Comparing Two Population Means (Dependent Samples): Learn It 4

  • Complete a two-sample [latex]t[/latex]-test for dependent population means from hypotheses to conclusions

Hypothesis Testing for Dependent Samples (continued)

The third step in hypothesis testing is to calculate a test statistic, which we will utilize to find the P-value, write a conclusion, and make an inference about the population.

test statistic ([latex]t[/latex])

The notations for the summary statistics used to compare paired populations/samples are shown in the following table. We will use [latex]d[/latex] to represent the difference variable.

Summary Statistics Notation
Population Mean of Difference [latex]\mu_d[/latex]
Sample Mean of Difference [latex]\bar{d}[/latex]
Population Standard Deviation of Difference [latex]\sigma_d[/latex]
Sample Standard Deviation of Difference [latex]s_d[/latex]

 

The test statistic for the dependent (paired) t-test is calculated using the following formulas:

 

[latex]\text{standard error of the difference}=\dfrac{s_d}{\sqrt{n}}[/latex]

 

[latex]\text{test statistic }(t)=\dfrac{\text{estimator - null value}}{\text{standard error of estimator}}=\dfrac{\bar{d}-\text{null value}}{\text{standard error of difference}}[/latex]

Use the following steps:

Step 1: Click on the tab Two Dependent Samples.

Step 2: In the “Dataset” drop-down menu, choose “Reaction Times (Paired Experiment).”

Step 3: In the left column, go to the drop-down menu for “Type of Inference” and select “Significance Test.”


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When thinking about the difference variable, we need to use a different calculation for the standard deviation of the estimate. The standard deviation of the difference in the sample means, [latex]\bar{x}_1-\bar{x}_2[/latex], is NOT the same as the standard deviation of the difference variable, denoted using [latex]s_d[/latex].

Take advantage of the statistical tool to calculate the standard deviation of the difference in the sample means.