Question Type 12: Initiating the two sample t-test with the hypothesis and critical region Exercises

In comparing two independent normal populations with equal variances, you collect the following small samples:

• Sample 1: 9 observations
• Sample 2: 13 observations

At the 5% significance level, test whether their means differ. State $H_0$, $H_1$, and give the two-sided critical region for the pooled two-sample $t$-test.

Two independent normal populations with equal variances are sampled with $n_1 = 18$ and $n_2 = 18$. At the 2.5% significance level, test whether population 1 has a lower mean than population 2. State $H_0$, $H_1$, and the left-tailed critical region for the pooled two-sample $t$-test.

Two independent samples of completion times (minutes) were recorded:

• Method A: 22, 21, 19, 23, 20, 18, 21, 20, 19, 22, 20
• Method B: 24, 23, 21, 25, 22, 19, 24, 22, 21, 23, 22

Assuming normal populations with equal variances, test at the 10% level whether Method A has a lower mean time than Method B. State $H_0$, $H_1$, and the critical region for the two-sample t-test.

Two independent simple random samples of sizes $n_A = 8$ and $n_B = 8$ are taken from normal populations with equal variances. At the $1\%$ significance level, test whether the population means are different by stating $H_0$, $H_1$, and the two-sided critical region for the two-sample $t$-test.

Two independent normal populations are compared with Welch’s two-sample $t$-test. The summary statistics are as follows:

Group A: $n_A = 6, s_A = 8$ Group B: $n_B = 7, s_B = 3$

At the $5\%$ significance level, test for a difference in means by stating the null and alternative hypotheses, computing the Welch degrees of freedom, and determining the two-sided critical region.

Two independent classes took the same short math quiz. The scores (out of 10) were:

• Class A: 5, 7, 6, 8, 7, 5, 9, 6, 7, 8, 6, 7
• Class B: 4, 6, 5, 7, 5, 4, 6, 5, 5, 6, 4, 5

Assuming normal populations with equal variances, at the 5% significance level, state suitable null and alternative hypotheses to test whether Class A’s mean is higher than Class B’s, and determine the critical region for the two-sample $t$-test.

Two independent production lines output lengths (cm) of a part:

• Line 1: 12, 13, 11, 15, 14, 12, 16, 13, 14, 12
• Line 2: 10, 14, 12, 13, 11, 12, 13, 12, 11, 13

Assuming normal populations with equal variances, at the 5% significance level, state $H_0$ and $H_1$ to test whether their means differ, and give the two-sided critical region for the two-sample $t$-test.

Two independent samples of sizes $n_1 = 16$ and $n_2 = 16$ will be compared. At the 1% significance level, assuming normal populations with equal variances, state the null and alternative hypotheses, $H_0$ and $H_1$, to test if the first population mean exceeds the second. Furthermore, determine the degrees of freedom and state the right-tailed critical region for the two-sample $t$-test.

Two labs measured the concentration (ppm) of the same compound on independent samples:

• Lab X ($n=13$): 31, 29, 33, 28, 30, 32, 29, 31, 30, 28, 32, 30, 29
• Lab Y ($n=13$): 30, 28, 31, 27, 29, 30, 28, 30, 29, 27, 31, 29, 28

At the 10% significance level, assuming normal populations with equal variances, test whether the labs differ in mean concentration by stating $H_0$, $H_1$, and the two-sided critical region.

Two independent normal populations with equal variances are compared using samples of sizes $n_1 = 14$ and $n_2 = 14$. At the $2\%$ significance level, test for a difference in means.

State $H_0$ and $H_1$, and determine the two-sided critical region for the pooled two-sample $t$-test.

Two independent normal populations are compared using Welch’s two-sample $t$-test. The summary statistics for the two groups are given below:

Group 1: $n_1 = 9$, sample standard deviation $s_1 = 4$ Group 2: $n_2 = 12$, sample standard deviation $s_2 = 6$

At the 5% significance level, test whether the means differ by stating the null and alternative hypotheses, and determining the two-sided critical region using the Welch–Satterthwaite degrees of freedom.

Two independent normal populations with equal variances are sampled with $n_1 = 16$ and $n_2 = 16$. At the 10% significance level, test whether population 1 has a higher mean than population 2. State $H_0$, $H_1$, and the right-tailed critical region for the pooled two-sample $t$-test.