions Testing and Pollution Control : Undertaking emissions testing on vehicles influences pollution control measures.
• Material Strength Testing and Product Durability : Testing the strength of materials affects predictions about product durability.

How do you know if a hypothesis is two-tailed?

To determine if a hypothesis is two-tailed, you must look at the nature of the prediction. A two-tailed hypothesis is neutral concerning the direction of the predicted relationship or difference between groups. It simply predicts a difference or relationship without specifying whether it will be positive, negative, greater, or lesser. The hypothesis tests for effects in both directions.

What is one-tailed and two-tailed Hypothesis test with example?

In hypothesis testing, the choice between a one-tailed and a two-tailed test is determined by the nature of the research question.

One-tailed hypothesis: This tests for a specific direction of the effect. It predicts the direction of the relationship or difference between groups. For example, a one-tailed hypothesis might state: “The new drug will reduce symptoms more effectively than the standard treatment.”

Two-tailed hypothesis: This doesn’t specify the direction. It predicts that there will be a difference, but it doesn’t forecast whether the difference will be positive or negative. For example, a two-tailed hypothesis might state: “The new drug will have a different effect on symptoms compared to the standard treatment.”

What is a two-tailed hypothesis in psychology?

In psychology, a two-tailed hypothesis is frequently used when researchers are exploring new areas or relationships without a strong prior basis to predict the direction of findings. For instance, a psychologist might use a two-tailed hypothesis to explore whether a new therapeutic method has different outcomes than a traditional method, without predicting whether the outcomes will be better or worse.

What does a two-tailed alternative hypothesis look like?

A two-tailed alternative hypothesis is generally framed to show that a parameter is simply different from a certain value, without specifying the direction of the difference. Using mathematical notation, for a population mean (μ) and a proposed value (k), the two-tailed hypothesis would look like: H1: μ ≠ k.

How do you write a Two-Tailed hypothesis statement? – A Step by Step Guide

• Identify the Variables: Start by identifying the independent and dependent variables you want to study.
• Formulate a Relationship: Consider the potential relationship between these variables without setting a direction.
• Avoid Directional Language: Words like “increase”, “decrease”, “more than”, or “less than” should be avoided as they point to a one-tailed hypothesis.
• Keep it Simple: The statement should be clear, concise, and to the point.
• Use Neutral Language: For instance, words like “affects”, “influences”, or “has an impact on” can be used to indicate a relationship without specifying a direction.
• Finalize the Statement: Once the relationship is clear in your mind, form a coherent sentence that describes the relationship between your variables.

Tips for Writing Two Tailed Hypothesis

• Start Broad: Given that you’re not seeking a specific direction, it’s okay to start with a broad idea.
• Be Objective: Avoid letting any biases or expectations shape your hypothesis.
• Stay Informed: Familiarize yourself with existing research on the topic to ensure your hypothesis is novel and not inadvertently directional.
• Seek Feedback: Share your hypothesis with colleagues or mentors to ensure it’s indeed non-directional.
• Revisit and Refine: As with any research process, be open to revisiting and refining your hypothesis as you delve deeper into the literature or collect preliminary data.

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• Instructive
• Professional

10 Examples of Public speaking

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What are three examples of two-tailed hypothesis tests?

Table of Contents

A two-tailed hypothesis test is a statistical method used to determine if there is a significant difference between two groups or variables. It involves testing the null hypothesis that there is no difference between the two groups against the alternative hypothesis that there is a difference. Three examples of two-tailed hypothesis tests include: comparing the mean test scores of students who received a new teaching method versus those who did not, analyzing the effectiveness of two different medications on treating a certain illness, and examining the relationship between income and job satisfaction among employees in two different industries. In each of these examples, the two-tailed hypothesis test would help determine if there is a significant difference between the two groups being compared.

Two-Tailed Hypothesis Tests: 3 Example Problems

In statistics, we use to determine whether some claim about a is true or not.

Whenever we perform a hypothesis test, we always write a null hypothesis and an alternative hypothesis, which take the following forms:

H 0 (Null Hypothesis): Population parameter = ≤, ≥ some value

H A (Alternative Hypothesis): Population parameter <, >, ≠ some value

There are two types of hypothesis tests:

• One-tailed test : Alternative hypothesis contains either < or > sign
• Two-tailed test : Alternative hypothesis contains the ≠ sign

In a two-tailed test , the alternative hypothesis always contains the not equal ( ≠ ) sign.

This indicates that we’re testing whether or not some effect exists, regardless of whether it’s a positive or negative effect.

Check out the following example problems to gain a better understanding of two-tailed tests.

Example 1: Factory Widgets

Suppose it’s assumed that the average weight of a certain widget produced at a factory is 20 grams. However, one engineer believes that a new method produces widgets that weigh less than 20 grams.

To test this, he can perform a one-tailed hypothesis test with the following null and alternative hypotheses:

• H 0 (Null Hypothesis): μ = 20 grams
• H A (Alternative Hypothesis): μ ≠ 20 grams

This is an example of a two-tailed hypothesis test because the alternative hypothesis contains the not equal “≠” sign. The engineer believes that the new method will influence widget weight, but doesn’t specify whether it will cause average weight to increase or decrease.

To test this, he uses the new method to produce 20 widgets and obtains the following information :

• n = 20 widgets
• x = 19.8 grams
• s = 3.1 grams

Plugging these values into the , we obtain the following results:

• t-test statistic: -0.288525
• two-tailed p-value: 0.776

Since the p-value is not less than .05, the engineer fails to reject the null hypothesis.

He does not have sufficient evidence to say that the true mean weight of widgets produced by the new method is different than 20 grams.

Example 2: Plant Growth

Suppose a standard fertilizer has been shown to cause a species of plants to grow by an average of 10 inches. However, one botanist believes a new fertilizer causes this species of plants to grow by an average amount different than 10 inches.

To test this, she can perform a one-tailed hypothesis test with the following null and alternative hypotheses:

• H 0 (Null Hypothesis): μ = 10 inches
• H A (Alternative Hypothesis): μ ≠ 10 inches

This is an example of a two-tailed hypothesis test because the alternative hypothesis contains the not equal “≠” sign. The botanist believes that the new fertilizer will influence plant growth, but doesn’t specify whether it will cause average growth to increase or decrease.

To test this claim, she applies the new fertilizer to a simple random sample of 15 plants and obtains the following information :

• n = 15 plants
• x = 11.4 inches
• s = 2.5 inches
• t-test statistic: 2.1689
• two-tailed p-value: 0.0478

Since the p-value is less than .05, the botanist rejects the null hypothesis.

She has sufficient evidence to conclude that the new fertilizer causes an average growth that is different than 10 inches.

Example 3: Studying Method

A professor believes that a certain studying technique will influence the mean score that her students receive on a certain exam, but she’s unsure if it will increase or decrease the mean score, which is currently 82.

To test this, she lets each student use the studying technique for one month leading up to the exam and then administers the same exam to each of the students .

She then performs a hypothesis test using the following hypotheses:

• H 0 : μ = 82
• H A : μ ≠ 82

This is an example of a two-tailed hypothesis test because the alternative hypothesis contains the not equal “≠” sign. The professor believes that the studying technique will influence the mean exam score, but doesn’t specify whether it will cause the mean score to increase or decrease.

To test this claim, the professor has 25 students use the new studying method and then take the exam. He collects the following data on the exam scores for this sample of students :

• t-test statistic: 3.6586
• two-tailed p-value: 0.0012

Since the p-value is less than .05, the professor rejects the null hypothesis.

She has sufficient evidence to conclude that the new studying method produces exam scores with an average score that is different than 82.

Additional Resources

The following tutorials provide additional information about hypothesis testing:

Related terms:

• What are three example problems of one-tailed hypothesis tests?
• What are the key differences between left-tailed and right-tailed tests?
• What does Two-Tailed Hypothesis Tests: 3 Example Problems mean?
• What is the difference between a Left Tailed Test and a Right Tailed Test?
• One Tailed Hypothesis
• Two Tailed Hypothesis
• What are three examples of one sample t-tests?
• Is it appropriate to use Proc Univariate for conducting normality tests, following the SAS principle of “SAS: Use Proc Univariate for Normality Tests”?
• What is the null hypothesis for linear regression and how does it relate to the alternative hypothesis?
• ONE-TAILED TEST

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