Table of Contents
CFA Level 1 – Quantitative Methods, Session 2 – Reading 11
(Notes, Practice Questions, Sample Questions)
1. Which of the following statements about testing a hypothesis using a Z-test is least accurate?
A)If the calculated Z-statistic lies outside the critical Z-statistic range, the null hypothesis can be rejected.
B)The calculated Z-statistic determines the appropriate significance level to use.
C)The confidence interval for a two-tailed test of a population mean at the 5% level of significance is that the sample mean falls between ±1.96 σ/√n of the null hypothesis value
[Explanation]: (B) The significance level is chosen before the test so the calculated Z-statistic can be compared to an appropriate critical value
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2. Susan Bellows is comparing the return on equity for two industries. She is convinced that the return on equity for the discount retail industry (DR) is greater than that of the luxury retail (LR) industry. What are the hypotheses for a test of her comparison of return on equity?
A)H0: µDR = µLR versus Ha: µDR ≠ µLR.
B)H0: µDR ≤ µLR versus Ha: µDR > µLR.
C)H0: µDR = µLR versus Ha: µDR < µLR
[Explanation]: (B) The alternative hypothesis is determined by the theory or the belief. The researcher specifies the null as the hypothesis that she wishes to reject (in favor of the alternative). Note that this is a one-sided alternative because of the “greater than” belief
3. In the process of hypothesis testing, what is the proper order for these steps?
A)State the hypotheses. Specify the level of significance. Collect the sample and calculate the test statistics. Make a decision.
B)Collect the sample and calculate the sample statistics. State the hypotheses. Specify the level of significance. Make a decision.
C)Specify the level of significance. State the hypotheses. Make a decision. Collect the sample and calculate the sample statistics
[Explanation]: (A) The hypotheses must be established first. Then the test statistic is chosen and the level of significance is determined. Following these steps, the sample is collected, the test statistic is calculated, and the decision is made
4. The first step in the process of hypothesis testing is:
A)to state the hypotheses.
B)selecting the test statistic.
C)the collection of the sample
[Explanation]: (A) The researcher must state the hypotheses prior to the collection and analysis of the data. More importantly, it is necessary to know the hypotheses before selecting the appropriate test statistic
5. Which of the following statements least describes the procedure for testing a hypothesis?
A)Compute the sample value of the test statistic, set up a rejection (critical) region, and make a decision.
B)Develop a hypothesis, compute the test statistic, and make a decision.
C)Select the level of significance, formulate the decision rule, and make a decision
[Explanation]: (A) Depending upon the author there can be as many as seven steps in hypothesis testing which are:
Stating the hypotheses.
Identifying the test statistic and its probability distribution.
Specifying the significance level.
Stating the decision rule.
Collecting the data and performing the calculations.
Making the statistical decision.
Making the economic or investment decision
6. Which of the following is the correct sequence of events for testing a hypothesis?
A)State the hypothesis, select the level of significance, formulate the decision rule, compute the test statistic, and make a decision.
B)State the hypothesis, select the level of significance, compute the test statistic, formulate the decision rule, and make a decision.
C)State the hypothesis, formulate the decision rule, select the level of significance, compute the test statistic, and make a decision
[Explanation]: (A) Depending upon the author there can be as many as seven steps in hypothesis testing which are:
Stating the hypotheses.
Identifying the test statistic and its probability distribution.
Specifying the significance level.
Stating the decision rule.
Collecting the data and performing the calculations.
Making the statistical decision.
Making the economic or investment decision
7. Which of the following statements about hypothesis testing is most accurate?
A)If you can disprove the null hypothesis, then you have proven the alternative hypothesis.
B)The power of a test is one minus the probability of a Type I error.
C)The probability of a Type I error is equal to the significance level of the test.
[Explanation]: (C) The probability of getting a test statistic outside the critical value(s) when the null is true is the level of significance and is the probability of a Type I error. The power of a test is 1 minus the probability of a Type II error. Hypothesis testing does not prove a hypothesis, we either reject the null or fail to reject it
8. An analyst conducts a two-tailed z-test to determine if small cap returns are significantly different from 10%. The sample size was 200. The computed z-statistic is 2.3. Using a 5% level of significance, which statement is most accurate?
A)You cannot determine what to do with the information given.
B)Reject the null hypothesis and conclude that small cap returns are significantly different from 10%.
C)Fail to reject the null hypothesis and conclude that small cap returns are close enough to 10% that we cannot say they are significantly different from 10%
[Explanation]: (B) At the 5% level of significance the critical z-statistic for a two-tailed test is 1.96 (assuming a large sample size). The null hypothesis is H0: x = 10%. The alternative hypothesis is HA: x ≠ 10%. Because the computed z-statistic is greater than the critical z-statistic (2.33 > 1.96), we reject the null hypothesis and we conclude that small cap returns are significantly different than 10%
9. An analyst conducts a two-tailed test to determine if mean earnings estimates are significantly different from reported earnings. The sample size is greater than 25 and the computed test statistic is 1.25. Using a 5% significance level, which of the following statements is most accurate?
A)The analyst should reject the null hypothesis and conclude that the earnings estimates are significantly different from reported earnings.
B)The analyst should fail to reject the null hypothesis and conclude that the earnings estimates are not significantly different from reported earnings.
C)To test the null hypothesis, the analyst must determine the exact sample size and calculate the degrees of freedom for the test
[Explanation]: (B) The null hypothesis is that earnings estimates are equal to reported earnings. To reject the null hypothesis, the calculated test statistic must fall outside the two critical values. IF the analyst tests the null hypothesis with a z-statistic, the crtical values at a 5% confidence level are ±1.96. Because the calculated test statistic, 1.25, lies between the two critical values, the analyst should fail to reject the null hypothesis and conclude that earnings estimates are not significantly different from reported earnings. If the analyst uses a t-statistic, the upper critical value will be even greater than 1.96, never less, so even without the exact degrees of freedom the analyst knows any t-test would fail to reject the null
10. Given the following hypothesis:
The null hypothesis is H0 : µ = 5
The alternative is H1 : µ ≠ 5
The mean of a sample of 17 is 7
The population standard deviation is 2.0
What is the calculated z-statistic?
A)4.00.
B)4.12.
C)8.00.
[Explanation]: (B) The z-statistic is calculated by subtracting the hypothesized parameter from the parameter that has been estimated and dividing the difference by the standard error of the sample statistic. Here, the test statistic = (sample mean − hypothesized mean) / (population standard deviation / (sample size)1/2 = (X − μ) / (σ / n1/2) = (7 − 5) / (2 / 171/2) = (2) / (2 / 4.1231) = 4.12
11. 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
HA: 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
[Explanation]: (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
12. 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
[Explanation]: (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
13. 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, tn-1 = 3.4. The null and alternative hypotheses are:
A)H0: µ = 100; Ha: µ ≠ 100.
B)H0: µ ≤ 100; Ha: µ > 100.
C)H0: X ≤ 100; Ha: X > 100
[Explanation]: (B) 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
14. 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, tn-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
[Explanation]: (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
15. If the null hypothesis is H0: ρ ≤ 0, what is the appropriate alternative hypothesis?
A)Ha: ρ ≠ 0.
B)Ha: ρ > 0.
C)Ha: ρ < 0
[Explanation]: (B) The alternative hypothesis must include the possible outcomes the null does not
16. Jo Su believes that there should be a negative relation between returns and systematic risk. She intends to collect data on returns and systematic risk to test this theory. What is the appropriate alternative hypothesis?.
A)Ha: ρ > 0.
B)Ha: ρ ≠ 0.
C)Ha: ρ < 0
[Explanation]: (C) The alternative hypothesis is determined by the theory or the belief. The researcher specifies the null as the hypothesis that she wishes to reject (in favor of the alternative). The theory in this case is that the correlation is negative
17. Jill Woodall believes that the average return on equity in the retail industry, µ, is less than 15%. What are the null (H0) and alternative (Ha) hypotheses for her study?
A)H0: µ ≤ 0.15 versus Ha: µ > 0.15.
B)H0: µ < 0.15 versus Ha: µ ≥ 0.15.
C)H0: µ ≥ 0.15 versus Ha: µ < 0.15.
[Explanation]: (C) This is a one-sided alternative because of the “less than” belief
18. Brian Ci believes that the average return on equity in the airline industry, µ, is less than 5%. What are the appropriate null (H0) and alternative (Ha) hypotheses to test this belief?
A)H0: µ ≥ 0.05 versus Ha: µ < 0.05.
B)H0: µ < 0.05 versus Ha: µ ≥ 0.05.
C)H0: µ < 0.05 versus Ha: µ > 0.05.
[Explanation]: (A) The alternative hypothesis is determined by the theory or the belief. The researcher specifies the null as the hypothesis that he wishes to reject (in favor of the alternative). Note that this is a one-sided alternative because of the “less than” belief
19. George Appleton believes that the average return on equity in the amusement industry, µ, is greater than 10%. What is the null (H0) and alternative (Ha) hypothesis for his study?
A)H0: > 0.10 versus Ha: ≤ 0.10.
B)H0: > 0.10 versus Ha: < 0.10.
C)H0: ≤ 0.10 versus Ha: > 0.10.
[Explanation]: (C) The alternative hypothesis is determined by the theory or the belief. The researcher specifies the null as the hypothesis that he wishes to reject (in favor of the alternative). Note that this is a one-sided alternative because of the “greater than” belief
20. James Ambercrombie believes that the average return on equity in the utility industry, µ, is greater than 10%. What are the null (H0) and alternative (Ha) hypotheses for his study?
A)H0: µ ≤ 0.10 versus Ha: µ > 0.10.
B)H0: µ < 0.10 versus Ha: µ > 0.10.
C)H0: µ > 0.10 versus Ha: µ < 0.10
[Explanation]: (A) This is a one-sided alternative because of the “greater than” belief.
21. Which one of the following is the most appropriate set of hypotheses to use when a researcher is trying to demonstrate that a return is greater than the risk-free rate? The null hypothesis is framed as a:
A)less than statement and the alternative hypothesis is framed as a greater than or equal to statement.
B)less than or equal to statement and the alternative hypothesis is framed as a greater than statement.
C)greater than statement and the alternative hypothesis is framed as a less than or equal to statement.
[Explanation]: (B) If a researcher is trying to show that a return is greater than the risk-free rate then this should be the alternative hypothesis. The null hypothesis would then take the form of a less than or equal to statement
22. Which one of the following best characterizes the alternative hypothesis? The alternative hypothesis is usually the:
A)hoped-for outcome.
B)hypothesis to be proved through statistical testing.
C)hypothesis that is accepted after a statistical test is conducted
[Explanation]: (A) The alternative hypothesis is typically the hypothesis that a researcher hopes to support after a statistical test is carried out. We can reject or fail to reject the null, not ‘prove’ a hypothesis
23. Jill Woodall believes that the average return on equity in the retail industry, µ, is less than 15%. What is null (H0) and alternative (Ha) hypothesis for her study?
A)H0: µ ≥ 0.15 versus Ha: µ < 0.15.
B)H0: µ < 0.15 versus Ha: µ = 0.15.
C)H0: µ = 0.15 versus Ha: µ ≠ 0.15
[Explanation]: (A) This is a one-sided alternative because of the “less than” belief. We expect to reject the null
24. James Ambercrombie believes that the average return on equity in the utility industry, µ, is greater than 10%. What is null (H0) and alternative (Ha) hypothesis for his study?
A)H0: µ = 0.10 versus Ha: µ ≠ 0.10.
B)H0: µ ≥ 0.10 versus Ha: µ < 0.10.
C)H0: µ ≤ 0.10 versus Ha: µ > 0.10
[Explanation]: (C) This is a one-sided alternative because of the “greater than” belief. We expect to reject the null
25. What is the most common formulation of null and alternative hypotheses?
A)Less than for the null and greater than for the alternative.
B)Equal to for the null and not equal to for the alternative.
C)Greater than or equal to for the null and less than for the alternative
[Explanation]: (B) The most common set of hypotheses will take the form of an equal to statement for the null and a not equal to statement for the alternative
26. Robert Patterson, an options trader, believes that the return on options trading is higher on Mondays than on other days. In order to test his theory, he formulates a null hypothesis. Which of the following would be an appropriate null hypothesis? Returns on Mondays are:
A)not greater than returns on other days.
B)greater than returns on other days.
C)less than returns on other days
[Explanation]: (A) An appropriate null hypothesis is one that the researcher wants to reject. If Patterson believes that the returns on Mondays are greater than on other days, he would like to reject the hypothesis that the opposite is true–that returns on Mondays are not greater than returns on other days
27. For a two-tailed test of hypothesis involving a z-distributed test statistic and a 5% level of significance, a calculated z-statistic of 1.5 indicates that:
A)the null hypothesis cannot be rejected.
B)the null hypothesis is rejected.
C)the test is inconclusive.
[Explanation]: (A) For a two-tailed test at a 5% level of significance the calculated z-statistic would have to be greater than the critical z value of 1.96 for the null hypothesis to be rejected
28. A pitching machine is calibrated to deliver a fastball at a speed of 98 miles per hour. Every day, a technician samples the speed of twenty-five fastballs in order to determine if the machine needs adjustment. Today, the sample showed a mean speed of 99 miles per hour with a standard deviation of 1.75 miles per hour. Assume the population is normally distributed. At a 95% confidence level, what is the t-value in relation to the critical value?
A)The critical value exceeds the t-value by 1.3 standard deviations.
B)The t-value exceeds the critical value by 1.5 standard deviations.
C)The t-value exceeds the critical value by 0.8 standard deviations
[Explanation]: (C) t = (99 – 98) / (1.75 / √25) = 2.86. The critical value for a two-tailed test at the 95% confidence level with 24 degrees of freedom is ±2.06 standard deviations. Therefore, the t-value exceeds the critical value by 0.8 standard deviations
29. Ron Jacobi, manager with the Toulee Department of Natural Resources, is responsible for setting catch-and-release limits for Lake Norby, a large and popular fishing lake. For the last two months he has been sampling to determine whether the average length of Northern Pike in the lake exceeds 18 inches (using a significance level of 0.05). Assume that the p-value is 0.08. In concluding that the average size of the fish exceeds 18 inches, Jacobi:
A)makes a Type I error.
B)makes a Type II error.
C)is correct
[Explanation]: (A) This statement is an example of a Type I error, or rejection of a hypothesis when it is actually true (also known as the significance level of the test). Here, Ho: μ = 18 inches and Ha: μ > 18 inches. When the p-value is greater than the significance level (0.08 > 0.05), we should fail to reject the null hypothesis. Since Jacobi rejected Ho when it was true, he made a Type 1 error.
The other statements are incorrect. Type II errors occur when you fail to reject a hypothesis when it is actually false (also known as the power of the test)
30. Kyra Mosby, M.D., has a patient who is complaining of severe abdominal pain. Based on an examination and the results from laboratory tests, Mosby states the following diagnosis hypothesis: Ho: Appendicitis, HA: Not Appendicitis. Dr. Mosby removes the patient’s appendix and the patient still complains of pain. Subsequent tests show that the gall bladder was causing the problem. By taking out the patient’s appendix, Dr. Mosby:
A)made a Type II error.
B)is correct.
C)made a Type I error
[Explanation]: (A) This statement is an example of a Type II error, which occurs when you fail to reject a hypothesis when it is actually false (also known as the power of the test).
The other statements are incorrect. A Type I error is the rejection of a hypothesis when it is actually true (also known as the significance level of the test)
31. Which of the following statements about hypothesis testing is least accurate?
A)The null hypothesis is a statement about the value of a population parameter.
B)If the alternative hypothesis is Ha: µ > µ0, a two-tailed test is appropriate.
C)A Type II error is failing to reject a false null hypothesis.
[Explanation]: (B) The hypotheses are always stated in terms of a population parameter. Type I and Type II are the two types of errors you can make – reject a null hypothesis that is true or fail to reject a null hypothesis that is false. The alternative may be one-sided (in which case a > or < sign is used) or two-sided (in which case a ≠ is used)
32. Which of the following statements about hypothesis testing is most accurate? A Type I error is the probability of:
A)failing to reject a false hypothesis.
B)rejecting a true alternative hypothesis.
C)rejecting a true null hypothesis
[Explanation]: (C) The Type I error is the error of rejecting the null hypothesis when, in fact, the null is true
33. Which of the following statements about hypothesis testing is least accurate?
A)A Type I error is the probability of rejecting the null hypothesis when the null hypothesis is false.
B)The significance level is the probability of making a Type I error.
C)A Type II error is the probability of failing to reject a null hypothesis that is not true.
[Explanation]: (A) A Type I error is the probability of rejecting the null hypothesis when the null hypothesis is true
34. John Jenkins, CFA, is performing a study on the behavior of the mean P/E ratio for a sample of small-cap companies. Which of the following statements is most accurate?
A)One minus the confidence level of the test represents the probability of making a Type II error.
B)The significance level of the test represents the probability of making a Type I error.
C)A Type I error represents the failure to reject the null hypothesis when it is, in truth, false.
[Explanation]: (B) A Type I error is the rejection of the null when the null is actually true. The significance level of the test (alpha) (which is one minus the confidence level) is the probability of making a Type I error. A Type II error is the failure to reject the null when it is actually false
35. A Type II error:
A)fails to reject a false null hypothesis.
B)fails to reject a true null hypothesis.
C)rejects a true null hypothesis.
[Explanation]: (A) A Type II error is defined as accepting the null hypothesis when it is actually false. The chance of making a Type II error is called beta risk
36. If we fail to reject the null hypothesis when it is false, what type of error has occured?
A)Type II.
B)Type III.
C)Type I.
[Explanation]: (A) A Type II error is defined as failing to reject the null hypothesis when it is actually false
37. Abby Ness is an analyst for a firm that specializes in evaluating firms involved in mineral extraction. Ness believes that the earnings of copper extracting firms are more volatile than those of bauxite extraction firms. In order to test this, Ness examines the volatility of returns for 31 copper firms and 25 bauxite firms. The standard deviation of earnings for copper firms was $2.69, while the standard deviation of earnings for bauxite firms was $2.92. Ness’s Null Hypothesis is σ12 = σ22. Based on the samples, can we reject the null hypothesis at a 95% confidence level using an F-statistic and why? Null is:
A)rejected. The F-value exceeds the critical value by 0.849.
B)not rejected. The critical value exceeds the F-value by 0.71.
C)rejected. The F-value exceeds the critical value by 0.71.
[Explanation]: (B) F = s12 / s22 = $2.922 / $2.692 = 1.18
From an F table, the critical value with numerator df = 24 and denominator df = 30 is 1.89
38. In order to test if Stock A is more volatile than Stock B, prices of both stocks are observed to construct the sample variance of the two stocks. The appropriate test statistics to carry out the test is the:
A)Chi-square test.
B)t test.
C)F test.
[Explanation]: (C) The F test is used to test the differences of variance between two samples
39. Which of the following statements about parametric and nonparametric tests is least accurate?
A)The test of the difference in means is used when you are comparing means from two independent samples.
B)Nonparametric tests rely on population parameters.
C)The test of the mean of the differences is used when performing a paired comparison
[Explanation]: (B) Nonparametric tests are not concerned with parameters; they make minimal assumptions about the population from which a sample comes. It is important to distinguish between the test of the difference in the means and the test of the mean of the differences. Also, it is important to understand that parametric tests rely on distributional assumptions, whereas nonparametric tests are not as strict regarding distributional properties
40. Which of the following statements about parametric and nonparametric tests is least accurate?
A)Parametric tests are most appropriate when a population is heavily skewed.
B)Nonparametric tests have fewer assumptions than parametric tests.
C)Nonparametric tests are often used in conjunction with parametric tests
[Explanation]: (A) For a distribution that is non-normally distributed, a nonparametric test may be most appropriate. A nonparametric test tends to make minimal assumptions about the population, while parametric tests rely on assumptions regarding the distribution of the population. Both kinds of tests are often used in conjunction with one another
