Q: For the scenario comparing the two investment portfolios, which of the following is the correct formula for a 98% confidence interval for the difference in means ($$ \mu_1$ - \mu_2$ $$)?

$$ (\bar{x}_1 - \bar{x}_2) \pm z^* \times SE $$

Question 1

A financial analyst wants to estimate the difference in mean annual returns between a growth portfolio and an income portfolio. The growth portfolio has a sample size of 50, mean return of 8%, and standard deviation of 10%. The income portfolio has a sample size of 60, mean return of 6%, and standard deviation of 12%. What is the correct formula for the standard error (SE) of the difference in sample means?

Accepted Answer

$$ SE = \sqrt{ \frac{10^2$}{50} + \frac{12^2$}{60} } $$

Question 2

Given the following data for two independent samples: Growth Portfolio (n=50, mean=8%, SD=10%) and Income Portfolio (n=60, mean=6%, SD=12%), what is the point estimate for the difference in mean returns?

Accepted Answer

$$ 2\% $$

Question 3

For the scenario comparing the two investment portfolios, which of the following is the correct formula for a 98% confidence interval for the difference in means ($$ \mu_1$ - \mu_2$ $$)?

Accepted Answer

$$ (\bar{x}_1 - \bar{x}_2) \pm z^* 	imes SE $$

Question 4

Suppose you calculated the standard error for the difference in means between the two portfolios as $$2.07. $$What is the margin of error for a 98% confidence interval? (Use $$z^* = 2.33$$ for 98% confidence.)

Accepted Answer

$$ 4.83 $$

Question 5

A company wants to test if there is a significant difference in mean productivity between employees at two offices. The null hypothesis is:

Accepted Answer

$$ H_0$: \mu_1$ - \mu_2$ = 0 $$

Question 6

Given the following data: Silicon Valley (n=40, mean=150, SD=20), Austin (n=40, mean=140, SD=25), what is the test statistic for testing the difference in means? (Use the formula $$ t = \frac{\bar{x}_1 - \bar{x}_2}{SE} $$ and calculate $$ SE $$ accordingly.)

Accepted Answer

$$ t = \frac{150 - 140}{\sqrt{\frac{20^2$}{40} + \frac{25^2$}{40}}} $$

Question 7

If the calculated test statistic for the productivity difference is $$2.13$$ and the critical value for $$\alpha = 0.05 ($$two-tailed) is $$1.99$$, what is the correct conclusion?

Accepted Answer

Reject the null hypothesis; there is a significant difference in mean productivity.

Question 8

Which of the following assumptions is NOT required for the two-sample $$t-$$test comparing means of two independent samples?

Accepted Answer

The two samples must be paired.

Question 9

In the context of the investment portfolios, which of the following would increase the width of the confidence interval for the difference in means?

Accepted Answer

Increasing the confidence level from 95% to 99%.

Question 10

A financial analyst constructs a 98% confidence interval for the difference in mean returns between two portfolios and finds the interval is (–1.0%, 5.0%). What does this interval suggest?

Accepted Answer

There is not enough evidence to conclude a significant difference in mean returns.

Question 11

If the sample sizes for both offices in the productivity study were doubled, what would happen to the standard error of the difference in means?

Accepted Answer

It would decrease.

Question 12

Which of the following best describes the meaning of a 98% confidence interval for the difference in means?

Accepted Answer

If we repeated the sampling many times, about 98% of the intervals would contain the true difference in means.

Question 13

A company tests $$H_0$$: \$$mu_1$$ - \$$mu_2 = 0$ at $$\alpha = 0.05$$ and obtains a p-value of 0.03$. What is the correct interpretation?

Accepted Answer

There is sufficient evidence to reject the null hypothesis at the 5% significance level.

Question 14

Which of the following scenarios would violate the independence assumption required for the two-sample $$t-$$test?

Accepted Answer

Selecting the same employees for both samples.

Question 15

If the standard deviations of the two samples are very different, which adjustment should be considered when performing a two-sample $$t-$$test?

Accepted Answer

Use the unequal variances (Welch's) $$t-$$test.

Question 16

A 98% confidence interval for the difference in means is calculated as $$(1.5, 6.5). $$What does this imply about the statistical significance at the 2% level?

Accepted Answer

There is a statistically significant difference in means at the 2% significance level.

Guided Practice: Comparing Means and Confidence Intervals in Statistics

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