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Question 1

Two independent samples of exam scores were collected with the following results:

Sample 1: $n_1 = 25$ , $\bar{x}_1 = 100$ , $s_1 = 12$ Sample 2: $n_2 = 30$ , $\bar{x}_2 = 95$ , $s_2 = 10$

Assuming the populations are normally distributed with equal variances, perform a $t$ -test at the $10\%$ significance level to determine if there is a difference between the mean scores of the two groups.

State the hypotheses, calculate the two-tailed $p$ -value, and state your conclusion in context.

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Question 2

This question requires performing a two-sample $t$ -test for the equality of means from two independent normal populations, assuming equal variances. Students must calculate the pooled standard deviation estimate, the $t$ -statistic, identify the critical value for a one-tailed test, and make a statistical conclusion.

Two independent random samples are taken from populations with normal distributions and equal variances.

Sample 1: $n_1 = 15, \bar{x}_1 = 7.2, s_1 = 1.5$ Sample 2: $n_2 = 15, \bar{x}_2 = 6.8, s_2 = 1.2$

Test, at the 5% level of significance, the null hypothesis $H_0: \mu_1 = \mu_2$ against the alternative hypothesis $H_1: \mu_1 > \mu_2$ .

Calculate the pooled standard deviation estimate and the $t$ -test statistic. Find the critical $t$ -value and state your conclusion.

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Question 3

Two independent samples are drawn from two populations which are assumed to be normally distributed with equal variances. The following data is collected from the samples:

Sample 1: $n_1 = 14$ , $\bar{x}_1 = 5.4$ , $s_1 = 0.8$ Sample 2: $n_2 = 16$ , $\bar{x}_2 = 5.1$ , $s_2 = 0.9$

Perform a pooled $t$ -test to determine if there is evidence at the $1\%$ significance level that $\mu_1 > \mu_2$ . Calculate the $p$ -value and state your conclusion.

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Question 4

The lifetimes of two types of industrial components were measured to compare their mean performance.

Sample 1 consisted of $n_1=18$ components with a mean lifetime $\bar{x}_1=120$ hours and a standard deviation $s_1=15$ hours.

Sample 2 consisted of $n_2=16$ components with a mean lifetime $\bar{x}_2=110$ hours and a standard deviation $s_2=20$ hours.

Assuming the lifetimes of both populations follow a normal distribution with equal variances, test the hypothesis $H_0: \mu_1 = \mu_2$ against $H_1: \mu_1 > \mu_2$ at the $5\%$ level of significance. Calculate the one-tailed $p$ -value and state the conclusion of the test.

[8]

Question 5

Two small independent samples are taken from two normally distributed populations with equal variances. The data collected are summarized below:

Sample 1: $n_1 = 12$ , $\bar{x}_1 = 45$ , $s_1 = 5$ Sample 2: $n_2 = 12$ , $\bar{x}_2 = 40$ , $s_2 = 4$

where $s_1$ and $s_2$ are the unbiased estimates of the population standard deviations.

Perform a two-tailed $t$ -test at the $5\%$ significance level to determine whether there is a significant difference between the population means $\mu_1$ and $\mu_2$ .

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Question 6

Two samples of weights provide the following data: Sample 1: $n_1 = 8$ , $\bar{x}_1 = 85$ , $s_1 = 6$ Sample 2: $n_2 = 10$ , $\bar{x}_2 = 80$ , $s_2 = 5$

At the $1\%$ significance level, test for a difference in means (two-tailed). Find the critical $t$ -value and conclude your findings.

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Question 7

A researcher compares the average reaction times of two groups. Sample 1: $n_1 = 12$ , $\bar{x}_1 = 24$ ms, $s_1 = 4$ ms; Sample 2: $n_2 = 15$ , $\bar{x}_2 = 20$ ms, $s_2 = 5$ ms. At the 5% significance level, test for a difference in means by finding the critical $t$ -value and stating your conclusion.

[7]

Question 8

Two independent samples are taken from normally distributed populations with equal variances. The sample statistics are as follows:

Sample 1: $n_1 = 20$ , $\bar{x}_1 = 200$ , $s_1 = 30$ Sample 2: $n_2 = 22$ , $\bar{x}_2 = 185$ , $s_2 = 25$

At the 1% significance level, test the hypothesis $H_0: \mu_1 = \mu_2$ against the alternative hypothesis $H_1: \mu_1 > \mu_2$ .

Find the critical $t$ -value and state your decision.

[6]

Question 9

Independent samples are taken from two normal populations with equal variances. The sample statistics are as follows:

Sample 1: $n_1 = 30, \bar{x}_1 = 54, s_1 = 8$ Sample 2: $n_2 = 28, \bar{x}_2 = 50, s_2 = 7$

A two-tailed $t$ -test is conducted at the 5% level of significance to determine if the population means differ significantly. Calculate the pooled standard deviation and the test statistic. Determine the critical $t$ -value and state the conclusion of the test, giving a reason for your answer.

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Question 10

Two independent samples yield the following data: Sample 1: $n_1 = 20, \bar{x}_1 = 15, s_1 = 3$ Sample 2: $n_2 = 18, \bar{x}_2 = 12, s_2 = 4$

Test $H_0: \mu_1 = \mu_2$ against $H_1: \mu_1 > \mu_2$ at the $5\%$ significance level. Calculate the $t$ -statistic, the one-tailed $p$ -value, and state your conclusion.

[7]

Question 11

Two independent samples are taken from normally distributed populations with equal variances. The sample statistics are as follows:

Sample 1: $n_1=16$ , $\bar{x}_1=250$ , $s_1=30$ Sample 2: $n_2=14$ , $\bar{x}_2=230$ , $s_2=25$

Test the hypothesis $H_0: \mu_1 = \mu_2$ against $H_1: \mu_1 \neq \mu_2$ at the $5\%$ significance level. Calculate the pooled standard deviation, the $t$ -statistic, and the $p$ -value to justify your conclusion.

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Question 12

Two independent samples are taken from populations where the variances are assumed to be equal. The sample data is as follows:

Sample 1: $n_1 = 9$ , $\bar{x}_1 = 50$ , $s_1 = 6$
Sample 2: $n_2 = 9$ , $\bar{x}_2 = 45$ , $s_2 = 7$

At the $5\%$ significance level, find the critical $t$ -value and state the conclusion for a two-tailed test of the null hypothesis $H_0: \mu_1 = \mu_2$ against the alternative hypothesis $H_1: \mu_1 \neq \mu_2$ .

[6]

Question Type 14: Calculating the critical value and/or p-value to conclude the two sample t-test Exercises