Q: If the sample sizes are $$n_1 = 15$ and n_2$ = 20$$, and the sample variances are $$s_1^2 = 4$ and s_2$^2 = 9$$, what is the pooled variance $$s_p^2$?

$$s_p^2$ = \frac{ (15-1) \times 4 + (20-1) \times 9 }{ 15 + 20 - 2 } = \frac{ 56 + 171 }{ 33 } = \frac{ 227 }{ 33 } \approx 6.88$$

Question 1

Which of the following is an appropriate scenario for using a two-sample t-test for the difference between means?

Accepted Answer

Comparing the mean scores of two independent groups when population variances are unknown and both populations are normally distributed.

Question 2

What is the null hypothesis $$H_0$$ when testing if there is a difference in mean mathematics test scores between students taught by two different teachers?

Accepted Answer

$$H_0$$: \$$mu_1$$ = \$$mu_2$$

Question 3

When performing a two-sample t-test with independent samples and unknown variances, which assumption must be met?

Accepted Answer

The samples must be random and independent.

Question 4

If the population variances are assumed to be equal in a two-sample t-test, which formula is used to estimate the standard error?

Accepted Answer

$$\sqrt{ s_p^2$ \left( \frac{1}{n_1$} + \frac{1}{n_2$} \right) }$$

Question 5

A car manufacturer claims its sedans have a lower mean driving cost per mile than its competitor. Which is the correct alternative hypothesis for this left-tailed test?

Accepted Answer

$$H_a: \mu_1$ \u003c \mu_2$

Question 6

Which formula is used to calculate the test statistic for a two-sample t-test when variances are not assumed equal?

Accepted Answer

$$t = \frac{ (\bar{x}_1 - \bar{x}_2) - (\mu_1$ - \mu_2$) }{ \sqrt{ \frac{s_1$^2}{n_1$} + \frac{s_2$^2}{n_2$} } }$$

Question 7

In a two-sample t-test, if the calculated test statistic falls in the rejection region, what is the correct conclusion?

Accepted Answer

Reject the null hypothesis.

Question 8

Which of the following best describes the pooled variance $$s_p^2$ used in a two-sample t-test with equal variances?

Accepted Answer

$$s_p^2$ = \frac{ (n_1$ - 1)s_1$^2 + (n_2$ - 1)s_2$^2 }{ n_1$ + n_2$ - 2 }$$

Question 9

A researcher wants to compare the mean recovery times for two different drugs using independent samples. The population variances are unknown and not assumed equal. Which test should be used?

Accepted Answer

Two-sample t-test for independent samples with unequal variances

Question 10

Which of the following is NOT an assumption of the two-sample t-test for the difference between means?

Accepted Answer

The samples are dependent.

Question 11

Suppose you conduct a two-sample t-test and obtain a p-value of 0.08. If your significance level is $$\alpha = 0.05$$, what is the correct interpretation?

Accepted Answer

There is not enough evidence to reject the null hypothesis at the 5% significance level.

Question 12

Which of the following real-world scenarios would require a two-tailed two-sample t-test?

Accepted Answer

Testing if there is any difference in average blood pressure between patients treated with two different medications.

Question 13

If the sample sizes are $$n_1 = 15$ and n_2$ = 20$$, and the sample variances are $$s_1^2 = 4$ and s_2$^2 = 9$$, what is the pooled variance $$s_p^2$?

Accepted Answer

$$s_p^2$ = \frac{ (15-1) 	imes 4 + (20-1) 	imes 9 }{ 15 + 20 - 2 } = \frac{ 56 + 171 }{ 33 } = \frac{ 227 }{ 33 } \approx 6.88$$

Question 14

Which of the following best describes the degrees of freedom for a two-sample t-test with unequal variances?

Accepted Answer

$$df = \frac{ \left( \frac{s_1$^2}{n_1$} + \frac{s_2$^2}{n_2$} \$$right)^2$$ }{ \frac{ (\frac{s_1$^2}{n_1$})^2$ }{ n_1$ - 1 } + \frac{ (\frac{s_2$^2}{n_2$})^2$ }{ n_2$ - 1 } }$$

Hypothesis Testing with Two Samples: Two-Sample t-Tests for Means

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