# 10.6: Dependent t-test Exercises

• • Michelle Oja
• Taft College
$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

Let's try some Exercises to think about dependendent sample t-tests, and practice calculations.

## Exercise $$\PageIndex{1}$$

What is the difference between a one-sample $$t$$-test and a dependent samples $$t$$-test? How are they alike?

A one-sample $$t$$-test uses raw scores to compare an average that is compared to the mean of a population. A dependent samples $$t$$-test uses two raw scores from a pair to calculate difference scores and test for an average difference score. What is different is where the two means being compared come from, and the equation. What is similar is that two means are being compared. Also, the calculations, steps, and interpretation are similar for each.

## Exercise $$\PageIndex{2}$$

What are difference scores and why do we calculate them?

Difference scores indicate change or discrepancy relative to a single person or pair of people by subtracting one score from another. We calculate them to eliminate individual differences in our study of change or agreement.

## Exercise $$\PageIndex{4}$$

What is the null hypothesis for a depedent samples t-test in words and symbols?

• Null Hypothesis: The average score at Time 1 will be similar to the average score at Time 2.
• Symbols: $$\bar{X}_{\text {T1}} = \bar{X}_{\text {T2}}$$

## Exercise $$\PageIndex{5}$$

A researcher hypothesizes that whether explaining the processes of statistics decreases trust in computer algorithms. He wants to test for a difference at the $$α$$ = 0.05 level. He gathers pretest and posttest data from 35 people who took a statistics course, and finds that the average difference score is $$\overline{X_{D}}$$ = 12.10 with a standard deviation of $$s_{D}$$= 17.39. Conduct a hypothesis test to answer the research question.

Step 1:

• Research Hypothesis: People score higher on average on trust of computer algorithms BEFORE they take a statistics course compared to their average trust of computer algorithms AFTER taking a statistics course.
• Symbols: $$\bar{X}_{\text {T1}} > \bar{X}_{\text {T2}}$$
• Null Hypothesis: People score similarly on trust of computer algorithms BEFORE they take a statistics course compared to their average trust of computer algorithms AFTER taking a statistics course.
• Symbols: $$\bar{X}_{\text {T1}} = \bar{X}_{\text {T2}}$$

Step 2:

df = N - 1 = 35 - 1 = 34

The critical t-value from the table for df = 1.697 (df = 30 is the one in the table).

Step 3: $$\overline{X_{D}}$$ = 12.10, $$s_{\overline{X_{D}}}$$= 2.94, $$t$$ = 4.12.

Step 4:

Critical < |Calculated| = Reject null = means are different = p<.05

Based on opinions from 35 people, we can conclude that people trust algorithms more ($$\overline{X_{D}}$$= 12.10) after learning statistics (because the mean of the difference is positive); this is opposite of what was hypothesized ($$t(34) = 4.12, p < .05$$).

Note, the actual means for the pretest and posttest are missing again, so they cannot be included.