Question Type 4: Performing a hypothesis test comparing two means for paired samples Exercises

Blood glucose levels are measured for 30 patients before and after taking a new drug. The mean difference (before $-$ after) is found to be $\bar{d} = 5$ mg/dL with a sample standard deviation of differences $s_d = 9$ mg/dL.

State the hypotheses and perform a $t$-test at the 1% significance level to determine if the drug lowers blood glucose.

A coach records the sprint times, in seconds, of 7 athletes before and after completing a training program. The results are shown in the table below:

Perform a paired $t$-test at the 5% significance level to determine if the training program significantly reduces sprint times. State the null and alternative hypotheses clearly.

In a study of memory, 20 participants learn word lists by method 1 then by method 2. The mean difference in recall (method 2 $-$ method 1) is $\bar{d}=2.5$ points, with a standard deviation of differences $s_d=3.1$.

Test at the $1\%$ significance level (two-tailed) whether the recall scores for method 2 differ significantly from method 1. [6 marks]

Perform a two-tailed paired $t$-test at the 5% significance level to determine whether the mean calorie consumption differs when watching horror versus romance movies.

The blood pressure of 12 patients is measured before and after taking drug X. The differences in blood pressure ($d = \text{after} - \text{before}$) are calculated. The sample mean difference is $\bar{d} = -5 \text{ mmHg}$ and the sample standard deviation of the differences is $s_d = 6 \text{ mmHg}$.

Conduct a one-tailed paired $t$-test at the 5% level of significance to determine whether drug X lowers blood pressure.

Eight subjects have their weights recorded before and after a 4-week diet:

(85, 82), (90, 88), (78, 75), (110, 105), (95, 93), (100, 98), (72, 70), (88, 85).

Test at the $\alpha=0.05$ significance level whether the diet leads to a significant weight reduction (one-tailed).

A study was conducted to determine if a specific treatment reduces pollutant levels in soil. Fifteen soil samples were taken, and pollutant levels were measured before and after the treatment. The mean difference in pollutant levels (before $-$ after) was found to be $\bar{d} = 3.2$ ppm, with a sample standard deviation of the differences $s_d = 5.5$ ppm.

At a $10\%$ significance level, test the claim that the treatment reduces pollution levels using a one-tailed paired $t$-test.

Eighteen participants take two recall tests (old and new). The mean difference is $\bar{d} = \text{old} - \text{new} = -1.8$ items, and the sample standard deviation of the differences is $s_d = 2.5$.

Test at the 5% significance level, using a one-tailed test, whether the new test yields a significantly higher recall (i.e., $\mu_d < 0$).

A cognitive researcher measures error counts under two conditions, A and B, for $16$ participants. The mean difference is $\bar{d} = A - B = -2$ errors, with sample standard deviation $s_d = 5$. Perform a two-tailed paired $t$-test at the $\alpha = 0.05$ significance level to decide if a difference exists between the two conditions.

Using the summary statistics provided, test at the 1% significance level whether mean calorie consumption is higher when watching romance movies than horror movies. Use a one-tailed paired t-test.

The reaction times, in seconds, for 10 subjects were measured before (standard) and after consuming a caffeine drink. The results are shown in the following table.

Test at the 5% significance level whether caffeine reduces reaction time.

A study measures screen time (in hours) before and after an intervention for 25 subjects. The sample mean difference is $\bar{d} = \text{before} - \text{after} = 0.5$ hours and the sample standard deviation of the differences is $s_d = 1.2$ hours.

Conduct a two-tailed paired $t$-test at the $5\%$ significance level.