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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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Formulation of hypothesis and testing

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Hypothesis Testing – Two Samples

Nov 22, 2012

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Hypothesis Testing – Two Samples. Chapters 12 & 13. Chapter Topics. Comparing Two Independent Samples Independent samples Z test for the difference in two means Pooled-variance t test for the difference in two means F Test for the Difference in Two Variances

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• independent populations
• different data sources
• research questions
• reject region

Presentation Transcript

Hypothesis Testing – Two Samples Chapters 12 & 13

Chapter Topics • Comparing Two Independent Samples • Independent samples Z test for the difference in two means • Pooled-variance t test for the difference in two means • F Test for the Difference in Two Variances • Comparing Two Related Samples • Paired-sample Z test for the mean difference • Paired-sample t test for the mean difference • Two-sample Z test for population proportions • Independent and Dependent Samples

Comparing Two Independent Samples • Different Data Sources • Unrelated • Independent • Sample selected from one population has no effect or bearing on the sample selected from the other population • Use the Difference between 2 Sample Means • Use Z Test or t Test for Independent Samples

Independent Sample Z Test (Variances Known) • Assumptions • Samples are randomly and independently drawn from normal distributions • Population variances are known • Test Statistic

t Test for Independent Samples (Variances Unknown) • Assumptions • Both populations are normally distributed • Samples are randomly and independently drawn • Population variances are unknown but assumed equal • If both populations are not normal, need large sample sizes

Developing the t Test for Independent Samples • Setting Up the Hypotheses H0: m 1 = m 2H1: m 1¹m 2 H0: m 1 -m 2 = 0 H1: m 1 - m 2 ¹ 0 Two Tail OR H0: m 1£ m 2H1: m 1 > m 2 H0: m 1 - m 2£ 0 H1: m 1 - m 2> 0 Right Tail OR H0: m 1³ m 2 H0: m 1 - m 2³ 0 H1: m 1 - m 2 < 0 Left Tail OR H1: m 1 < m 2

Developing the t Test for Independent Samples (continued) • Calculate the Pooled Sample Variance as an Estimate of the Common Population Variance

Developing the t Test for Independent Samples (continued) • Compute the Sample Statistic Hypothesized difference

You’re a financial analyst for Charles Schwab. Is there a difference in average dividend yield between stocks listed on the NYSE & NASDAQ? You collect the following data: NYSENASDAQNumber 21 25 Sample Mean 3.27 2.53 Sample Std Dev 1.30 1.16 Assuming equal variances, isthere a difference in average yield (a= 0.05)? t Test for Independent Samples: Example © 1984-1994 T/Maker Co.

Calculating the Test Statistic

H0: m1 - m2 = 0 i.e. (m1 = m2) H1: m1 - m2 ¹ 0 i.e. (m1¹ m2) a= 0.05 df = 21 + 25 - 2 = 44 Critical Value(s): Solution Test Statistic: Decision: Conclusion: Reject H Reject at a = 0.05. Reject H 0 0 .025 .025 There is evidence of a difference in means. t -2.0154 0 2.0154 2.03

p -Value Solution (p-Value is between .02 and .05) < (a = 0.05) Reject. p-Value 2 is between .01 and .025 Reject Reject a2 =.025 0 Z -2.0154 2.0154 2.03 Test Statistic 2.03 is in the Reject Region

You’re a financial analyst for Charles Schwab. You collect the following data: NYSENASDAQNumber 21 25 Sample Mean 3.27 2.53 Sample Std Dev 1.30 1.16 Example You want to construct a 95% confidence interval for the difference in population average yields of the stocks listed on NYSE and NASDAQ. © 1984-1994 T/Maker Co.

Example: Solution

Independent Sample (Two Sample) t- Test in JMP • Independent Sample t Test with Variances Known • Analyze | Fit Y by X | Measurement in Y box (Continuous) | Grouping Variable in X box (Nominal) |  | Means/Anova/Pooled t

Comparing Two Related Samples • Test the Means of Two Related Samples • Paired or matched • Repeated measures (before and after) • Use difference between pairs • Eliminates Variation between Subjects

Z Test for Mean Difference (Variance Known) • Assumptions • Both populations are normally distributed • Observations are paired or matched • Variance known • Test Statistic

t Test for Mean Difference (Variance Unknown) • Assumptions • Both populations are normally distributed • Observations are matched or paired • Variance unknown • If population not normal, need large samples • Test Statistic

Assume you work in the finance department. Is the new financial package faster (a=0.05 level)? You collect the following processing times: Dependent-Sample t Test: Example UserExisting System (1)New Software (2)DifferenceDi C.B. 9.98 Seconds 9.88 Seconds .10 T.F. 9.88 9.86 .02 M.H. 9.84 9.75 .09 R.K. 9.99 9.80 .19 M.O. 9.94 9.87 .07 D.S. 9.84 9.84 .00 S.S. 9.86 9.87 - .01 C.T. 10.12 9.98 .14 K.T. 9.90 9.83 .07 S.Z. 9.91 9.86 .05

Is the new financial package faster (0.05 level)? Dependent-Sample t Test: Example Solution H0: mD £0 H1: mD>0 Reject a =.05 D = .072 a =.05 Critical Value=1.8331df = n - 1 = 9 1.8331 3.66 Decision: Reject H0 t Stat. in the rejection zone. Test Statistic Conclusion: The new software package is faster.

Confidence Interval Estimate for of Two Dependent Samples • Assumptions • Both populations are normally distributed • Observations are matched or paired • Variance is unknown • Confidence Interval Estimate:

Assume you work in the finance department. You want to construct a 95% confidence interval for the mean difference in data entry time. You collect the following processing times: Example UserExisting System (1)New Software (2)DifferenceDi C.B. 9.98 Seconds 9.88 Seconds .10 T.F. 9.88 9.86 .02 M.H. 9.84 9.75 .09 R.K. 9.99 9.80 .19 M.O. 9.94 9.87 .07 D.S. 9.84 9.84 .00 S.S. 9.86 9.87 - .01 C.T. 10.12 9.98 .14 K.T. 9.90 9.83 .07 S.Z. 9.91 9.86 .05

F Test for Difference in Two Population Variances • Test for the Difference in 2 Independent Populations • Parametric Test Procedure • Assumptions • Both populations are normally distributed • Test is not robust to this violation • Samples are randomly and independently drawn

The F Test Statistic = Variance of Sample 1 n1 - 1 = degrees of freedom = Variance of Sample 2 n2 - 1 = degrees of freedom 0 F

Developing the F Test • Hypotheses • H0: s12 = s22 • H1: s12¹ s22 • Test Statistic • F = S12 /S22 • Two Sets of Degrees of Freedom • df1 = n1 - 1; df2 = n2 - 1 • Critical Values: FL( ) and FU( ) FL = 1/FU* (*degrees of freedom switched) Reject H0 Reject H0 Do Not Reject a/2 a/2 0 FL FU F n1 -1, n2 -1 n1 -1 , n2 -1

Developing the F Test • Easier Way • Put the largest in the num. • Test Statistic • F = S12 /S22 Reject H0 Do Not Reject a 0 F F

Assume you are a financial analyst for Charles Schwab. You want to compare dividend yields between stocks listed on the NYSE & NASDAQ. You collect the following data: NYSENASDAQNumber 21 25 Mean 3.27 2.53 Std Dev 1.30 1.16 Is there a difference in the variances between the NYSE & NASDAQ at the a = 0.05 level? F Test: An Example © 1984-1994 T/Maker Co.

F Test: Example Solution • Finding the Critical Values for a = .05

H0:s12 = s22 H1: s12¹s22 a=.05 df1=20 df2=24 Critical Value(s): F Test: Example Solution Test Statistic: Decision: Conclusion: Do not reject at a = 0.05. Reject .05 There is insufficient evidence to prove a difference in variances. F 0 2.33 1.25

F Test: One-Tail H0: s12³s22 H0: s12£ s22 or • H1: s12 > s22 • H1: s12< s22 a = .05 Degrees of freedom switched Reject Reject a = .05 a = .05 F F 0 0

F Test: One-Tail • Easier Way • Put the largest in the num. • Test Statistic • F = S12 /S22 Reject H0 Do Not Reject a 0 F F

Z Test for Differences in Two Proportions (Independent Samples) • What is It Used For? • To determine whether there is a difference between 2 population proportions and whether one is larger than the other • Assumptions: • Independent samples • Population follows binomial distribution • Sample size large enough: np 5 and n(1-p) 5 for each population

Z Test Statistic

The Hypotheses for the Z Test Research Questions No Difference Prop 1 Prop 2  Prop 1  Prop 2 Hypothesis Any Difference Prop 1 < Prop 2 Prop 1 > Prop 2 - p p = 0   p - p 0 p - p 0 H 1 2 1 2 1 2 0 - p - p < 0 p p > 0  p - p 0 H 1 2 1 2 1 2 1

As personnel director, you want to test the perception of fairness of two methods of performance evaluation. 63 of 78 employees rated Method 1 as fair. 49 of 82 rated Method 2 as fair. At the 0.01 significance level, is there a difference in perceptions? Z Test for Differences in Two Proportions: Example 

H0: p1 - p2 = 0 H1: p1 - p2 0  = 0.01 n1 = 78 n2 = 82 Critical Value(s): Z Test for Differences in Two Proportions: Solution Test Statistic: Decision: Conclusion: Z  2 . 90 Reject at  = 0.01. Reject H Reject H 0 0 There is evidence of a difference in proportions. .005 .005 0 Z -2.58 2.58

Confidence Interval for Differences in Two Proportions • The Confidence Interval for Differences in Two Proportions

Confidence Interval for Differences in Two Proportions: Example As personnel director, you want to find out the perception of fairness of two methods of performance evaluation. 63 of 78 employees rated Method 1 as fair. 49 of 82 rated Method 2 as fair. Construct a 99% confidence interval for the difference in two proportions. 

Confidence Interval for Differences in Two Proportions: Solution We are 99% confident that the difference between two proportions is somewhere between 0.0294 and 0.3909.

Z Test for Differences in Two Proportions (Dependent Samples) • What is It Used For? • To determine whether there is a difference between 2 population proportions and whether one is larger than the other • Assumptions: • Dependent samples • Population follows binomial distribution • Sample size large enough: np 5 and n(1-p) 5 for each population

Z Test Statistic for Dependent Samples • This Z can be used when a+d>10 • If 10<a+d<20, use the correction in the text

Unit Summary • Compared Two Independent Samples • Performed Z test for the differences in two means • Performed t test for the differences in two means • Performed Z test for differences in two proportions • Addressed F Test for Difference in Two Variances

Unit Summary • Compared Two Related Samples • Performed dependent sample Z tests for the mean difference • Performed dependent sample t tests for the mean difference • Performed Z tests for proportions using dependent samples

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