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Reading 6: The Time Value of Money Question #1 of 90 Question ID: 1203913 John is getting a $25,000 loan, with an 8% annual interest rate to be paid in 48 equal monthly installments If the rst payment is due at the end of the rst month, the principal and interest values for the rst payment are closest to: Principal Interest A) $443.65 $166.67 B) $410.32 $200.00 C) $443.65 $200.00 Explanation Calculate the payment rst: N = 48; I/Y = 8/12 = 0.667; PV = 25,000; FV = 0; CPT PMT = 610.32 Interest = 0.006667 × 25,000 = $166.67; Principal = 610.32 – 166.67 = $443.65 (Study Session 2, Module 6.2, LOS 6.f) Question #2 of 90 Question ID: 1203853 Given: an 11% annual rate compounded quarterly for years; compute the future value of $8,000 today A) $9,857 B) $9,939 C) $8,962 Explanation Divide the interest rate by the number of compound periods and multiply the number of years by the number of compound periods I = 11 / = 2.75; N = (2)(4) = 8; PV = 8,000 (Study Session 2, Module 6.1, LOS 6.d) Question #3 of 90 Question ID: 1203876 What is the maximum an investor should be willing to pay for an annuity that will pay out $10,000 at the beginning of each of the next 10 years, given the investor wants to earn 12.5%, compounded annually? A) $62,285 B) $55,364 C) $52,285 Explanation Using END mode, the PV of this annuity due is $10,000 plus the present value of a 9-year ordinary annuity: N=9; I/Y=12.5; PMT=-10,000; FV=0; CPT PV=$52,285; $52,285 + $10,000 = $62,285 Or set your calculator to BGN mode then N=10; I/Y=12.5; PMT=-10,000; FV=0; CPT PV= $62,285 (Study Session 2, Module 6.3, LOS 6.e) Question #4 of 90 Question ID: 1203903 The First State Bank is willing to lend $100,000 for years at a 12% rate of interest, with the loan to be repaid in equal semi-annual payments Given the payments are to be made at the end of each 6-month period, how much will each loan payment be? A) $16,104 B) $32,925 C) $25,450 Explanation N = × = 8; I/Y = 12/2 = 6; PV = -100,000; FV = 0; CPT → PMT = 16,103.59 (Study Session 2, Module 6.2, LOS 6.f) Question #5 of 90 Question ID: 1203895 Suppose you are going to deposit $1,000 at the start of this year, $1,500 at the start of next year, and $2,000 at the start of the following year in an savings account How much money will you have at the end of three years if the rate of interest is 10% each year? A) $5,346.00 B) $4,000.00 C) $5,750.00 Explanation Future value of  $1,000 for periods at 10% = 1,331 Future value of $1,500 for periods at 10% = 1,815 Future value of $2,000 for period at 10% = 2,200 Total = $5,346 N = 3; PV = -$1,000; I/Y = 10%; CPT → FV = $1,331 N = 2; PV = -$1,500; I/Y = 10%; CPT → FV = $1,815 N = 1; PV = -$2,000; I/Y = 10%; CPT → FV = $2,200 (Study Session 2, Module 6.3, LOS 6.e) Question #6 of 90 Question ID: 1203845 Peter Wallace wants to deposit $10,000 in a bank certi cate of deposit (CD) Wallace is considering the following banks: Bank A o ers 5.85% annual interest compounded annually Bank B o ers 5.75% annual interest rate compounded monthly Bank C o ers 5.70% annual interest compounded daily Which bank o ers the highest e ective interest rate and how much? A) Bank C, 5.87% B) Bank B, 5.90% C) Bank A, 5.85% Explanation E ective interest rates: Bank A = 5.85 (already annual compounding) Bank B, nominal = 5.75; C/Y = 12; e ective = 5.90 Bank C, nominal = 5.70, C/Y = 365; e ective = 5.87 Hence Bank B has the highest e ective interest rate (Study Session 2, Module 6.1, LOS 6.c) Question #7 of 90 Question ID: 1203833 Vega research has been conducting investor polls for Third State Bank They have found the most investors are not willing to tie up their money in a 1-year (2-year) CD unless they receive at least 1.0% (1.5%) more than they would on an ordinary savings account If the savings account rate is 3%, and the bank wants to raise funds with 2-year CDs, the yield must be at least: A) 4.5%, and this represents a discount rate B) 4.5%, and this represents a required rate of return C) 4.0%, and this represents a required rate of return Explanation Since we are taking the view of the minimum amount required to induce investors to lend funds to the bank, this is best described as a required rate of return Based upon the numerical information, the rate must be 4.5% (= 3.0 + 1.5) (Study Session 2, Module 6.1, LOS 6.a) Question #8 of 90 Question ID: 1203856 Jamie Morgan needs to accumulate $2,000 in 18 months If she can earn 6% at the bank, compounded quarterly, how much must she deposit today? A) $1,832.61 B) $1,840.45 C) $1,829.08 Explanation Each quarter of a year is comprised of months thus N = 18 / = 6; I/Y = / = 1.5; PMT = 0; FV = 2,000; CPT → PV = $1,829.08 (Study Session 2, Module 6.1, LOS 6.d) Question #9 of 90 Question ID: 1152242 Which of the following statements about compounding and interest rates is least accurate? A) Present values and discount rates move in opposite directions B) On monthly compounded loans, the e ective annual rate (EAR) will exceed the annual percentage rate (APR) C) All else equal, the longer the term of a loan, the lower will be the total interest you pay Explanation Since the proportion of each payment going toward the principal decreases as the original loan maturity increases, the total dollars interest paid over the life of the loan also increases (Study Session 2, Module 6.2, LOS 6.f) Question #10 of 90 Question ID: 1203866 Consider a 10-year annuity that promises to pay out $10,000 per year; given this is an ordinary annuity and that an investor can earn 10% on her money, the future value of this annuity, at the end of 10 years, would be: A) $175,312.00 B) $159,374.00 C) $110.000 Explanation N = 10; I/Y = 10; PMT = -10,000; PV = 0; CPT → FV = $159,374 (Study Session 2, Module 6.3, LOS 6.e) Question #11 of 90 Question ID: 1203889 Given an 8.5% discount rate, an asset that generates cash ows of $10 in Year 1, –$20 in Year 2, $10 in Year 3, and is then sold for $150 at the end of Year 4, has a present value of: A) $135.58 B) $163.42 C) $108.29 Explanation Using your cash ow keys, CF0 = 0; CF1 = 10; CF2 = –20; CF3 = 10; CF4 = 150; I/Y = 8.5; NPV = $108.29 (Study Session 2, Module 6.3, LOS 6.e) Question #12 of 90 Question ID: 1203852 If $1,000 is invested at the beginning of the year at an annual rate of 48%, compounded quarterly, what would that investment be worth at the end of the year? A) $1,048 B) $4,798 C) $1,574 Explanation N = × = 4; I/Y = 48/4 = 12; PMT = 0; PV = –1,000; CPT → FV = 1,573.52 (Study Session 2, Module 6.1, LOS 6.d) Question #13 of 90 Question ID: 1203915 Three years from now, an investor will deposit the rst of eight $1,000 payments into a special fund The fund will earn interest at the rate of 5% per year until the third deposit is made Thereafter, the fund will return a reduced interest rate of 4% compounded annually until the nal deposit is made How much money will the investor have in the fund at the end of ten years assuming no withdrawals are made? A) $8,872.93 B) $9,549.11 C) $9,251.82 Explanation It's best to break this problem into parts to accommodate the change in the interest rate Money in the fund at the end of ten years based on deposits made with initial interest of 5%: (1) The total value in the fund at the end of the fth year is $3,152.50: PMT = −1,000; N = 3; I/Y =5; CPT → FV = $3,152.50 (calculator in END mode) (2) The $3,152.50 is now the present value and will then grow at 4% until the end of the tenth year We get: PV = −3,152.50; N = 5; I/Y = 4; PMT = −1,000; CPT → FV = $9,251.82 (Study Session 2, Module 6.3, LOS 6.e) Question #14 of 90 Question ID: 1203885 The value in years of $500 invested today at an interest rate of 6% compounded monthly is closest to: A) $760 B) $780 C) $750 Explanation PV = -500; N = × 12 = 84; I/Y = 6/12 = 0.5; compute FV = 760.18 (Study Session 2, Module 6.3, LOS 6.e) Question #15 of 90 Question ID: 1203840 A stated interest rate of 9% compounded semiannually results in an e ective annual rate closest to: A) 9.2% B) 9.1% C) 9.3% Explanation Semiannual rate = 0.09 / = 0.045 E ective annual rate = (1 + 0.045)2 – = 0.09203, or 9.203% (Study Session 2, Module 6.1, LOS 6.c) Question #16 of 90 Question ID: 1203859 Bill Jones is creating a charitable trust to provide six annual payments of $20,000 each, beginning next year How much must Jones set aside now at 10% interest compounded annually to meet the required disbursements? A) $87,105.21 B) $95,815.74 C) $154,312.20 Explanation N = 6, PMT = -$20,000, I/Y = 10%, FV = 0, Compute PV → $87,105.21 (Study Session 2, Module 6.3, LOS 6.e) Question #17 of 90 Question ID: 1203847 As the number of compounding periods increases, what is the e ect on the annual percentage rate (APR) and the e ective annual rate (EAR)? A) APR remains the same, EAR increases B) APR increases, EAR remains the same C) APR increases, EAR increases Explanation The APR remains the same since the APR is computed as (interest per period) × (number of compounding periods in year) As the frequency of compounding increases, the interest rate per period decreases leaving the original APR unchanged However, the EAR increases with the frequency of compounding (Study Session 2, Module 6.1, LOS 6.c) Question #18 of 90 Question ID: 1203873 An investor wants to receive $1,000 at the beginning of each of the next ten years with the rst payment starting today If the investor can earn 10 percent interest, what must the investor put into the account today in order to receive this $1,000 cash ow stream? A) $6,759 B) $6,145 C) $7,145 Explanation This is an annuity due problem There are several ways to solve this problem Method 1: PV of rst $1,000 = $1,000 PV of next payments at 10% = 5,759.02 Sum of payments = $6,759.02 Method 2: Put calculator in BGN mode N = 10; I = 10; PMT = -1,000; CPT → PV = 6,759.02 Note: make PMT negative to get a positive PV Don't forget to take your calculator out of BGN mode Method 3: You can also nd the present value of the ordinary annuity $6,144.57 and multiply by + k to add one year of interest to each cash ow $6,144.57 × 1.1 = $6,759.02 (Study Session 2, Module 6.3, LOS 6.e) Question #19 of 90 Question ID: 1203862 An annuity will pay eight annual payments of $100, with the rst payment to be received one year from now If the interest rate is 12% per year, what is the present value of this annuity? A) $496.76 B) $1,229.97 C) $556.38 Explanation N = 8; I/Y = 12%; PMT = -$100; FV = 0; CPT → PV = $496.76 (Study Session 2, Module 6.3, LOS 6.e) Question #20 of 90 Question ID: 1203846 A local loan shark o ers for on payday What it involves is that you borrow $4 from him and repay $5 on the next payday (one week later) What would the stated annual interest rate be on this loan, with weekly compounding? Assuming 52 weeks in one year, what is the e ective annual interest rate on this loan? Select the respective answer choices closest to your numbers A) 25%; 300% B) 1,300%; 10,947,544% C) 25%; 1,300% Explanation Stated Weekly Rate= 5/4 – = 25% Stated Annual Rate = 1,300% Annual E ective Interest Rate = (1 + 0.25)52 – = 109,476.44 – = 10,947,544% (Study Session 2, Module 6.1, LOS 6.c) Question #21 of 90 Question ID: 1152243 Elise Corrs, hedge fund manager and avid downhill skier, was recently granted permission to take a month sabbatical During the sabbatical, (scheduled to start in 11 months), Corrs will ski at approximately 12 resorts located in the Austrian, Italian, and Swiss Alps Corrs estimates that she will need $6,000 at the beginning of each month for expenses that month (She has already nanced her initial travel and equipment costs.) Her nancial planner estimates that she will earn an annual rate of 8.5% during her savings period and an annual rate of return during her sabbatical of 9.5% How much does she need to put in her savings account at the end of each month for the next 11 months to ensure the cash ow she needs over her sabbatical? Each month, Corrs should save approximately: A) $2,070 B) $2,080 C) $2,065 Explanation This is a two-step problem First, we need to calculate the present value of the amount she needs over her sabbatical (This amount will be in the form of an annuity due since she requires the payment at the beginning of the month.) Then, we will use future value formulas to determine how much she needs to save each month Step 1: Calculate present value of amount required during the sabbatical Using a nancial calculator: Set to BEGIN Mode, then N = 4; I/Y = 9.5 / 12 = 0.79167; PMT = 6,000; FV = 0; CPT → PV = -23,719 Step 2: Calculate amount to save each month Using a nancial calculator: Make sure it is set to END mode, then N = 11; I/Y = 8.5 / 12.0 = 0.70833; PV = 0; FV = 23,719; CPT → PMT= -2,081, or approximately $2,080 (Study Session 2, Module 6.2, LOS 6.f) Question #22 of 90 As the number of compounding periods increases, what is the e ect on the EAR? EAR: A) increases at an increasing rate B) does not increase C) increases at a decreasing rate Explanation Question ID: 1203849 There is an upper limit to the EAR as the frequency of compounding increases In the limit, with continuous compounding the EAR = eAPR –1 Hence, the EAR increases at a decreasing rate (Study Session 2, Module 6.1, LOS 6.c) Question #23 of 90 Question ID: 1203880 If $10,000 is invested in a mutual fund that returns 12% per year, after 30 years the investment will be worth: A) $299,599.00 B) $300,000.00 C) $10,120.00 Explanation FV = 10,000(1.12)30 = 299,599 Using TI BAII Plus: N = 30; I/Y = 12; PV = -10,000; CPT → FV = 299,599 (Study Session 2, Module 6.3, LOS 6.e) Question #24 of 90 Question ID: 1203899 Find the future value of the following uneven cash ow stream Assume end of the year payments The discount rate is 12% Year -2,000 Year -3,000 Year 6,000 Year 25,000 Year 30,000 A) $58,164.58 B) $33,004.15 C) $65,144.33 Explanation C) rejecting a true null hypothesis Explanation The Type II error is the error of failing to reject a null hypothesis that is not true (Study Session 3, Module 11.1, LOS 11.c) Question #65 of 97 Question ID: 1204369 A researcher is testing whether the average age of employees in a large rm is statistically di erent from 35 years (either above or below) A sample is drawn of 250 employees and the researcher determines that the appropriate critical value for the test statistic is 1.96 The value of the computed test statistic is 4.35 Given this information, which of the following statements is least accurate? The test: A) has a signi cance level of 95% B) indicates that the researcher will reject the null hypothesis C) indicates that the researcher is 95% dent that the average employee age is di erent than 35 years Explanation This test has a signi cance level of 5% The relationship between dence and signi cance is: signi cance level = – dence level We know that the signi cance level is 5% because the sample size is large and the critical value of the test statistic is 1.96 (2.5% of probability is in both the upper and lower tails) (Study Session 3, Module 11.1, LOS 11.c) Question #66 of 97 Question ID: 1204373 If a two-tailed hypothesis test has a 5% probability of rejecting the null hypothesis when the null is true, it is most likely that the: A) signi cance level of the test is 5% B) probability of a Type I error is 2.5% C) power of the test is 95% Explanation Rejecting the null hypothesis when it is true is a Type I error The probability of a Type I error is the signi cance level of the test The power of a test is one minus the probability of a Type II error, which cannot be calculated from the information given (Study Session 3, Module 11.1, LOS 11.c) Question #67 of 97 Question ID: 1204419 Which of the following statements about test statistics is least accurate? A) In the case of a test of the di erence in means of two independent samples, we use a tdistributed test statistic B) In a test of the population mean, if the population variance is unknown, we should use a tdistributed test statistic C) In a test of the population mean, if the population variance is unknown and the sample is small, we should use a z-distributed test statistic Explanation If the population sampled has a known variance, the z-test is the correct test to use In general, a t-test is used to test the mean of a population when the population is unknown Note that in special cases when the sample is extremely large, the z-test may be used in place of the t-test, but the t-test is considered to be the test of choice when the population variance is unknown A t-test is also used to test the di erence between two population means while an F-test is used to compare di erences between the variances of two populations (Study Session 3, Module 11.2, LOS 11.g) Question #68 of 97 Question ID: 1204397 An analyst calculates that the mean of a sample of 200 observations is The analyst wants to determine whether the calculated mean, which has a standard error of the sample statistic of 1, is signi cantly di erent from at the 5% level of signi cance Which of the following statements is least accurate?: A) The mean observation is signi cantly di erent from 7, because the calculated Z-statistic is less than the critical Z-statistic B) The null hypothesis would be: H0: mean = C) The alternative hypothesis would be Ha: mean > Explanation The way the question is worded, this is a two tailed test.The alternative hypothesis is not Ha: M > because in a two-tailed test the alternative is =, while < and > indicate one-tailed tests A test statistic is calculated by subtracting the hypothesized parameter from the parameter that has been estimated and dividing the di erence by the standard error of the sample statistic Here, the test statistic = (sample mean – hypothesized mean) / (standard error of the sample statistic) = (5 - 7) / (1) = -2 The calculated Z is -2, while the critical value is -1.96 The calculated test statistic of -2 falls to the left of the critical Z-statistic of -1.96, and is in the rejection region Thus, the null hypothesis is rejected and the conclusion is that the sample mean of is signi cantly di erent than What the negative sign shows is that the mean is less than 7; a positive sign would indicate that the mean is more than The way the null hypothesis is written, it makes no di erence whether the mean is more or less than 7, just that it is not (Study Session 3, Module 11.1, LOS 11.d) Question #69 of 97 Question ID: 1204407 An analyst conducts a two-tailed test to determine if mean earnings estimates are signi cantly di erent from reported earnings The sample size is greater than 25 and the computed test statistic is 1.25 Using a 5% signi cance level, which of the following statements is most accurate? A) The analyst should reject the null hypothesis and conclude that the earnings estimates are signi cantly di erent from reported earnings B) To test the null hypothesis, the analyst must determine the exact sample size and calculate the degrees of freedom for the test C) The analyst should fail to reject the null hypothesis and conclude that the earnings estimates are not signi cantly di erent from reported earnings Explanation 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% dence 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 signi cantly di erent 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 (Study Session 3, Module 11.2, LOS 11.g) Question #70 of 97 Question ID: 1204427 Student's t-Distribution Level of Signi cance for One-Tailed Test df 0.100 0.050 0.025 0.01 0.005 0.0005 Level of Signi cance for Two-Tailed Test df 0.20 0.10 0.05 0.02 0.01 0.001 10 1.372 1.812 2.228 2.764 3.169 4.587 11 1.363 1.796 2.201 2.718 3.106 4.437 12 1.356 1.782 2.179 2.681 3.055 4.318 22 1.321 1.717 2.074 2.508 2.819 3.792 23 1.319 1.714 2.069 2.500 2.807 3.768 24 1.318 1.711 2.064 2.492 2.797 3.745 Roy Fisher, CFA, wants to determine whether there is a signi cant di erence, at the 5% signi cance level, between the mean monthly return on Stock GHI and the mean monthly return on Stock JKL Fisher assumes the variances of the two stocks' returns are equal Using the last 12 months of returns on each stock, Fisher calculates a t-statistic of 2.0 for a test of equality of means Based on this result, Fisher's test: A) rejects the null hypothesis, and Fisher can conclude that the means are equal B) fails to reject the null hypothesis C) rejects the null hypothesis, and Fisher can conclude that the means are not equal Explanation The null hypothesis for a test of equality of means is H0: μ1 − μ2 = Assuming the variances are equal, degrees of freedom for this test are (n1 + n2 − 2) = 12 + 12 − = 22 From the table of critical values for Student's t-distribution, the critical value for a two-tailed test at the 5% signi cance level for df = 22 is 2.074 Because the calculated t-statistic of 2.0 is less than the critical value, this test fails to reject the null hypothesis that the means are equal (Study Session 3, Module 11.3, LOS 11.h) Question #71 of 97 Question ID: 1204391 For a two-tailed test of hypothesis involving a z-distributed test statistic and a 5% level of signi cance, a calculated z-statistic of 1.5 indicates that: A) the null hypothesis is rejected B) the null hypothesis cannot be rejected C) the test is inconclusive Explanation For a two-tailed test at a 5% level of signi cance the calculated z-statistic would have to be greater than the critical z value of 1.96 for the null hypothesis to be rejected (Study Session 3, Module 11.1, LOS 11.c) Question #72 of 97 Question ID: 1204403 A p-value of 0.02% means that a researcher: A) can reject the null hypothesis at both the 5% and 1% signi cance levels B) can reject the null hypothesis at the 5% signi cance level but cannot reject at the 1% signi cance level C) cannot reject the null hypothesis at either the 5% or 1% signi cance levels Explanation A p-value of 0.02% means that the smallest signi cance level at which the hypothesis can be rejected is 0.0002, which is smaller than 0.05 or 0.01 Therefore the null hypothesis can be rejected at both the 5% and 1% signi cance levels (Study Session 3, Module 11.2, LOS 11.f) Question #73 of 97 Question ID: 1204412 Segment of the table of critical values for Student's t-distribution: Level of Signi cance for a One-Tailed Test df 0.050 0.025 Level of Signi cance for a Two-Tailed Test df 0.10 0.05 16 1.746 2.120 17 1.740 2.110 18 1.734 2.101 19 1.729 2.093 Simone Mak is a television network advertising executive One of her responsibilities is selling commercial spots for a successful weekly sitcom If the average share of viewers for this season exceeds 8.5%, she can raise the advertising rates by 50% for the next season The population of viewer shares is normally distributed A sample of the past 18 episodes results in a mean share of 9.6% with a standard deviation of 10.0% If Mak is willing to make a Type error with a 5% probability, which of the following statements is most accurate? A) With an unknown population variance and a small sample size, Mak cannot test a hypothesis based on her sample data B) Mak cannot charge a higher rate next season for advertising spots based on this sample C) The null hypothesis Mak needs to test is that the mean share of viewers is greater than 8.5% Explanation Mak cannot conclude with 95% dence that the average share of viewers for the show this season exceeds 8.5 and thus she cannot charge a higher advertising rate next season Hypothesis testing process: Step 1: State the hypothesis Null hypothesis: mean ≤ 8.5%; Alternative hypothesis: mean > 8.5% Step 2: Select the appropriate test statistic Use a t statistic because we have a normally distributed population with an unknown variance (we are given only the sample variance) and a small sample size (less than 30) If the population were not normally distributed, no test would be available to use with a small sample size Step 3: Specify the level of signi cance The signi cance level is the probability of a Type I error, or 0.05 Step 4: State the decision rule This is a one-tailed test The critical value for this question will be the tstatistic that corresponds to a signi cance level of 0.05 and n-1 or 17 degrees of freedom Using the ttable, we determine that we will reject the null hypothesis if the calculated test statistic is greater than the critical value of 1.74 Step 5: Calculate the sample (test) statistic The test statistic = t = (9.6 – 8.5) / (10.0 / √18) = 0.4667 (Note: Remember to use standard error in the denominator because we are testing a hypothesis about the population mean based on the mean of 18 observations.) Step 6: Make a decision The calculated statistic is less than the critical value Mak cannot conclude with 95% dence that the mean share of viewers exceeds 8.5% and thus she cannot charge higher rates Note: By eliminating the two incorrect choices, you can select the correct response to this question without performing the calculations (Study Session 3, Module 11.2, LOS 11.g) Question #74 of 97 Question ID: 1204384 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 dence level of the test represents the probability of making a Type II error B) The signi cance 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 A Type I error is the rejection of the null when the null is actually true The signi cance level of the test (alpha) (which is one minus the dence 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 (Study Session 3, Module 11.1, LOS 11.c) Question #75 of 97 Question ID: 1204410 Student's t-Distribution Level of Signi cance for One-Tailed Test df 0.100 0.050 0.025 0.01 0.005 0.0005 Level of Signi cance for Two-Tailed Test df 0.20 0.10 0.05 0.02 0.01 0.001 18 1.330 1.734 2.101 2.552 2.878 3.922 19 1.328 1.729 2.093 2.539 2.861 3.883 20 1.325 1.725 2.086 2.528 2.845 3.850 21 1.323 1.721 2.080 2.518 2.831 3.819 In a two-tailed test of a hypothesis concerning whether a population mean is zero, Jack Olson computes a tstatistic of 2.7 based on a sample of 20 observations where the distribution is normal If a 5% signi cance level is chosen, Olson should: A) fail to reject the null hypothesis that the population mean is not signi cantly di erent from zero B) reject the null hypothesis and conclude that the population mean is not signi cantly di erent from zero C) reject the null hypothesis and conclude that the population mean is signi cantly di erent from zero Explanation At a 5% signi cance 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 zstatistic of 1.960 may be used) Because the critical t-statistic of 2.093 is to the left of the calculated tstatistic 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 signi cantly di erent from zero (Study Session 3, Module 11.2, LOS 11.g) Question #76 of 97 Question ID: 1204362 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) greater than statement and the alternative hypothesis is framed as a less than or equal to statement B) less than or equal to statement and the alternative hypothesis is framed as a greater than statement C) less than statement and the alternative hypothesis is framed as a greater than or equal to statement Explanation 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 (Study Session 3, Module 11.1, LOS 11.b) Question #77 of 97 Question ID: 1204402 A researcher determines that the mean annual return over the last 10 years for an investment strategy was greater than that of an index portfolio of equal risk with a statistical signi cance level of 1% To determine whether the abnormal portfolio returns to the strategy are economically meaningful, it would be most appropriate to additionally account for: A) only the transaction costs and tax e ects of the strategy B) the transaction costs, tax e ects, and risk of the strategy C) only the transaction costs of the strategy Explanation A statistically signi cant excess of mean strategy return over the return of an index or benchmark portfolio may not be economically meaningful because of 1) the transaction costs of implementing the strategy, 2) the increase in taxes incurred by using the strategy, 3) the risk of the strategy Although the market risk of the strategy portfolios is matched to that of the index portfolio, variability in the annual strategy returns introduces additional risk that must be considered before we can determine whether the results of the analysis are economically meaningful, that is, whether we should invest according to the strategy (Study Session 3, Module 11.2, LOS 11.e) Question #78 of 97 Question ID: 1204445 Student's t-distribution, level of signi cance for a two-tailed test: df 0.20 0.10 0.05 0.02 0.01 0.001 16 1.337 1.746 2.120 2.583 2.921 4.015 17 1.333 1.740 2.110 2.567 2.898 3.965 18 1.330 1.734 2.101 2.552 2.878 3.922 19 1.328 1.729 2.093 2.539 2.861 3.883 20 1.325 1.725 2.086 2.528 2.845 3.850 Based on a sample correlation coe cient of −0.525 from a sample size of 19, an analyst calculates a tstatistic of −−−− −0.525√19−2 −−−−−−−−−− √1−(−0.525) = −2.5433 The analyst can reject the hypothesis that the population correlation coe cient equals zero: A) at a 1% signi cance level B) at a 2% signi cance level, but not at a 1% signi cance level C) at a 5% signi cance level, but not at a 2% signi cance level Explanation With 19 − = 17 degrees of freedom, the critical values are plus-or-minus 2.110 at a 5% signi cance level, 2.567 at a 2% signi cance level, and 2.898 at a 1% signi cance level Because the t-statistic of −2.5433 is less than −2.110, the hypothesis can be rejected at a 5% signi cance level Because the t-statistic is greater than −2.567, the hypothesis cannot be rejected at a 2% signi cance level (or any smaller signi cance level) (Study Session 3, Module 11.3, LOS 11.k) Question #79 of 97 Question ID: 1204432 Joe Sutton is evaluating the e ects of the 1987 market decline on the volume of trading Speci cally, he wants to test whether the decline a ected 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: µd ≠ µd0 versus Ha: µd = µd0 B) H0: µd = µd0 versus Ha: µd > µd0 C) H0: µd = µd0 versus Ha: µd ≠ µd0 Explanation 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 (Study Session 3, Module 11.3, LOS 11.i) Question #80 of 97 Question ID: 1204438 Which of the following statements about the variance of a normally distributed population is least accurate? A) The Chi-squared distribution is a symmetric distribution B) The test of whether the population variance equals σ02 requires the use of a Chi-squared distributed test statistic, [(n − 1)s2] / σ02 C) A test of whether the variance of a normally distributed population is equal to some value σ 2, the hypotheses are: H0: σ2 = σ02, versus Ha: σ2 ≠ σ02 Explanation The Chi-squared distribution is not symmetrical, which means that the critical values will not be numerically equidistant from the center of the distribution, though the probability on either side of the critical values will be equal (that is, if there is a 5% level of signi cance and a two-sided test, 2.5% will lie outside each of the two critical values) (Study Session 3, Module 11.3, LOS 11.j) Question #81 of 97 Question ID: 1204413 Student's t-Distribution Level of Signi cance for One-Tailed Test df 0.100 0.050 0.025 0.01 0.005 0.0005 Level of Signi cance for Two-Tailed Test df 0.20 0.10 0.05 0.02 0.01 0.001 40 1.303 1.684 2.021 2.423 2.704 3.551 Ken Wallace is interested in testing whether the average price to earnings (P/E) of rms in the retail industry is 25 Using a t-distributed test statistic and a 5% level of signi cance, the critical values for a sample of 41 rms is (are): A) -1.685 and 1.685 B) -1.96 and 1.96 C) -2.021 and 2.021 Explanation There are 41 − = 40 degrees of freedom and the test is two-tailed Therefore, the critical t-values are ± 2.021 The value 2.021 is the critical value for a one-tailed probability of 2.5% (Study Session 3, Module 11.2, LOS 11.g) Question #82 of 97 Question ID: 1204377 A survey is taken to determine whether the average starting salaries of CFA charterholders is equal to or greater than $58,500 per year What is the test statistic given a sample of 175 newly acquired CFA charterholders with a mean starting salary of $67,000 and a standard deviation of $5,200? A) 1.63 B) -1.63 C) 21.62 Explanation With a large sample size (175) the z-statistic is used The z-statistic is calculated by subtracting the hypothesized parameter from the parameter that has been estimated and dividing the di erence 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) = (67,000 – 58,500) / (5,200 / 1751/2) = (8,500) / (5,200 / 13.22) = 21.62 (Study Session 3, Module 11.1, LOS 11.c) Question #83 of 97 Question ID: 1204434 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) F test B) t test C) Chi-square test Explanation The F test is used to test the di erences of variance between two samples (Study Session 3, Module 11.3, LOS 11.j) Question #84 of 97 Question ID: 1204395 If the null hypothesis is innocence, then the statement "It is better that the guilty go free, than the innocent are punished" is an example of preferring a: A) higher level of signi cance B) type II error over a type I error C) type I error over a type II error Explanation The statement shows a preference for accepting the null hypothesis when it is false (a type II error), over rejecting it when it is true (a type I error) (Study Session 3, Module 11.1, LOS 11.d) Question #85 of 97 Question ID: 1204381 Which of the following statements regarding hypothesis testing is least accurate? A) A type I error is acceptance of a hypothesis that is actually false B) A type II error is the acceptance of a hypothesis that is actually false C) The signi cance level is the risk of making a type I error Explanation A type I error is the rejection of a hypothesis that is actually true (Study Session 3, Module 11.1, LOS 11.c) Question #86 of 97 Question ID: 1204447 Which of the following statements about parametric and nonparametric tests is least accurate? A) Nonparametric tests rely on population parameters B) The test of the mean of the di erences is used when performing a paired comparison C) The test of the di erence in means is used when you are comparing means from two independent samples Explanation 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 di erence in the means and the test of the mean of the di erences Also, it is important to understand that parametric tests rely on distributional assumptions, whereas nonparametric tests are not as strict regarding distributional properties (Study Session 3, Module 11.3, LOS 11.l) Question #87 of 97 If the null hypothesis is H0: ρ ≤ 0, what is the appropriate alternative hypothesis? A) Ha: ρ < B) Ha: ρ > C) Ha: ρ ≠ Explanation The alternative hypothesis must include the possible outcomes the null does not (Study Session 3, Module 11.1, LOS 11.a) Question ID: 1204353 Question #88 of 97 Question ID: 1204433 An analyst has calculated the sample variances for two random samples from independent normally distributed populations The test statistic for the hypothesis that the true population variances are equal is a(n): A) F-statistic B) t-statistic C) chi square statistic Explanation The ratio of the two sample variances follows an F distribution (Study Session 3, Module 11.3, LOS 11.j) Question #89 of 97 Question ID: 1204423 An analyst is testing to see if the mean of a population is less than 133 A random sample of 50 observations had a mean of 130 Assume a standard deviation of The test is to be made at the 1% level of signi cance The analyst should: A) reject the null hypothesis B) fail to reject the null hypothesis C) accept the null hypothesis Explanation The null hypothesis is that the mean is greater than or equal to 133 The test statistic = (sample mean – hypothesized mean) / ((sample standard deviation / (sample size)1/2)) = (130 – 133) / (5 / 501/2) = (-3) / (5 / 7.0711) = -4.24 The critical value for a one-tailed test at a 1% level of signi cance is -2.33 The calculated test statistic of -4.24 falls to the left of the critical value of -2.33, and is in the rejection region Thus, the null hypothesis can be rejected at the 1% signi cance level (Study Session 3, Module 11.2, LOS 11.g) Question #90 of 97 Question ID: 1204424 An analyst conducts a two-tailed z-test to determine if small cap returns are signi cantly di erent from 10% The sample size was 200 The computed z-statistic is 2.3 Using a 5% level of signi cance, which statement is most accurate? A) Reject the null hypothesis and conclude that small cap returns are signi cantly di erent from 10% B) You cannot determine what to with the information given C) Fail to reject the null hypothesis and conclude that small cap returns are close enough to 10% that we cannot say they are signi cantly di erent from 10% Explanation At the 5% level of signi cance 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 signi cantly di erent than 10% (Study Session 3, Module 11.2, LOS 11.g) Question #91 of 97 Question ID: 1204411 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 signi cance level to use C) The dence interval for a two-tailed test of a population mean at the 5% level of signi cance is that the sample mean falls between ±1.96 σ/√n of the null hypothesis value Explanation The signi cance level is chosen before the test so the calculated Z-statistic can be compared to an appropriate critical value (Study Session 3, Module 11.2, LOS 11.g) Question #92 of 97 Question ID: 1204448 A test of whether a mutual fund's performance rank in one period provides information about the fund's performance rank in a subsequent period is best described as a: A) parametric test B) nonparametric test C) mean-rank test Explanation A rank correlation test is best described as a nonparametric test (Study Session 3, Module 11.3, LOS 11.l) Question #93 of 97 Question ID: 1204444 To test a hypothesis that the population correlation coe cient of two variables is equal to zero, an analyst collects a sample of 24 observations and calculates a sample correlation coe cient of 0.37 Can the analyst test this hypothesis using only these two inputs? A) Yes B) No, because the sample standard deviations of the two variables are also required C) No, because the sample means of the two variables are also required Explanation The t-statistic for a test of the population correlation coe cient is correlation coe cient and n is the sample size − − − − r√n−2 −− −− √1−r2 , where r is the sample (Study Session 3, Module 11.3, LOS 11.k) Question #94 of 97 Question ID: 1204378 A survey is taken to determine whether the average starting salaries of CFA charterholders is equal to or greater than $59,000 per year What is the test statistic given a sample of 135 newly acquired CFA charterholders with a mean starting salary of $64,000 and a standard deviation of $5,500? A) 10.56 B) -10.56 C) 0.91 Explanation With a large sample size (135) the z-statistic is used The z-statistic is calculated by subtracting the hypothesized parameter from the parameter that has been estimated and dividing the di erence 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) = (64,000 – 59,000) / (5,500 / 1351/2) = (5,000) / (5,500 / 11.62) = 10.56 (Study Session 3, Module 11.1, LOS 11.c) Question #95 of 97 Question ID: 1204375 Identify the error type associated with the level of signi cance and the meaning of a percent signi cance level Error type α = 0.05 means there is a percent probability of A) Type I error failing to reject a true null hypothesis B) Type II error rejecting a true null hypothesis C) Type I error rejecting a true null hypothesis Explanation The signi cance level is the risk of making a Type error and rejecting the null hypothesis when it is true (Study Session 3, Module 11.1, LOS 11.c) Question #96 of 97 Question ID: 1204365 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 This is a one-sided alternative because of the "less than" belief (Study Session 3, Module 11.1, LOS 11.b) Question #97 of 97 Question ID: 1204393 The power of the test is: A) the probability of rejecting a false null hypothesis B) the probability of rejecting a true null hypothesis C) equal to the level of dence Explanation This is the de nition of the power of the test: the probability of correctly rejecting the null hypothesis (rejecting the null hypothesis when it is false) (Study Session 3, Module 11.1, LOS 11.d)