Q: A random sample of 100 people from population 1 yields $$\hat{p}_1 = 0.48$$, and a random sample of 100 from population 2 yields $$\hat{p}_2 = 0.40. $$What is the standard error for the difference in sample proportions?

$$\sqrt{\frac{0.48 \times 0.52}{100} + \frac{0.40 \times 0.60}{100}}$$

Question 1

A company wants to compare the average delivery times of two courier services by sending two documents to each of 10 district offices, one with each service. Which statistical test is most appropriate for analyzing the difference in delivery times?

Accepted Answer

Paired t-test

Question 2

Suppose you are comparing the mean driving distances of two brands of golf balls using independent samples. The sample sizes are $$n_1 = 15$ and n_2$ = 10$$, with sample standard deviations $$s_1 = 15$ and s_2$ = 20. If $$you assume equal population variances, what is the formula for the pooled variance estimate $$s_{pooled}^2$$?

Accepted Answer

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

Question 3

A random sample of 100 people from population 1 yields $$\hat{p}_1 = 0.48$$, and a random sample of 100 from population 2 yields $$\hat{p}_2 = 0.40. $$What is the standard error for the difference in sample proportions?

Accepted Answer

$$\sqrt{\frac{0.48 	imes 0.52}{100} + \frac{0.40 	imes 0.60}{100}}$$

Question 4

Which of the following is NOT an assumption required for the classical two-sample t-test for means with independent samples?

Accepted Answer

The two samples are paired

Question 5

In the Par Golf, Inc. case, the 95% confidence interval for the difference in mean driving distances (Par minus Competitor) was (2.5, 31.5) yards. What does this interval mean in context?

Accepted Answer

We are 95% confident that Par's golf ball travels between 2.5 and 31.5 yards farther than the competitor's on average.

Question 6

Which formula would you use to construct a 95% confidence interval for the difference in means when population variances are unknown and assumed unequal?

Accepted Answer

$$\bar{X}_1 - \bar{X}_2 \pm t^* \sqrt{\frac{s_1$^2}{n_1$} + \frac{s_2$^2}{n_2$}}$$

Question 7

A study compares spending per pupil in 35 urban and 35 rural schools. Which test is most appropriate if the data are quantitative and variances are not assumed equal?

Accepted Answer

Two-sample t-test for means, unequal variance

Question 8

Which of the following scenarios best describes when to use a paired t-test?

Accepted Answer

Comparing before-and-after measurements on the same subjects

Question 9

If you are constructing a confidence interval for $$p_1$$ - $$p_2$$ using two independent samples, which formula should you use?

Accepted Answer

$$\hat{p}_1 - \hat{p}_2 \pm Z^* \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1$} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2$}}$$

Question 10

A manufacturer wants to test if a new machine (A) completes a task faster than the old machine (B). They collect independent samples of completion times. Which hypothesis test should they use?

Accepted Answer

Two-sample t-test for means

Question 11

Which of the following is the correct formula for the test statistic in a two-sample t-test for means with equal unknown variances?

Accepted Answer

$$t_{obs} = \frac{\bar{x}_1 - \bar{x}_2 - \#}{s_p \sqrt{\frac{1}{n_1$} + \frac{1}{n_2$}}}$$

Question 12

In a hypothesis test comparing two means with unequal variances, which method is used to approximate the degrees of freedom?

Accepted Answer

Satterthwaite approximation

Question 13

A 95% confidence interval for $$\mu_1$ - \mu_2$ is (-3, 5). What can you conclude?

Accepted Answer

There is no statistically significant difference between $$\mu_1$ and $$\$$mu_2 at $$the 5% level.

Question 14

Which of the following is a key difference between paired and independent samples?

Accepted Answer

Paired samples involve matching or repeated measures on the same subjects, while independent samples do not.

Question 15

A company wants to estimate the difference in customer satisfaction proportions between two stores. Which statistical method should be used?

Accepted Answer

Two-sample z-interval for proportions

Question 16

Which of the following is a correct interpretation of a 90% confidence interval for $$p_1$$ - $$p_2$$ that does not include zero?

Accepted Answer

There is evidence at the 10% significance level that the population proportions are different.

Question 17

Which of the following is NOT a condition for using the two-sample z-interval for proportions?

Accepted Answer

Population variances are equal

Question 18

A researcher wants to compare the mean test scores of two independent groups, but the population variances are unknown and unequal. Which test statistic should be used?

Accepted Answer

Two-sample t-test with Satterthwaite approximation

Comparing Two Groups: Means and Proportions in Business Statistics

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