Quantitative Methods for Valuation
Reading 11: Hypothesis Testing
1. Which of the following statements is most likely regarding a one-tailed hypothesis test where the alternate hypothesis is that the population mean is greater than the hypothesized mean?
A. Reject null when the test statistic is greater than the negative critical value.
B. Reject null when the test statistic is lower than the negative critical value.
C. Reject null when the test statistic is greater than the positive critical value.
2. Which of the following statements is least likely?
A. Decreasing the significance level increases the power of a hypothesis test.
B. Increasing the sample size increases the power of a hypothesis test.
C. Reducing the size of the “fail-to-reject-the-null region” increases the power of a hypothesis test.
3. Which of the following statements regarding the p-value is correct?
A. It is the maximum level of significance at which the null hypothesis is not rejected.
B. If the p-value is lower than the significance level, the null is accepted.
C. If the p-value is greater than the significance level, we fail to reject the null hypothesis.
4. A smaller sample size:
A. Reduces the size of the “fail-to-reject-the-null” region, reduces the probability of a Type II error, and increases the power of a test.
B. Increases the size of the “fail-to-reject-the-null” region, increases the probability of a Type II error, and reduces the power of a test.
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C. Reduces the size of the “fail-to-reject-the-null” region, increases the probability of a Type II error, and reduces the power of a test.
5. To use the t-test with a pooled variance:
A. The samples must be dependent.
B. The variances must be assumed unequal.
C. The populations must at least approximately follow the normal distribution.
6. To use the paired comparisons test:
A. The samples must be dependent.
B. The variances must be assumed equal.
C. The chi-square stat must be calculated.
7. Which feature distinguishes the F-test from other hypothesis tests?
A. The rejection region always lies in one tail.
B. The rejection region always lies in the lower tail.
C. There are no degrees of freedom for the F-test.
8. Which stat is most likely used to test the equality of the variances of two normally distributed, independent populations?
A. Chi-square stat B. t-stat
C. F-stat
9. Failure to reject an incorrect null hypothesis is a(n):
A. Correct decision
B. Incorrect decision (Type II error) C. Incorrect decision (Type I error)
10. The mean monthly return on a security over the past 32 weeks is 0.05% and the standard deviation of returns is 0.15%. The analyst wants to test whether this security’s returns have been positive at the 5% level of significance. Based on this information which of the following is most likely?
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Hypothesis Test Null hypothesis A. One-tailed test H0: µ0 = 0
B. Two-tailed test H0: µ0 > 0 C. One-tailed test H0: µ0 ≤ 0
11. Suppose the p-value of a two-tailed hypothesis test is given as 0.02.
Would the analyst reject the null hypothesis at the 5% and at the 1%
level of significance?
0.05 0.01 A. Yes Yes B. No No C. Yes No
Use the following information to answer Questions 12 to 17:
A manufacturer claims that the life of its batteries is normally distributed with a mean of 20 hours. For a random sample of 64 batteries, it is observed that the average life of the batteries in the sample is 19 hours with a standard deviation of 3.25 hours. Determine whether the average life of batteries manufactured by this company is greater than 20 hours at the 5% significance level.
12. The null hypothesis for this test would be structured as:
A. H0: µ0 ≤ 20 B. H0: µ0 > 20 C. H0: µ0 ≥ 20
13. The alternate hypothesis for this test would be structured as:
A. Ha: µ0 ≤ 20 B. Ha: µ0 > 20 C. Ha: µ0 ≥ 20
14. This test is most likely an example of a:
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A. One-tailed t-test B. Two-tailed z-test C. Two-tailed t-test
15. The test-statistic for this hypothesis test equals:
A. −2.462 B. −2.84 C. +2.84
16. Assuming that the t-distribution is used for this hypothesis test, the critical value(s) is/are:
A. −1.67 B. +1.67
C. −2.00 and +2.00
17. The conclusion of this test is that at the 5% significance level:
A. We fail to reject the null hypothesis.
B. We reject the null hypothesis.
C. We accept the alternate hypothesis.
Use the following information to answer Questions 18 to 23:
Test the accuracy of a claim made by AB Associates that the investment strategies they follow result in standard deviation of monthly returns of 0.58%. Use the 5% level of significance. ZX performance data for the last 31 months has a standard deviation of 0.55%.
18. The null hypothesis for this test would be structured as:
A. H0: σ2 ≤ 0.003364 B. H0: σ2 = 0.00003364 C. H0: σ2 = 0.0058
19. The alternate hypothesis for this test would be structured as:
A. Ha: σ2 ≠ 0.0058
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B. Ha: σ2 ≠ 0.003364 C. Ha: σ2 ≠ 0.00003364 20. This test is an example of a:
A. One-tailed t-test
B. Two-tailed chi-square test C. Two-tailed t-test
21. The test-statistic for this hypothesis test equals:
A. 26.98 B. 27.87 C. 33.36
22. For this hypothesis test, the critical value(s) is/are:
A. 43.77
B. 16.79 and 46.98 C. −2.04 and +2.04
23. The conclusion of this test is that at the 5% significance level:
A. We fail to reject the null hypothesis.
B. We accept the null hypothesis.
C. We reject the null hypothesis.
24. Alexis is conducting research on the stock market of an emerging economy. She believes that the mean daily return on the market’s all-share index is statistically significantly different from zero. She
randomly selects 50 stocks that are traded on the country’s stock exchange and calculates their average daily return to be 0.3%. The index that comprises all the shares in the country has a daily standard deviation of 0.2%. At the 5% level of significance, Alexis would most likely:
A. Reject the null hypothesis, and conclude that the mean daily return is not statistically significantly different from zero
B. Fail to reject the null hypothesis, and conclude that the mean daily
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return is not statistically significantly different from zero.
C. Reject the null hypothesis, and conclude that the mean daily return is statistically significantly different from zero.
25. Donald wants to determine whether, on an average, a player scores more than 10 goals in a season of soccer. He selects 40 players in his sample and calculates a sample mean of 11 goals a season. Given a population standard deviation of 3 goals a season, at the 5% level of significance Donald should most likely:
A. Reject the null hypothesis, and conclude that a player scores more than 10 goals in a season of football.
B. Fail to reject the null hypothesis, and conclude that a player does not score more than 10 goals in a season of football.
C. Fail to reject the null hypothesis, and conclude that a player scores 10 goals or more in a season of football.
26. Timothy believes that Mark Johnson scores less than 50 runs on average every time he plays a cricket match. He selects a sample of 30 games and calculates a sample mean of 48 runs.
The standard deviation of Johnson’s scores throughout his entire career is given as 9 runs. At the 10% level of significance, Timothy would most likely:
A. Reject the null hypothesis. He would conclude that Johnson scores 50 runs or more in any match.
B. Fail to reject the null hypothesis. He would conclude that Johnson does not score less than 50 runs on average.
C. Reject the null hypothesis, and conclude that Johnson scores 50 runs or more in any match.
27. Mike is testing whether a population’s mean equals 25. He computes a t-stat of 1.71 based on a sample of 20 observations assuming that the population is normally distributed. At the 10% level of significance, Mike should most likely:
A. Reject the null hypothesis, and conclude that the population mean is not significantly different from 25.
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B. Reject the null hypothesis, and conclude that the population mean is significantly different from 25.
C. Fail to reject the null hypothesis that the population mean equals 25.
28. Zona is evaluating whether the average age of employees at a
particular organization is greater than 30 years. She draws a sample of 100 employees and calculates their average age to be 31.1 years with a sample standard deviation of 6.3 years. At the 5% level of significance, Zona should most likely:
A. Reject the null hypothesis, and conclude that the average age is greater than 30 years.
B. Fail to reject the null hypothesis, and conclude that the average age is less than or equal to 30 years.
C. Fail to reject the null hypothesis, and conclude that the average age is greater than or equal to 30 years.
29. Carlos wants to test the hypothesis that a player scores 12 goals in a season of soccer. He selects 40 players and determines that these players average 13 goals a season. Assuming a population standard deviation of 4 goals, at the 10% level of significance Carlos should most likely:
A. Reject the null hypothesis, and conclude that the number of goals scored by a player in a season of football is different from 12.
B. Fail to reject the null hypothesis, and conclude that the number of goals scored by a player in a season of football is different from 12.
C. Fail to reject the null hypothesis, and conclude that the number of goals scored by a player in a season of football is 12.
Use the following information to answer Questions 30 to 33:
Alex is deciding between investing in 10 stocks with the highest dividend yields on the FTSE and investing in all 100 stocks in the FTSE 100. He calculated the following information based on data from the 30 years between 1980 and 2009:
Strategy Mean
Return
Standard Deviation
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10 stocks with highest dividend yields
14.34% 17.40%
Buy-and-hold strategy on the FTSE 100
11.25% 15.42%
Difference 3.09% 5.52%
Alex is trying to determine whether the mean difference between the two strategies equals zero.
30. Which of the following most accurately describes the null and alternative hypothesis for this test?
A. Ha: µd > 0; H0: µd ≤ 0 B. H0: µd = 0; Ha: µd ≠ 0 C. H0: µd ≠ 0; Ha: µd = 0
31. The test statistic stat for testing this hypothesis is most likely:
A. z-stat B. t-stat C. F-stat
32. The rejection points for the hypothesis test at 10% level of significance are closest to:
A. t < −1.6973 and t > 1.6973 B. t < −1.3114 and t > 1.3114 C. t < −1.6991 and t > 1.6991
33. Based on the calculated t-stat, Alex will most likely:
A. Fail to reject the null hypothesis, and conclude that the difference in mean returns is not statistically significant.
B. Reject the null hypothesis, and conclude that the difference in mean returns is statistically significant.
C. Reject the null hypothesis, and conclude that the difference in mean returns is not statistically significant.
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Use the following information to answer Questions 34 to 37:
Brian wants to investigate whether the population variance of returns on the FTSE 100 changed after the financial crisis of 2008. He gathered the following monthly returns information for 2 years before and after the crisis:
Mean Monthly
Return Variance of
Returns Before the financial
crisis 1.932% 20.436
After the financial crisis 1.214% 9.826
34. Which of the following most accurately describes the null and alternative hypothesis for this study?
A. H0: σ2Before = σ2After; Ha: σ2Before ≠ σ2After B. H0: σ2Before ≠ σ2After; Ha: σ2Before = σ2After C. H0: σ2Before ≥ σ2After; Ha: σ2Before < σ2After
35. The test statistic for conducting this hypothesis test is most likely:
A. z-stat B. t-stat C. F-stat
36. The critical value(s) for this test at the 5% level of significance is (are) closest to:
A. 2.01 B. 2.31 C. ±35.17
37. Brian should most likely:
A. Reject the null hypothesis, and conclude that the population variance of returns is the same in the pre and post-crisis periods.
B. Fail to reject the null hypothesis, and conclude that the population
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variance of returns is not the same in the pre and post-crisis periods.
C. Fail to reject the null hypothesis, and conclude that the population variance of returns is the same in the pre- and post crisis periods.
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Reading 11: Hypothesis Testing
1. Which of the following statements is most likely regarding a one-tailed hypothesis test where the alternate hypothesis is that the population mean is greater than the hypothesized mean?
A. Reject null when the test statistic is greater than the negative critical value.
B. Reject null when the test statistic is lower than the negative critical value.
C. Reject null when the test statistic is greater than the positive critical value.
Answer: C
When trying to determine whether the population mean is greater than the hypothesized mean, you reject the null hypothesis when the test stat is greater than the positive critical value.
2. Which of the following statements is least likely?
A. Decreasing the significance level increases the power of a hypothesis test.
B. Increasing the sample size increases the power of a hypothesis test.
C. Reducing the size of the “fail-to-reject-the-null region” increases the power of a hypothesis test.
Answer: A
Decreasing the significance level increases the probability of a Type II error, reduces the probability of a Type I error, and therefore, reduces the power of a test.
3. Which of the following statements regarding the p-value is correct?
A. It is the maximum level of significance at which the null hypothesis is not rejected.
B. If the p-value is lower than the significance level, the null is accepted.
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C. If the p-value is greater than the significance level, we fail to reject the null hypothesis.
Answer: C
The p-value represents the lowest level of significance at which the null hypothesis can be rejected. If the p-value is greater than the
significance level, we fail to reject the null hypothesis.
4. A smaller sample size:
A. Reduces the size of the “fail-to-reject-the-null” region, reduces the probability of a Type II error, and increases the power of a test.
B. Increases the size of the “fail-to-reject-the-null” region, increases the probability of a Type II error, and reduces the power of a test.
C. Reduces the size of the “fail-to-reject-the-null” region, increases the probability of a Type II error, and reduces the power of a test.
Answer: B
A smaller sample size increases the standard error. Therefore, it increases the size of the fail-to-reject-the-null region, increases the probability of a Type II error, and reduces the power of a hypothesis test.
5. To use the t-test with a pooled variance:
A. The samples must be dependent.
B. The variances must be assumed unequal.
C. The populations must at least approximately follow the normal distribution.
Answer: C
A t-test is used with a pooled variance when the populations are at least approximately normally distributed, the samples are
independent, and their variances are assumed equal.
6. To use the paired comparisons test:
A. The samples must be dependent.
B. The variances must be assumed equal.
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C. The chi-square stat must be calculated.
Answer: A
The paired comparisons test is used to test the mean of the difference between observations of two samples when they are dependent.
7. Which feature distinguishes the F-test from other hypothesis tests?
A. The rejection region always lies in one tail.
B. The rejection region always lies in the lower tail.
C. There are no degrees of freedom for the F-test.
Answer: A
For an F-test, whether it is one-tailed or two-tailed, the rejection region always lies in the upper tail.
8. Which stat is most likely used to test the equality of the variances of two normally distributed, independent populations?
A. Chi-square stat B. t-stat
C. F-stat Answer: C
When testing the equality of the variances of two populations, the F-test is used.
9. Failure to reject an incorrect null hypothesis is a(n):
A. Correct decision
B. Incorrect decision (Type II error) C. Incorrect decision (Type I error) Answer: B
A Type II error is one where we fail to reject the null hypothesis when it is false.
10. The mean monthly return on a security over the past 32 weeks is 0.05% and the standard deviation of returns is 0.15%. The analyst wants to test whether this security’s returns have been positive at the
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5% level of significance. Based on this information which of the following is most likely?
Hypothesis Test Null hypothesis A. One-tailed test H0: µ0 = 0
B. Two-tailed test H0: µ0 > 0 C. One-tailed test H0: µ0 ≤ 0 Answer: C
Since we are testing whether security returns are greater than zero (the alternate hypothesis), the null hypothesis for this test would be that security returns are lower than or equal to zero. This is an example of a one-tailed test.
11. Suppose the p-value of a two-tailed hypothesis test is given as 0.02.
Would the analyst reject the null hypothesis at the 5% and at the 1%
level of significance?
0.05 0.01 A. Yes Yes B. No No C. Yes No Answer: C
The p-value represents the lowest level of significance at which the null hypothesis can be rejected. A p-value of 0.02 means that the null
hypothesis can be rejected at the 5% significance level, but not at the 1% significance level.
Use the following information to answer Questions 12 to 17:
A manufacturer claims that the life of its batteries is normally distributed with a mean of 20 hours. For a random sample of 64 batteries, it is
observed that the average life of the batteries in the sample is 19 hours with a standard deviation of 3.25 hours. Determine whether the average life of batteries manufactured by this company is greater than 20 hours at the 5% significance level.
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12. The null hypothesis for this test would be structured as:
A. H0: µ0 ≤ 20 B. H0: µ0 > 20 C. H0: µ0 ≥ 20 Answer: A
13. The alternate hypothesis for this test would be structured as:
A. Ha: µ0 ≤ 20 B. Ha: µ0 > 20 C. Ha: µ0 ≥ 20 Answer: B
The statement whose validity we are evaluating is whether the average life on the manufacturer’s batteries is greater than 20 hours.
14. This test is most likely an example of a:
A. One-tailed t-test B. Two-tailed z-test C. Two-tailed t-test Answer: A
The sample size is large and this is a one-tailed test. The population variance is not known, so the t-test must be used.
15. The test-statistic for this hypothesis test equals:
A. −2.462 B. −2.84 C. +2.84 Answer: A
Test-statistic = (19 − 20) / (3.25/8) = −2.462
16. Assuming that the t-distribution is used for this hypothesis test, the critical value(s) is/are:
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A. −1.67 B. +1.67
C. −2.00 and +2.00 Answer: B
Since we are testing whether the average life of the batteries is
greater than 20 hours, the relevant critical value for this test equals +1.67 at the 5% significance level.
17. The conclusion of this test is that at the 5% significance level:
A. We fail to reject the null hypothesis.
B. We reject the null hypothesis.
C. We accept the alternate hypothesis.
Answer: A
Since the test is statistic lower than the critical t-value for this test, we fail to reject the null hypothesis.
Use the following information to answer Questions 18 to 23:
Test the accuracy of a claim made by AB Associates that the investment strategies they follow result in standard deviation of monthly returns of 0.58%. Use the 5% level of significance. ZX performance data for the last 31 months has a standard deviation of 0.55%.
18. The null hypothesis for this test would be structured as:
A. H0: σ2 ≤ 0.003364 B. H0: σ2 = 0.00003364 C. H0: σ2 = 0.0058 Answer: B
The status quo is that the variance of returns equals (0.0058)2 or 0.00003364.
19. The alternate hypothesis for this test would be structured as:
A. Ha: σ2 ≠ 0.0058
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B. Ha: σ2 ≠ 0.003364 C. Ha: σ2 ≠ 0.00003364 Answer: C
The statement whose validity we are trying to establish is whether the variance of monthly returns is different from 0.00003364.
20. This test is an example of a:
A. One-tailed t-test
B. Two-tailed chi-square test C. Two-tailed t-test
Answer: B
A two-tailed chi-square test is used to determine whether the variance
A two-tailed chi-square test is used to determine whether the variance