CFA Study Session 3: Quantitative Methods Application (practice questions)
In which of the following situations would a non-parametric test of a hypothesis most likely be used? A. The sample data are ranked according to magnitude. B. The sample data come from a normally distributed population. C. The test validity depends on many assumptions about the nature of the population.
A. The sample data are ranked according to magnitude.
A chi-square test is most appropriate for tests concerning: A. a single variance. B. differences between two population means with variances assumed to be equal. C. differences between two population means with variances assumed to not be equal.
A. a single variance.
When evaluating mean differences between two dependent samples, the most appropriate test is a: A. chi-square test. B. paired comparisons test. C. z-test.
A. chi-square test.
Which of the following should be used to test the difference between the variances of two normally distributed populations? A. t-test B. F-test C. Paired comparisons test
B. F-test
An analyst is examining the monthly returns for two funds over one year. Both funds' returns are non-normally distributed. To test whether the mean return of one fund is greater than the mean return of the other fund, the analyst can use: A. a parametric test only. B. a nonparametric test only. C. both parametric and nonparametric tests.
B. a nonparametric test only.
A fund manager reported a 2% mean quarterly return over the past ten years for its entire base of 250 client accounts that all follow the same investment strategy. A consultant employing the manager for 45 client accounts notes that their mean quarterly returns were 0.25% less over the same period. The consultant tests the hypothesis that the return disparity between the returns of his clients and the reported returns of the fund manager's 250 client accounts are significantly different from zero. Assuming normally distributed populations with unknown population variances, the most appropriate test statistic is: A. a paired comparisons t-test. B. a t-test of the difference between the two population means. C. an approximate t-test of mean differences between the two populations.
B. a t-test of the difference between the two population means.
A standard lookback call option on stock has a value at maturity equal to (Value of the stock at maturity - Minimum value of stock during the life of the option prior to maturity) or $0, whichever is greater. If the minimum value reached prior to maturity was $20.11 and the value of the stock at maturity is $23, for example, the call is worth $23 − $20.11 = $2.89. Briefly discuss how you might use Monte Carlo simulation in valuing a lookback call option.
calculate the average value of a stock over a simulation trial, then calculate the terminal values of the call, given the minimum value of the simulation trial
Although he knows security returns are not independent, a colleague makes the claim that because of the central limit theorem, if we diversify across a large number of investments, the portfolio standard deviation will eventually approach zero as n becomes large. Is he correct?
no
Which of the following is characteristic of the normal distribution? A. Asymmetry B. Kurtosis of 3 C. Definitive limits or boundaries
B. Kurtosis of 3
Assume that monthly returns are normally distributed with a mean of 1 percent and a sample standard deviation of 4 percent. The population standard deviation is unknown. Construct a 95 percent confidence interval for the sample mean of monthly returns if the sample size is 24.
(-.689, 2.689)
Find the area under the normal curve up to z = 0.36; that is, find P(Z ≤ 0.36). Interpret this value.
0.6406
You are examining the record of an investment newsletter writer who claims a 70 percent success rate in making investment recommendations that are profitable over a one-year time horizon. You have the one-year record of the newsletter's seven most recent recommendations. Four of those recommendations were profitable. If all the recommendations are independent and the newsletter writer's skill is as claimed, what is the probability of observing four or fewer profitable recommendations out of seven in total?
35.29%
Define the term "binomial random variable." Describe the types of problems for which the binomial distribution is used.
A random variable defined as a the number of successes in trials that produce one or two outcomes. Used to make probability statements regarding anything with successes or failures (two possible outcomes)
Assume that the equity risk premium is normally distributed with a population mean of 6 percent and a population standard deviation of 18 percent. Over the last four years, equity returns (relative to the risk-free rate) have averaged −2.0 percent. You have a large client who is very upset and claims that results this poor should never occur. Evaluate your client's concerns. A. Construct a 95 percent confidence interval around the population mean for a sample of four-year returns. B. What is the probability of a −2.0 percent or lower average return over a four-year period?
A. (-11.64, 23.64) B. .1867
The price of a stock at t = 0 is $208.25 and at t = 1 is $186.75. The continuously compounded rate of return for the stock from t = 0 to t = 1 is closest to: A. -10.90%. B. -10.32%. C.11.51%.
A. -10.90%.
A client has a portfolio of common stocks and fixed-income instruments with a current value of £1,350,000. She intends to liquidate £50,000 from the portfolio at the end of the year to purchase a partnership share in a business. Furthermore, the client would like to be able to withdraw the £50,000 without reducing the initial capital of £1,350,000. The following table shows four alternative asset allocations. *REFER TO PAGE 571* Address the following questions (assume normality for Parts B and C): A. Given the client's desire not to invade the £1,350,000 principal, what is the shortfall level, RL? Use this shortfall level to answer Part B. B. According to the safety-first criterion, which of the allocations is the best? C. What is the probability that the return on the safety-first optimal portfolio will be less than the shortfall level, RL?
A. .037 B. Allocation C C. .2981
*REFER TO PAGES 569-570* A. Your hedging horizon is five days, and your liquidity pool is $2,000 per contract. You estimate that the standard deviation of daily price changes for the contract is $450. What is the probability that you will exhaust your liquidity pool in the five-day period? B. Suppose your hedging horizon is 20 days, but all the other facts given in Part A remain the same. What is the probability that you will exhaust your liquidity pool in the 20-day period?
A. .0466 B. .3222
A random number between zero and one is generated according to a continuous uniform distribution. What is the probability that the first number generated will have a value of exactly 0.30? A. 0% B. 30% C. 70%
A. 0%
*REFER TO PAGE 572* The probability that X will take on a value of either 2 or 4 is closest to: A. 0.20. B. 0.35. C. 0.85.
A. 0.20.
Find the reliability factors based on the t-distribution for the following confidence intervals for the population mean (df = degrees of freedom, n = sample size): A. A 99 percent confidence interval, df = 20. B. A 90 percent confidence interval, df = 20. C. A 95 percent confidence interval, n = 25. D. A 95 percent confidence interval, n = 16.
A. 2.845 B. 1.725 C. 2.064 D. 2.131
For a sample size of 65 with a mean of 31 taken from a normally distributed population with a variance of 529, a 99% confidence interval for the population mean will have a lower limit closest to: A. 23.64. B. 25.41. C. 30.09.
A. 23.64.
Petra Munzi wants to know how value managers performed last year. Munzi estimates that the population cross-sectional standard deviation of value manager returns is 4 percent and assumes that the returns are independent across managers. Munzi wants to build a 95 percent confidence interval for the mean return. A. How large a random sample does Munzi need if she wants the 95 percent confidence interval to have a total width of 1 percent? B. Munzi expects a cost of about $10 to collect each observation. If she has a $1,000 budget, will she be able to construct the confidence interval she wants?
A. 246 B. No (10*246>1000)
Peter Biggs wants to know how growth managers performed last year. Biggs assumes that the population cross-sectional standard deviation of growth manager returns is 6 percent and that the returns are independent across managers. A. How large a random sample does Biggs need if he wants the standard deviation of the sample means to be 1 percent? B. How large a random sample does Biggs need if he wants the standard deviation of the sample means to be 0.25 percent?
A. 36 B. 576
The weekly closing prices of Mordice Corporation shares are as follows: 1 August: 112 8 August: 160 15 August: 120 The continuously compounded return of Mordice Corporation shares for the period August 1 to August 15 is closest to: A. 6.90% B. 7.14% C. 8.95%
A. 6.90%
State the approximate probability that a normal random variable will fall within the following intervals: A. Mean plus or minus one standard deviation. B. Mean plus or minus two standard deviations. C. Mean plus or minus three standard deviations.
A. 68% B. 95% C. 99%
Over the last 10 years, a company's annual earnings increased year over year seven times and decreased year over year three times. You decide to model the number of earnings increases for the next decade as a binomial random variable. A. What is your estimate of the probability of success, defined as an increase in annual earnings? *For Parts B, C, and D of this problem, assume the estimated probability is the actual probability for the next decade.* B. What is the probability that earnings will increase in exactly 5 of the next 10 years? C. Calculate the expected number of yearly earnings increases during the next 10 years. D. Calculate the variance and standard deviation of the number of yearly earnings increases during the next 10 years. E. The expression for the probability function of a binomial random variable depends on two major assumptions. In the context of this problem, what must you assume about annual earnings increases to apply the binomial distribution in Part B? What reservations might you have about the validity of these assumptions?
A. 7/10= .7 B. .1029 C. 7 D. 2.1 ((10)(.7)(1-.7)) E. 1. probability of increased earnings in constant from year to year 2. Each year's returns are independent trials
*REFER TO PAGE 571* A portfolio has an expected mean return of 8 percent and standard deviation of 14 percent. The probability that its return falls between 8 and 11 percent is closest to: A. 8.3% B. 14.8%. C. 58.3%.
A. 8.3%
For a small sample with unknown variance, which of the following tests of a hypothesis concerning the population mean is most appropriate? A. A t-test if the population is normally distributed B. A t-test if the population is non-normally distributed C. A z-test regardless of the normality of the population distribution
A. A t-test if the population is normally distributed
An analyst develops the following capital market projections. Stocks return: 10% Stocks standard deviation: 15% Bonds return: 2% Bonds standard deviation: 5% Assuming the returns of the asset classes are described by normal distributions, which of the following statements is correct? A. Bonds have a higher probability of a negative return than stocks. B. On average, 99% of stock returns will fall within two standard deviations of the mean. C. The probability of a bond return less than or equal to 3% is determined using a Z-score of 0.25.
A. Bonds have a higher probability of a negative return than stocks.
A. Define Monte Carlo simulation and explain its use in finance. B. Compared with analytical methods, what are the strengths and weaknesses of Monte Carlo simulation for use in valuing securities?
A. Computer simulations that represent complex financial simulations B. Strengths: price complex financial securities Weaknesses: provides only statistical estimates, not actual results; analytic methods can offer more insight to cause and effect relationships
In the step "stating a decision rule" in testing a hypothesis, which of the following elements must be specified? A. Critical value B. Power of a test C. Value of a test statistic
A. Critical value (rejection point)
Willco is a manufacturer in a mature cyclical industry. During the most recent industry cycle, its net income averaged $30 million per year with a standard deviation of $10 million (n = 6 observations). Management claims that Willco's performance during the most recent cycle results from new approaches and that we can dismiss profitability expectations based on its average or normalized earnings of $24 million per year in prior cycles. A. With μ as the population value of mean annual net income, formulate null and alternative hypotheses consistent with testing Willco management's claim. B. Assuming that Willco's net income is at least approximately normally distributed, identify the appropriate test statistic. C. Identify the rejection point or points at the 0.05 level of significance for the hypothesis tested in Part A. D. Determine whether or not to reject the null hypothesis at the 0.05 significance level.
A. Ho: μ < or = 24 Ha: μ > 24 B. t-statistics because unknown variance t5 C. t> 2.015 D. 1.47; do not reject the null hypothesis because 1.47< 2.015
Investment analysts often use earnings per share (EPS) forecasts. One test of forecasting quality is the zero-mean test, which states that optimal forecasts should have a mean forecasting error of 0. (Forecasting error = Predicted value of variable − Actual value of variable.) *REFER TO PAGE 665* You have collected data (shown in the table above) for two analysts who cover two different industries: Analyst A covers the telecom industry; Analyst B covers automotive parts and suppliers. A. With μ as the population mean forecasting error, formulate null and alternative hypotheses for a zero-mean test of forecasting quality. B. For Analyst A, using both a t-test and a z-test, determine whether to reject the null at the 0.05 and 0.01 levels of significance. C. For Analyst B, using both a t-test and a z-test, determine whether to reject the null at the 0.05 and 0.01 levels of significance.
A. Ho: μ = 0 Ha: μ does not equal 0 B. T,.05= 1.984; T,.01= 2.626; Z,.05=1.96; Z,.01= 2.575; T= 5.025-- reject null C. T,.05= 1.980; T,.01= 2.617; Z,.05= 1.96; Z,.01= 2.575; T= 2.44-- reject null
Ten analysts have given the following fiscal year earnings forecasts for a stock: 1.40 1 1.43 1 1.44 3 1.45 2 1.47 1 1.48 1 1.50 1 Because the sample is a small fraction of the number of analysts who follow this stock, assume that we can ignore the finite population correction factor. Assume that the analyst forecasts are normally distributed. A. What are the mean forecast and standard deviation of forecasts? B. Provide a 95 percent confidence interval for the population mean of the forecasts.
A. mean= 1.45 s= .0278 B. (1.43, 1.47)
Reviewing the EPS forecasting performance data for Analysts A and B, you want to investigate whether the larger average forecast errors of Analyst A are due to chance or to a higher underlying mean value for Analyst A. Assume that the forecast errors of both analysts are normally distributed and that the samples are independent. *REFER TO PAGE 665* Formulate null and alternative hypotheses consistent with determining whether the population mean value of Analyst A's forecast errors (μ1) are larger than Analyst B's (μ2). A. Identify the test statistic for conducting a test of the null hypothesis formulated in Part A. B. Identify the rejection point or points for the hypothesis tested in Part A, at the 0.05 level of significance. C .Determine whether or not to reject the null hypothesis at the 0.05 level of significance.
A. Ho: μ1 - μ2 < or = 0 Ha: μ1 - μ2 > 0 B. 220 C. t > 1.653 D. reject Ho because 2.351 > 1.653
The table below gives data on the monthly returns on the S&P 500 and small-cap stocks for the period January 1960 through December 1999 and provides statistics relating to their mean differences. *REFER TO PAGE 666* Let μd stand for the population mean value of difference between S&P 500 returns and small-cap stock returns. Use a significance level of 0.05 and suppose that mean differences are approximately normally distributed. A. Formulate null and alternative hypotheses consistent with testing whether any difference exists between the mean returns on the S&P 500 and small-cap stocks. B. Determine whether or not to reject the null hypothesis at the 0.05 significance level for the January 1960 to December 1999 period. C. Determine whether or not to reject the null hypothesis at the 0.05 significance level for the January 1960 to December 1979 subperiod. D. Determine whether or not to reject the null hypothesis at the 0.05 significance level for the January 1980 to December 1999 subperiod.
A. Ho: μd = 0 Ha: μd does not = 0; t> 1.96 or t< -1.96 B. -1.5065; fail to reject the null hypothesis C. -2.421; reject the null hypothesis D. .58; fail to reject the null hypothesis
During a 10-year period, the standard deviation of annual returns on a portfolio you are analyzing was 15 percent a year. You want to see whether this record is sufficient evidence to support the conclusion that the portfolio's underlying variance of return was less than 400, the return variance of the portfolio's benchmark. A. Formulate null and alternative hypotheses consistent with the verbal description of your objective. B. Identify the test statistic for conducting a test of the hypotheses in Part A. C. Identify the rejection point or points at the 0.05 significance level for the hypothesis tested in Part A. D. Determine whether the null hypothesis is rejected or not rejected at the 0.05 level of significance.
A. Ho: σ² > or = 400 Ha: σ² < 400 B. 9 C. X2< 3.325 D. 5.0625; fail to reject the null hypothesis
You are investigating whether the population variance of returns on the S&P 500/BARRA Growth Index changed subsequent to the October 1987 market crash. You gather the following data for 120 months of returns before October 1987 and for 120 months of returns after October 1987. You have specified a 0.05 level of significance. Before October 1987: N= 120 Return= 1.416 Variance= 22.367 After October 1987: N= 120 Return= 1.436 Variance= 15.795 A. Formulate null and alternative hypotheses consistent with the verbal description of the research goal. B. Identify the test statistic for conducting a test of the hypotheses in Part A. C. Determine whether or not to reject the null hypothesis at the 0.05 level of significance. (Use the F-tables in the back of this volume.)
A. Ho: σ²before = σ²after Ha; σ²before does not = σ²after B. F= s²before/ s²after (1.416) C. (120,120,.025) = 1.43; fail to reject the null hypothesis because 1.43> 1.416
Which of the following statements is correct with respect to the null hypothesis? A. It is considered to be true unless the sample provides evidence showing it is false. B. It can be stated as "not equal to" provided the alternative hypothesis is stated as "equal to." C. In a two-tailed test, it is rejected when evidence supports equality between the hypothesized value and population parameter.
A. It is considered to be true unless the sample provides evidence showing it is false.
A European put option on stock conveys the right to sell the stock at a prespecified price, called the exercise price, at the maturity date of the option. The value of this put at maturity is (exercise price - stock price) or $0, whichever is greater. Suppose the exercise price is $100 and the underlying stock trades in ticks of $0.01. At any time before maturity, the terminal value of the put is a random variable. A. Describe the distinct possible outcomes for terminal put value. (Think of the put's maximum and minimum values and its minimum price increments.) B. Is terminal put value, at a time before maturity, a discrete or continuous random variable? C. Letting Y stand for terminal put value, express in standard notation the probability that terminal put value is less than or equal to $24. No calculations or formulas are necessary.
A. Minimum: $0 Maximum: $100 B. Discrete random variable because you can specify its nearest values C. P(Y< or = 24)
An analyst is examining a large sample with an unknown population variance. To test the hypothesis that the historical average return on an index is less than or equal to 6%, which of the following is the most appropriate test? A. One-tailed z-test B. Two-tailed z-test C. One-tailed F-test
A. One-tailed z-test
Which of the following is a Type I error? A. Rejecting a true null hypothesis B. Rejecting a false null hypothesis C. Failing to reject a false null hypothesis
A. Rejecting a true null hypothesis
Suppose we take a random sample of 30 companies in an industry with 200 companies. We calculate the sample mean of the ratio of cash flow to total debt for the prior year. We find that this ratio is 23 percent. Subsequently, we learn that the population cash flow to total debt ratio (taking account of all 200 companies) is 26 percent. What is the explanation for the discrepancy between the sample mean of 23 percent and the population mean of 26 percent? A. Sampling error. B. Bias. C. A lack of consistency.
A. Sampling error.
Identify the appropriate test statistic or statistics for conducting the following hypothesis tests. (Clearly identify the test statistic and, if applicable, the number of degrees of freedom. For example, "We conduct the test using an x-statistic with y degrees of freedom.") A. H0: μ = 0 versus Ha: μ ≠ 0, where μ is the mean of a normally distributed population with unknown variance. The test is based on a sample of 15 observations. B. H0: μ = 0 versus Ha: μ ≠ 0, where μ is the mean of a normally distributed population with unknown variance. The test is based on a sample of 40 observations. C. H0: μ ≤ 0 versus Ha: μ > 0, where μ is the mean of a normally distributed population with known variance σ2. The sample size is 45. D. H0: σ2 = 200 versus Ha: σ2 ≠ 200, where σ2 is the variance of a normally distributed population. The sample size is 50. E. H 0 : σ 2 1 = σ 2 2 versus H a : σ 2 1 ≠ σ 2 2 , where σ 2 1 is the variance of one normally distributed population and σ 2 2 is the variance of a second normally distributed population. The test is based on two independent random samples. F. H0: (Population mean 1) − (Population mean 2) = 0 versus Ha: (Population mean 1) − (Population mean 2) ≠ 0, where the samples are drawn from normally distributed populations with unknown variances. The observations in the two samples are correlated. G. H0: (Population mean 1) − (Population mean 2) = 0 versus Ha: (Population mean 1) − (Population mean 2) ≠ 0, where the samples are drawn from normally distributed populations with unknown but assumed equal variances. The observations in the two samples (of size 25 and 30, respectively) are independent.
A. T-statistic with 14 DF, no practical alternative with such a small sample size B. T-stat (more conservative than a Z-score) with 39 DF, possible to use Z if normally distributed C. z-stat because normal distribution with a known variance D. chi-squared with 49 DF because question is in regards to variance E. F-stat because comparing unknown variances F. T-stat for a paired, correlated observation G. T-state for a pooled population estimate
Which of the following events can be represented as a Bernoulli trial? A. The flip of a coin B. The closing price of a stock C. The picking of a random integer between 1 and 10
A. The flip of a coin
Which of the following statements on p-value is correct? A. The p-value is the smallest level of significance at which H0 can be rejected. B. The p-value indicates the probability of making a Type II error. C. The lower the p-value, the weaker the evidence for rejecting the H0.
A. The p-value is the smallest level of significance at which H0 can be rejected.
Which of the following characteristics of an investment study most likely indicates time-period bias? A. The study is based on a short time-series. B. Information not available on the test date is used. C. A structural change occurred prior to the start of the study's time series.
A. The study is based on a short time-series.
For a two-sided confidence interval, an increase in the degree of confidence will result in: A. a wider confidence interval. B. a narrower confidence interval. C. no change in the width of the confidence interval.
A. a wider confidence interval.
If an estimator is consistent, an increase in sample size will increase the: A. accuracy of estimates. B. efficiency of the estimator. C. unbiasedness of the estimator.
A. accuracy of estimates.
As the t-distribution's degrees of freedom decrease, the t-distribution most likely: A. exhibits tails that become fatter. B. approaches a standard normal distribution. C. becomes asymmetrically distributed around its mean value.
A. exhibits tails that become fatter.
Thirteen analysts have given the following fiscal-year earnings forecasts for a stock: 0.70 2 0.72 4 0.74 1 0.75 3 0.76 1 0.77 1 0.82 1 Because the sample is a small fraction of the number of analysts who follow this stock, assume that we can ignore the finite population correction factor. A. What are the mean forecast and standard deviation of forecasts? B. What aspect of the data makes us uncomfortable about using t-tables to construct confidence intervals for the population mean forecast?
A. mean= .74 s= .03266 B. the sample is small and seems to be abnormal. We can assume this sample is not normal
An investment consultant conducts two independent random samples of 5-year performance data for US and European absolute return hedge funds. Noting a 50 basis point return advantage for US managers, the consultant decides to test whether the two means are statistically different from one another at a 0.05 level of significance. The two populations are assumed to be normally distributed with unknown but equal variances. Results of the hypothesis test are contained in the tables below. *REFER TO PAGE 670* The results of the hypothesis test indicate that the: A. null hypothesis is not rejected. B. alternative hypothesis is statistically confirmed. C. difference in mean returns is statistically different from zero.
A. null hypothesis is not rejected.
A call option on a stock index is valued using a three-step binomial tree with an up move that equals 1.05 and a down move that equals 0.95. The current level of the index is $190, and the option exercise price is $200. If the option value is positive when the stock price exceeds the exercise price at expiration and $0 otherwise, the number of terminal nodes with a positive payoff is: A. one. B. two. C. three.
A. one.
The value of a test statistic is best described as the basis for deciding whether to: A. reject the null hypothesis. B. accept the null hypothesis. C. reject the alternative hypothesis.
A. reject the null hypothesis.
An analyst tests the profitability of a trading strategy with the null hypothesis being that the average abnormal return before trading costs equals zero. The calculated t-statistic is 2.802, with critical values of ± 2.756 at significance level α = 0.01. After considering trading costs, the strategy's return is near zero. The results are most likely: A. statistically but not economically significant. B. economically but not statistically significant. C. neither statistically nor economically significant.
A. statistically but not economically significant.
For each of the following hypothesis tests concerning the population mean, μ, state the rejection point condition or conditions for the test statistic (e.g., t > 1.25); n denotes sample size. A. H0: μ = 10 versus Ha: μ ≠ 10, using a t-test with n = 26 and α = 0.05 B. H0: μ = 10 versus Ha: μ ≠ 10, using a t-test with n = 40 and α = 0.01 C. H0: μ ≤ 10 versus Ha: μ > 10, using a t-test with n = 40 and α = 0.01 D. H0: μ ≤ 10 versus Ha: μ > 10, using a t-test with n = 21 and α = 0.05 E. H0: μ ≥ 10 versus Ha: μ < 10, using a t-test with n = 19 and α = 0.10 F. H0: μ ≥ 10 versus Ha: μ < 10, using a t-test with n = 50 and α = 0.05
A. t> 2.06 or t< -2.06 B. t> 2.7.08 or t< -2.708 C. t> 2.426 D. t> 1.725 E. t< -1.330 F. t< -1.677
For each of the following hypothesis tests concerning the population mean, μ, state the rejection point condition or conditions for the test statistic (e.g., z > 1.25); n denotes sample size. A. H0: μ = 10 vs Ha: μ ≠ 10, using a z-test with n = 50 and α = 0.01 B. H0: μ = 10 vs Ha: μ ≠ 10, using a z-test with n = 50 and α = 0.05 C. H0: μ = 10 vs Ha: μ ≠ 10, using a z-test with n = 50 and α = 0.10 D. H0: μ ≤ 10 vs Ha: μ > 10, using a z-test with n = 50 and α = 0.05
A. z> 2.575 or z< -2.575 B. z> 1.96 or z< -1.96 C. z> 1.645 or z< -1.645 D. z> 1.645
You are forecasting sales for a company in the fourth quarter of its fiscal year. Your low-end estimate of sales is €14 million, and your high-end estimate is €15 million. You decide to treat all outcomes for sales between these two values as equally likely, using a continuous uniform distribution. A. What is the expected value of sales for the fourth quarter? B. What is the probability that fourth-quarter sales will be less than or equal to €14,125,000?
A. €14.5m B. 12.50%
Which of the following tests of a hypothesis concerning the population mean is most appropriate? A. A z-test if the population variance is unknown and the sample is small B. A z-test if the population is normally distributed with a known variance C. A t-test if the population is non-normally distributed with unknown variance and a small sample
B. A z-test if the population is normally distributed with a known variance
All else equal, is specifying a smaller significance level in a hypothesis test likely to increase the probability of a: Type I error? Type II error? A No No B No Yes C Yes No
B No Yes
In a discrete uniform distribution with 20 potential outcomes of integers 1 to 20, the probability that X is greater than or equal to 3 but less than 6, P(3 ≤ X < 6), is: A. 0.10. B. 0.15. C. 0.20.
B. 0.15.
A sample mean is computed from a population with a variance of 2.45. The sample size is 40. The standard error of the sample mean is closest to: A. 0.039. B. 0.247. C. 0.387.
B. 0.247.
A portfolio manager annually outperforms her benchmark 60% of the time. Assuming independent annual trials, what is the probability that she will outperform her benchmark four or more times over the next five years? A. 0.26 B. 0.34 C. 0.48
B. 0.34
For a sample size of 17, with a mean of 116.23 and a variance of 245.55, the width of a 90% confidence interval using the appropriate t-distribution is closest to: A. 13.23. B. 13.27. C. 13.68.
B. 13.27.
*REFER TO PAGE 571* A portfolio has an expected return of 7% with a standard deviation of 13%. For an investor with a minimum annual return target of 4%, the probability that the portfolio return will fail to meet the target is closest to: A. 33%. B. 41%. C. 59%.
B. 41%.
Which of the following statements regarding a one-tailed hypothesis test is correct? A. The rejection region increases in size as the level of significance becomes smaller. B. A one-tailed test more strongly reflects the beliefs of the researcher than a two-tailed test. C. The absolute value of the rejection point is larger than that of a two-tailed test at the same level of significance.
B. A one-tailed test more strongly reflects the beliefs of the researcher than a two-tailed test.
Which of the following assets most likely requires the use of a multivariate distribution for modeling returns? A. A call option on a bond B. A portfolio of technology stocks C. A stock in a market index
B. A portfolio of technology stocks (group of related assets)
A client holding a £2,000,000 portfolio wants to withdraw £90,000 in one year without invading the principal. According to Roy's safety-first criterion, which of the following portfolio allocations is optimal? Allocation A: annual return: 6.5% standard deviation: 8.35% Allocation B: annual return: 7.5% standard deviation: 10.21% Allocation C: annual return: 8.5% standard deviation: 14.34% A. Allocation A B. Allocation B C. Allocation C
B. Allocation B
Which sampling bias is most likely investigated with an out-of-sample test? A. Look-ahead bias B. Data-mining bias C. Sample selection bias
B. Data-mining bias
Which parameter equals zero in a normal distribution? A. Kurtosis B. Skewness C. Standard deviation
B. Skewness
An increase in sample size is most likely to result in a: A. wider confidence interval. B. decrease in the standard error of the sample mean. C. lower likelihood of sampling from more than one population.
B. decrease in the standard error of the sample mean.
The level of significance of a hypothesis test is best used to: A. calculate the test statistic. B. define the test's rejection points. C. specify the probability of a Type II error.
B. define the test's rejection points.
A pooled estimator is used when testing a hypothesis concerning the: A. equality of the variances of two normally distributed populations. B. difference between the means of two at least approximately normally distributed populations with unknown but assumed equal variances. C. difference between the means of two at least approximately normally distributed populations with unknown and assumed unequal variances.
B. difference between the means of two at least approximately normally distributed populations with unknown but assumed equal variances.
A Type II error is best described as: A. rejecting a true null hypothesis. B. failing to reject a false null hypothesis. C. failing to reject a false alternative hypothesis.
B. failing to reject a false null hypothesis.
X is a discrete random variable with possible outcomes X = {1,2,3,4}. Three functions f(x), g(x), and h(x) are proposed to describe the probabilities of the outcomes in X. *REFER TO PAGE 572* The conditions for a probability function are satisfied by: A. f(x). B. g(x). C. h(x).
B. g(x).
In contrast to normal distributions, lognormal distributions: A. are skewed to the left. B. have outcomes that cannot be negative. C. are more suitable for describing asset returns than asset prices.
B. have outcomes that cannot be negative.
A population has a non-normal distribution with mean µ and variance σ2. The sampling distribution of the sample mean computed from samples of large size from that population will have: A. the same distribution as the population distribution. B. its mean approximately equal to the population mean. C. its variance approximately equal to the population variance.
B. its mean approximately equal to the population mean.
The value of the cumulative distribution function F(x), where x is a particular outcome, for a discrete uniform distribution: A. sums to 1. B. lies between 0 and 1. C. decreases as x increases.
B. lies between 0 and 1.
The probability of correctly rejecting the null hypothesis is the: A. p-value. B. power of a test. C. level of significance.
B. power of a test.
A report on long-term stock returns focused exclusively on all currently publicly traded firms in an industry is most likely susceptible to: A. look-ahead bias. B. survivorship bias. C. intergenerational data mining
B. survivorship bias.
For a small sample from a normally distributed population with unknown variance, the most appropriate test statistic for the mean is the: A. z-statistic. B. t-statistic. C. χ2 statistic.
B. t-statistic.
The power of a hypothesis test is: A. equivalent to the level of significance. B. the probability of not making a Type II error. C. unchanged by increasing a small sample size.
B. the probability of not making a Type II error.
An estimator with an expected value equal to the parameter that it is intended to estimate is described as: A. efficient. B. unbiased. C. consistent.
B. unbiased.
A stock is priced at $100.00 and follows a one-period binomial process with an up move that equals 1.05 and a down move that equals 0.97. If 1 million Bernoulli trials are conducted, and the average terminal stock price is $102.00, the probability of an up move (p) is closest to: A. 0.375. B. 0.500. C. 0.625.
C. 0.625.
If an analyst expects a portfolio to outperform its benchmark with a 75% success rate in any measurement period, and the portfolio meets that objective in three of four quarters, what is the probability that the realized portfolio performance over the year is at or below this expectation? A. 0.26 B. 0.42 C. 0.68
C. 0.68
A hypothesis test for a normally-distributed population at a 0.05 significance level implies a: A. 95% probability of rejecting a true null hypothesis. B. 95% probability of a Type I error for a two-tailed test. C. 5% critical value rejection region in a tail of the distribution for a one-tailed test.
C. 5% critical value rejection region in a tail of the distribution for a one-tailed test.
The total number of parameters that fully characterizes a multivariate normal distribution for the returns on two stocks is: A. 3. B. 4. C. 5.
C. 5.
Which of the following represents a correct statement about the p-value? A. The p-value offers less precise information than does the rejection points approach. B. A larger p-value provides stronger evidence in support of the alternative hypothesis. C. A p-value less than the specified level of significance leads to rejection of the null hypothesis.
C. A p-value less than the specified level of significance leads to rejection of the null hypothesis.
Which of the following statements is correct with respect to the p-value? A. It is a less precise measure of test evidence than rejection points. B. It is the largest level of significance at which the null hypothesis is rejected. C. It can be compared directly with the level of significance in reaching test conclusions.
C. It can be compared directly with the level of significance in reaching test conclusions.
The following table shows the significance level (α) and the p-value for three hypothesis tests. α p-value Test 1 0.05 0.10 Test 2 0.10 0.08 Test 3 0.10 0.05 The evidence for rejecting H0 is strongest for: A. Test 1. B. Test 2. C. Test 3.
C. Test 3.
Which of the following statements about hypothesis testing is correct? A. The null hypothesis is the condition a researcher hopes to support. B. The alternative hypothesis is the proposition considered true without conclusive evidence to the contrary. C. The alternative hypothesis exhausts all potential parameter values not accounted for by the null hypothesis.
C. The alternative hypothesis exhausts all potential parameter values not accounted for by the null hypothesis.
Which of the following is a continuous random variable? A. The value of a futures contract quoted in increments of $0.05 B. The total number of heads recorded in 1 million tosses of a coin. C. The rate of return on a diversified portfolio of stocks over a three-month period
C. The rate of return on a diversified portfolio of stocks over a three-month period
The best approach for creating a stratified random sample of a population involves: A. drawing an equal number of simple random samples from each subpopulation. B. selecting every kth member of the population until the desired sample size is reached. C. drawing simple random samples from each subpopulation in sizes proportional to the relative size of each subpopulation.
C. drawing simple random samples from each subpopulation in sizes proportional to the relative size of each subpopulation.
When making a decision in investments involving a statistically significant result, the: A. economic result should be presumed meaningful. B. statistical result should take priority over economic considerations. C. economic logic for the future relevance of the result should be further explored.
C. economic logic for the future relevance of the result should be further explored.
A limitation of Monte Carlo simulation is: A. its failure to do "what if" analysis. B. that it requires historical records of returns C. its inability to independently specify cause-and-effect relationships.
C. its inability to independently specify cause-and-effect relationships.
The lognormal distribution is a more accurate model for the distribution of stock prices than the normal distribution because stock prices are: A. symmetrical. B. unbounded. C. non-negative.
C. non-negative.
For a binomial random variable with five trials, and a probability of success on each trial of 0.50, the distribution will be: A. skewed. B. uniform. C. symmetric.
C. symmetric.
A Monte Carlo simulation can be used to: A. directly provide precise valuations of call options. B. simulate a process from historical records of returns. C. test the sensitivity of a model to changes in assumptions.
C. test the sensitivity of a model to changes in assumptions.
Alcorn Mutual Funds is placing large advertisements in several financial publications. The advertisements prominently display the returns of 5 of Alcorn's 30 funds for the past 1-, 3-, 5-, and 10-year periods. The results are indeed impressive, with all of the funds beating the major market indexes and a few beating them by a large margin. Is the Alcorn family of funds superior to its competitors?
No because it does not show the returns of all 30 funds, rather just a sample that could be skewed to show better results
Compare the standard normal distribution and Student's t-distribution.
Normal: 1 distribution, narrow confidence intervals, higher probabilities T: many distributions, wide confidence intervals, lower probabilities Have the same distribution for large numbers (high degrees of freedom)
