Q2. Joe Sutton is evaluating the effects of the 1987 market decline on the volume of trading. Specifically, he wants to test whether the decline affected trading volume. He selected a sample of 500 companies and collected data on the total annual volume for one year prior to the decline and for one year following the decline. What is the set of hypotheses that Sutton is testing?

A)   H₀: µ_d = µ_d0 versus H_a: µ_d ≠ µ_d0.

B)   H₀: µ_d ≠ µ_d0 versus H_a: µ_d = µ_d0.

C)   H₀: µ_d = µ_d0 versus H_a: µ_d > µ_d0.

Q3. An analyst wants to determine whether the monthly returns on two stocks over the last year were the same or not. What test should she use if she is willing to assume that the returns are normally distributed?

A)   A difference in means test only if the variances of monthly returns are equal for the two stocks.

B)   A paired comparisons test because the samples are not independent.

C)   A difference in means test with pooled variances from the two samples.

Q4. An analyst for the entertainment industry theorizes that betas for most firms in the industry are higher after September 11, 2001. She sampled 31 firms comparing their betas for the one-year period before and after this date. Based on this sample, she found that the mean differences in betas were 0.19, with a sample standard deviation of 0.11. Her null hypothesis is that the betas are the same before and after September 11. Based on the results of her sample, can we reject the null hypothesis at a 5% significance level and why? Null is:

A)   not rejected. The critical value exceeds the t-value by 7.58.

B)   rejected. The t-value exceeds the critical value by 5.67.

C)   rejected. The t-value exceeds the critical value by 7.58.

答案和详解如下：

Q2. Joe Sutton is evaluating the effects of the 1987 market decline on the volume of trading. Specifically, he wants to test whether the decline affected trading volume. He selected a sample of 500 companies and collected data on the total annual volume for one year prior to the decline and for one year following the decline. What is the set of hypotheses that Sutton is testing?

A)   H₀: µ_d = µ_d0 versus H_a: µ_d ≠ µ_d0.

B)   H₀: µ_d ≠ µ_d0 versus H_a: µ_d = µ_d0.

C)   H₀: µ_d = µ_d0 versus H_a: µ_d > µ_d0.

Correct answer is A)

This is a paired comparison because the sample cases are not independent (i.e., there is a before and an after for each stock). Note that the test is two-tailed, t-test.

Q3. An analyst wants to determine whether the monthly returns on two stocks over the last year were the same or not. What test should she use if she is willing to assume that the returns are normally distributed?

A)   A difference in means test only if the variances of monthly returns are equal for the two stocks.

B)   A paired comparisons test because the samples are not independent.

C)   A difference in means test with pooled variances from the two samples.

Correct answer is B)

A paired comparisons test must be used. The difference in means test requires that the samples be independent. Portfolio theory teaches us that returns on two stocks over the same time period are unlikely to be independent since both have some systematic risk.

Q4. An analyst for the entertainment industry theorizes that betas for most firms in the industry are higher after September 11, 2001. She sampled 31 firms comparing their betas for the one-year period before and after this date. Based on this sample, she found that the mean differences in betas were 0.19, with a sample standard deviation of 0.11. Her null hypothesis is that the betas are the same before and after September 11. Based on the results of her sample, can we reject the null hypothesis at a 5% significance level and why? Null is:

A)   not rejected. The critical value exceeds the t-value by 7.58.

B)   rejected. The t-value exceeds the critical value by 5.67.

C)   rejected. The t-value exceeds the critical value by 7.58.

Correct answer is C)         

The t-statistic for paired differences:

t = (d – u_{d 0}) / s_d and s_d = s_d / √n

t = 9.62 from a table with 30 df, the critical value = 2.042

Q6. If the null hypothesis is H₀: ρ ≤ 0, what is the appropriate alternative hypothesis?

A)     H_a: ρ ≠ 0.

B)     H_a: ρ > 0.

C)     H_a: ρ < 0.

Correct answer is B)

The alternative hypothesis must include the possible outcomes the null does not.

Q7. In order to test if the mean IQ of employees in an organization is greater than 100, a sample of 30 employees is taken and the sample value of the computed test statistic, t_n-1 = 1.2. If you choose a 5% significance level you should:

A)   fail to reject the null hypothesis and conclude that the population mean is not greater than 100.

B)   reject the null hypothesis and conclude that the population mean is greater than 100.

C)   fail to reject the null hypothesis and conclude that the population mean is greater than 100.

Correct answer is A)

At a 5% significance level, the critical t-statistic using the Student’s t distribution table for a one-tailed test and 29 degrees of freedom (sample size of 30 less 1) is 1.699 (with a large sample size the critical z-statistic of 1.645 may be used). Because the critical t-statistic is greater than the calculated t-statistic, meaning that the calculated t-statistic is not in the rejection range, we fail to reject the null hypothesis and we conclude that the population mean is not significantly greater than 100.

Q8. In order to test whether the mean IQ of employees in an organization is greater than 100, a sample of 30 employees is taken and the sample value of the computed test statistic, t_n-1 = 3.4. The null and alternative hypotheses are:

A)H₀: μ ≤ 100; H_a: μ > 100.

B)H₀: μ = 100; H_a: μ ≠ 100.

C)H₀: X ≤ 100; H_a: X > 100.

Correct answer is A)

The null hypothesis is that the theoretical mean is not significantly different from zero. The alternative hypothesis is that the theoretical mean is greater than zero.

Q9. In a two-tailed test of a hypothesis concerning whether a population mean is zero, Jack Olson computes a t-statistic of 2.7 based on a sample of 20 observations where the distribution is normal. If a 5% significance level is chosen, Olson should:

A)   reject the null hypothesis and conclude that the population mean is not significantly different from zero.

B)   fail to reject the null hypothesis that the population mean is not significantly different from zero.

C)   reject the null hypothesis and conclude that the population mean is significantly different from zero.

Correct answer is C)

At a 5% significance level, the critical t-statistic using the Student’s t-distribution table for a two-tailed test and 19 degrees of freedom (sample size of 20 less 1) is ± 2.093 (with a large sample size the critical z-statistic of 1.960 may be used). Because the critical t-statistic of 2.093 is to the left of the calculated t-statistic of 2.7, meaning that the calculated t-statistic is in the rejection range, we reject the null hypothesis and we conclude that the population mean is significantly different from zero.

Q10. What kind of test is being used for the following hypothesis and what would a z-statistic of 1.68 tell us about a hypothesis with the appropriate test and a level of significance of 5%, respectively?

H₀: B ≤ 0

H_A: B > 0

A)   One-tailed test; fail to reject the null.

B)   Two-tailed test; fail to reject the null.

C)   One-tailed test; reject the null.

Correct answer is C)

The way the alternative hypothesis is written you are only looking at the right side of the distribution. You are only interested in showing that B is greater than 0. You don't care if it is less than zero. For a one-tailed test at the 5% level of significance, the critical z value is 1.645. Since the test statistic of 1.68 is greater than the critical value we would reject the null hypothesis.

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