Practice Paired t-Test in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

A hypothesis test for the mean difference in a paired (matched) data design, where each subject provides two related measurements. The test analyzes the differences di=x1iโˆ’x2id_i = x_{1i} - x_{2i} as a single sample.

You want to know if a tutoring program improves math scores. Instead of comparing two separate groups, you test the SAME students before and after tutoring. Each student is their own control. By looking at the difference (after โˆ’- before) for each student, you eliminate individual variation and focus purely on the change.

Showing a random 20 of 50 problems.

Example 1

easy
True or false: a paired t-test is the same as a one-sample t-test on the differences.

Example 2

medium
A student computes the mean of the before group (7272) and the after group (7878) and tests the difference 66 with a two-sample test. Why is this wrong for paired data?

Example 3

hard
What happens to the paired t-test if you accidentally enter the data as beforeโˆ’after\text{before} - \text{after} instead of afterโˆ’before\text{after} - \text{before}?

Example 4

medium
A study measures the SAME 30 students before and after a course. Is a paired or two-sample t-test appropriate, and why?

Example 5

challenge
Derive: if X1i,X2iX_{1i}, X_{2i} are paired observations with Var(X1i)=Var(X2i)=ฯƒ2\text{Var}(X_{1i}) = \text{Var}(X_{2i}) = \sigma^2 and Cor(X1i,X2i)=ฯ\text{Cor}(X_{1i}, X_{2i}) = \rho, show that Var(di)=2ฯƒ2(1โˆ’ฯ)\text{Var}(d_i) = 2\sigma^2(1-\rho).

Example 6

easy
Why does a paired design often have more power than using two independent groups for the same subjects?

Example 7

hard
For a paired study with dห‰=0.5\bar{d} = 0.5, sd=2s_d = 2, what minimum sample size nn achieves โˆฃtโˆฃโ‰ฅ2|t| \ge 2?

Example 8

easy
For paired data, the null hypothesis is usually that the mean difference equals what value?

Example 9

easy
Why is matching subjects (e.g., before/after on the same person) called a paired design?

Example 10

medium
Fill in: the paired t-test requires the differences (not the original data) to be approximately ____.

Example 11

medium
Differences dd have dห‰=0\bar{d} = 0. Without computing further, what is the paired t-statistic and the likely conclusion?

Example 12

medium
A paired t-test gives t=3.2t = 3.2 on df=9df = 9, p-value =0.011= 0.011, at ฮฑ=0.05\alpha = 0.05. Conclude about the mean difference.

Example 13

easy
Five paired differences are: {2,โˆ’1,3,0,4}\{2, -1, 3, 0, 4\}. Calculate dห‰\bar{d} and sds_d, then set up the t-test statistic formula.

Example 14

challenge
A researcher uses a two-sample t-test on paired data and gets p-value 0.080.08; the correct paired test gives 0.020.02. Explain why the paired test found significance when the two-sample test did not.

Example 15

medium
A paired test gives t=2.5t = 2.5 on df=10df = 10. The critical value for ฮฑ=0.05\alpha = 0.05 two-sided is tโˆ—=2.228t^* = 2.228. Decide.

Example 16

medium
Build a 95%95\% CI for ฮผd\mu_d when dห‰=10\bar{d} = 10, sd=4s_d = 4, n=16n = 16. Use tโˆ—=2.131t^* = 2.131.

Example 17

easy
In a paired t-test with n=25n = 25 pairs, what are the degrees of freedom?

Example 18

easy
A paired design records each subject's before and after scores. What single quantity does the paired t-test analyze for each subject?

Example 19

hard
Twelve patients' cholesterol before and after a drug have dห‰=โˆ’8\bar{d} = -8 mg/dL, sd=12s_d = 12. Construct a 90%90\% CI for ฮผd\mu_d using tโˆ—=1.796t^* = 1.796.

Example 20

medium
Twins are matched and one of each pair gets a treatment. Each pair gives one difference. Is this a paired or two-sample design?
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