In a matched pairs study testing whether a new teaching method improves scores, which is the correct null hypothesis?

In a paired $t$-test, you calculated $t=1.82$. The test is right-tailed with $df=7$ and $\alpha =0.05$. The critical value is $t=1.895$. Should you reject or fail to reject the null hypothesis?

In a study on calcium supplements and blood pressure, $12$ men were given calcium supplements for $12$ weeks. The differences in diastolic blood pressure (after − before) were ranked, and the sum of positive ranks was $20$, while the sum of negative ranks was $16$. The critical value for this test at $\alpha =0.01$ is $25$. Should you reject the null hypothesis, and what does this decision mean in the context of the claim?

A psychologist is studying the difference in average reaction times (in milliseconds) between two age groups. The data collected are: Group $1$ (Young Adults): ${\bar{x}}_1=245$, $s_1=15$, $n_1=6$; Group $2$ (Older Adults): ${\bar{x}}_2=278$, $s_2=22$, $n_2=8$. Construct an $80\%$ confidence interval for the difference in mean reaction times (Young Adults $-$ Older Adults). Assume unequal variances and approximately normal populations.

A sports scientist claims that a new stretching routine increases vertical jump height in athletes. The vertical jump heights (in centimeters) for $8$ athletes are measured before and after following the routine. The following are the differences: $2$, $3$, $1$, $4$, $2$, $3$, $2$, $5$. At $\alpha =0.05$, is there sufficient evidence to support the scientist's claim? Assume the samples are random, dependent, and the population is normally distributed.

In a Wilcoxon Signed-Ranks Test, suppose you have $n$ nonzero paired differences and the sum of the negative ranks is $t$. What is the expression for the sum of the positive ranks in terms of $n$ and $t$?

A nutritionist tests a new diet plan on $12$ individuals and records the weight change (after − before) for each person. $6$ participants lost weight, $4$ participants gained weight, and $2$ participants had no change. What is the test statistic for the sign test?

A group of researchers believes that a new workout routine helps reduce blood pressure levels. In a study with $18$ participants:

$13$ had lower blood pressure after the program

$4$ had higher blood pressure

$1$ showed no change

Use a left-tailed sign test at $\alpha =0.05$. What is your decision, and what does it mean in context?

A nutritionist wants to determine if a new diet plan leads to weight loss. The weights in kilograms of six participants before and after following the diet for a month are as follows.

At the $0.05$ significance level, test the claim that the mean weight after the diet is less than before. Assume the paired differences are approximately normally distributed.

Two independent samples are taken to compare the average recovery times (in days) under two physical therapy protocols. Sample $1$ has $n_1=25$, sample mean ${\bar{x}}_1=10.5$, and sample standard deviation $s_1=\sqrt{12.5}$. Sample $2$ has $n_2=25$, sample mean ${\bar{x}}_2=8.5$, and sample standard deviation $s_2=\sqrt{12.5}$. Assume unequal variances. Perform a two-sided two-sample $t$-test for the difference in means ${\mu }_1-{\mu }_2$. What is the $P$-value for this test?