Quantitative Methods Qbank - 2

For a hypothesis test regarding a population parameter, an analyst has determined that the probability of failing to reject a false null hypothesis is 18%, and the probability of rejecting a true null hypothesis is 5%. The power of the test is: A) 0.82. B) 0.18. C) 0.95.

A The power of the test is 1 - the probability of failing to reject a false null (Type II error); 1 - 0.18 = 0.82.

If the significance level of a test is 0.05 and the probability of a Type II error is 0.15, what is the power of the test? A) 0.850. B) 0.950. C) 0.975.

A The power of a test is 1 - P(Type II error) = 1 - 0.15 = 0.85.

A Type I error is made when the researcher: A) rejects the null hypothesis when it is actually true. B) rejects the alternative hypothesis when it is actually true. C) fails to reject the null hypothesis when it is actually false.

A A Type I error is defined as rejecting the null hypothesis when it is actually true. It can be thought of as a false positive. A Type II error occurs when a researching fails to reject the null hypothesis when it is false. It can be thought of as a false negative.

Data shows that 75 out of 100 tourists who visit New York City visit the Empire State Building. It rains or snows in New York City one day in five. What is the joint probability that a randomly chosen tourist visits the Empire State Building on a day when it neither rains nor snows? A) 60%. B) 15%. C) 95%.

A A joint probability is the probability that two events occur when neither is certain or a given. Joint probability is calculated by multiplying the probability of each event together. (0.75) × (0.80) = 0.60 or 60%.

A p-value of 0.02% means that a researcher: A) can reject the null hypothesis at both the 5% and 1% significance levels. B) can reject the null hypothesis at the 5% significance level but cannot reject at the 1% significance level. C) cannot reject the null hypothesis at either the 5% or 1% significance levels.

A A p-value of 0.02% means that the smallest significance 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% significance levels.

A conditional expectation involves: A) refining a forecast because of the occurrence of some other event. B) determining the expected joint probability. C) calculating the conditional variance.

A Conditional expected values are contingent upon the occurrence of some other event. The expectation changes as new information is revealed.

A simple linear regression is a model of the relationship between: A) one dependent variable and one independent variable. B) one or more dependent variables and one or more independent variables. C) one dependent variable and one or more independent variables.

A Explanation A simple linear regression is a model of the relationship between one dependent variable and one independent variable. A multiple regression is a model of the relationship between one dependent variable and more than one independent variable.

For a parametric test of whether a correlation coefficient is equal to zero, it is least likely that: A) degrees of freedom are n - 1. B) the test statistic follows a t-distribution. C) the test statistic increases with a greater sample size.

A Explanation Degrees of freedom are n - 2 for a test of the hypothesis that correlation is equal to zero. The test statistic increases with sample size (degrees of freedom increase) and follows a t-distribution

William Adams wants to test whether the mean monthly returns over the last five years are the same for two stocks. If he assumes that the returns distributions are normal and have equal variances, the type of test and test statistic are best described as: A) paired comparisons test, t-statistic. B) paired comparisons test, F-statistic. C) difference in means test, t-statistic.

A Explanation Since the observations are likely dependent (both related to market returns), a paired comparisons (mean differences) test is appropriate and is based on a t-statistic.

The coefficient of determination for a linear regression is best described as the: A) percentage of the variation in the dependent variable explained by the variation of the independent variable. B) covariance of the independent and dependent variables. C) percentage of the variation in the independent variable explained by the variation of the dependent variable.

A Explanation The coefficient of determination for a linear regression describes the percentage of the variation in the dependent variable explained by the variation of the independent variable.

The estimated slope coefficient in a simple linear regression is: A) the ratio of the covariance of the regression variables to the variance of the independent variable. B) the predicted value of the dependent variable, given the actual value of the independent variable. C) the change in the independent variable, given a one-unit change in the dependent variable.

A Explanation The estimated slope coefficient in a simple linear regression is CovX,Y / σ^2X where Y is the dependent variable and X is the independent variable. The estimated slope coefficient is interpreted as the change in the dependent variable, given a one-unit change in the independent variable. The predicted value of the dependent variable must consider the estimated intercept term along with the estimated slope coefficient.

The test statistic for a Spearman rank correlation test for a sample size greater than 30 follows: A) a t-distribution. B) a normal distribution. C) a chi-square distribution.

A Explanation The test statistic for the Spearman rank correlation test follows a t-distribution.

An analyst wants to test a hypothesis concerning the population mean of monthly returns for a composite that has existed for 24 months. The analyst may appropriately use: A) a t-test but not a z-test if returns for the composite are normally distributed. B) either a t-test or a z-test if returns for the composite are normally distributed. C) a t-test but not a z-test, regardless of the distribution of returns for the composite.

A Explanation With a small sample size, a t-test may be used if the population is approximately normally distributed. If the population has a nonnormal distribution, no test statistic is available unless the sample size is large.

Brandee Shoffield is the public relations manager for Night Train Express, a local sports team. Shoffield is trying to sell advertising spots and wants to know if she can say with 90% confidence that average home game attendance is greater than 3,000. Attendance is approximately normally distributed. A sample of the attendance at 15 home games results in a mean of 3,150 and a standard deviation of 450. Which of the following statements is most accurate? A) The calculated test statistic is 1.291. B) Shoffield should use a two-tailed Z-test. C) With an unknown population variance and a small sample size, no statistic is available to test Shoffield's hypothesis.

A Here, we have a normally distributed population with an unknown variance (we are given only the sample standard deviation) and a small sample size (less than 30.) Thus, we will use the t-statistic. The test statistic = t = (3,150 - 3,000) / (450 / √ 15) = 1.291

Consider the following statement: In a simple linear regression, the appropriate degrees of freedom for the critical t-value used to calculate a confidence interval around both a parameter estimate and a predicted Y-value is the same as the number of observations minus two. The statement is: A) justified. B) not justified, because the appropriate of degrees of freedom used to calculate a confidence interval around a parameter estimate is the number of observations. C) not justified, because the appropriate of degrees of freedom used to calculate a confidence interval around a predicted Y-value is the number of observations.

A In simple linear regression, the appropriate degrees of freedom for both confidence intervals is the number of observations in the sample (n) minus two.

What is the appropriate alternative hypothesis to test the statistical significance of the intercept term in the following regression? Y = a1 + a2(X) + ε A) HA: a1 ≠0. B) HA: a1 > 0. C) HA: a2 ≠ 0.

A In this regression, a1 is the intercept term. To test the statistical significance means to test the null hypothesis that a1 is equal to zero, versus the alternative that a1 is not equal to zero.

Segment of the table of critical values for Student's t-distribution: Level of Significance for a One-Tailed Test df | 0.050 | 0.025 Level of Significance for a Two-Tailed Test df | 0.10 | 0.05 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 19 episodes results in a mean share of 9.6% with a standard deviation of 10.0%. If Mak is willing to make a Type 1 error with a 5% probability, which of the following statements is most accurate? A) Mak cannot charge a higher rate next season for advertising spots based on this sample. B) The null hypothesis Mak needs to test is that the mean share of viewers is greater than 8.5%. C) With an unknown population variance and a small sample size, Mak cannot test a hypothesis based on her sample data.

A Mak cannot conclude with 95% confidence 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 significance. The significance 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 t-statistic that corresponds to a significance level of 0.05 and n-1 or 18 degrees of freedom. Using the t-table, we determine that we will reject the null hypothesis if the calculated test statistic is greater than the critical value of 1.734. Step 5: Calculate the sample (test) statistic. The test statistic = t = (9.6 - 8.5) / (10.0 / √19) = 0.4795. (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% confidence that the mean share of viewers exceeds 8.5% and thus she cannot charge higher rates.

An analyst should use a t-test with n - 1 degrees of freedom for a test of: A) mean differences. B) equal variances. C) correlation.

A Mean-differences hypothesis tests are t-tests with n - 1 degrees of freedom. Tests of correlation are t-tests with n - 2 degrees of freedom. Tests for equality of variances are F-tests.

Given the following cash flow stream: End of Year | Annual Cash Flow 1 | $4,000 2 | $2,000 3 | -0- 4 | -$1,000 Using a 10% discount rate, the present value of this cash flow stream is: A) $4,606.00. B) $3,636.00. C) $3,415.00.

A PV(1): N = 1; I/Y = 10; FV = -4,000; PMT = 0; CPT → PV = 3,636 PV(2): N = 2; I/Y = 10; FV = -2,000; PMT = 0; CPT → PV = 1,653 PV(3): 0 PV(4): N = 4; I/Y = 10; FV = 1,000; PMT = 0; CPT → PV = -683 Total PV = 3,636 + 1,653 + 0 - 683 = 4,606

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.

A 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.

A researcher estimates a simple linear regression for two variables using 47 observations. The regression sum of squares is 0.0182 and the sum of squared errors is 0.0375. The coefficient of determination for the regression is closest to: A) 32.7%. B) 48.5%. C) 67.5%.

A The coefficient of determination = 0.0182 / (0.0182 + 0.0375) = 32.7%

An advantage of nonprobability sampling, as compared to probability sampling, is lower: A) cost. B) reliance on judgment. C) sampling error.

A The primary advantages of nonprobability sampling are lower cost and easier access to data, as compared to probability sampling. Nonprobability sampling relies more on the analyst's judgment, and because it typically results in a sample that is less random than probability sampling, it is subject to greater sampling error.

Which of the following statements about hypothesis testing is most accurate? A) The probability of a Type I error is equal to the significance level of the test. B) The power of a test is one minus the probability of a Type I error. C) If you can disprove the null hypothesis, then you have proven the alternative hypothesis.

A 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.

The percentage changes in annual earnings for a company are approximately normally distributed with a mean of 5% and a standard deviation of 12%. The probability that the average change in earnings over the next five years will be greater than 15.5% is closest to: A) 2.5%. B) 5.0%. C) 10.0%.

A The standard error of a 5-year average of earnings changes is 12% / √5 = 5.366 15.5% is (15.5 − 5) / 5.366 = 1.96 standard errors above the mean, and the probability of a 5-year average more than 1.96 standard errors above the mean is 2.5% for a normal distribution.

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) 4.12. B) 8.00. C) 4.00.

A 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 / n^[1/2]) = (7 − 5) / (2 / 17^[1/2]) = (2) / (2 / 4.1231) = 4.12.

An analyst is asked to calculate standard deviation using monthly returns over the last five years. These data are best described as: A) time series data. B) unstructured data. C) cross-sectional data.

A Time series data are taken at equally spaced intervals, such as monthly, quarterly, or annual. Cross sectional data are taken at a single point in time. An example of cross-sectional data is dividend yields on 500 stocks as of the end of a year.

Which of the following statements regarding hypothesis testing is least accurate? A) The significance level is the risk of making a type I error. B) A type I error is acceptance of a hypothesis that is actually false. C) A type II error is the acceptance of a hypothesis that is actually false.

B A type I error is the rejection of a hypothesis that is actually true.

For the last four years, the returns for XYZ Corporation's stock have been 10.4%, 8.1%, 3.2%, and 15.0%. The equivalent compound annual rate is: A) 8.9%. B) 9.1%. C) 9.2%.

B (1.104 × 1.081 × 1.032 × 1.15)^0.25 - 1 = 9.1%

A frequency polygon is best suited to summarizing: A) unstructured textual data. B) a distribution of numerical data. C) underlying trends over time.

B A frequency polygon depicts the shape and range of a distribution.

For a given stated annual rate of return, compared to the effective rate of return with discrete compounding, the effective rate of return with continuous compounding will be: A) the same. B) higher. C) lower.

B A higher frequency of compounding leads to a higher effective rate of return. The effective rate of return with continuous compounding will, therefore, be greater than any effective rate of return with discrete compounding.

An analyst is testing the hypothesis that the mean excess return from a trading strategy is less than or equal to zero. The analyst reports that this hypothesis test produces a p-value of 0.034. This result most likely suggests that the: A) best estimate of the mean excess return produced by the strategy is 3.4%. B) null hypothesis can be rejected at the 5% significance level. C) smallest significance level at which the null hypothesis can be rejected is 6.8%.

B A p-value of 0.035 means the hypothesis can be rejected at a significance level of 3.5% or higher. Thus, the hypothesis can be rejected at the 10% or 5% significance level, but cannot be rejected at the 1% significance level.

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) mean-rank test. B) nonparametric test. C) parametric test.

B A rank correlation test is best described as a nonparametric test.

Student's t-Distribution Level of Significance for One-Tailed Test df | 0.100 | 0.050 | 0.025 | 0.01 | 0.005 | 0.0005 Level of Significance for Two-Tailed Test df | 0.20 | 0.10 | 0.05 | 0.02 | 0.01 | 0.001 28 | 1.313 | 1.701 | 2.048 | 2.467 | 2.763 | 3.674 29 | 1.311 | 1.699 | 2.045 | 2.462 | 2.756 | 3.659 30 | 1.310 | 1.697 | 2.042 | 2.457 | 2.750 | 3.646 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 greater than 100. B) fail to reject the null hypothesis and conclude that the population mean is not greater than 100. C) reject the null hypothesis and conclude that the population mean is greater than 100.

B 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.

A contingency table can be used to test: A) a null hypothesis that rank correlations are equal to zero. B) whether multiple characteristics of a population are independent. C) the number of p-values from multiple tests that are less than adjusted critical values.

B Explanation A contingency table is used to determine whether two characteristics of a group are independent.

The appropriate test statistic for a test of the equality of variances for two normally distributed random variables, based on two independent random samples, is: A) the t-test. B) the F-test. C) the X2 test.

B Explanation The F-test is the appropriate test.

Use the following segment of the Student's t-distribution for this question. Level of Significance for One-Tailed Test df | 0.100 | 0.050 | 0.025 | 0.01 Level of Significance for Two-Tailed Test df | 0.20 | 0.10 | 0.05 | 0.02 11 | 1.363 | 1.796 | 2.201 | 2.718 12 | 1.356 | 1.782 | 2.179 | 2.681 13 | 1.350 | 1.771 | 2.160 | 2.650 14 | 1.345 | 1.761 | 2.145 | 2.624 15 | 1.341 | 1.753 | 2.131 | 2.602 From a sample of 14 observations, an analyst calculates a t-statistic to test a hypothesis that the population mean is equal to zero. If the analyst chooses a 5% significance level, the appropriate critical value is: A) less than 1.80. B) greater than 2.15. C) between 1.80 and 2.15.

B Explanation This is a two-tailed test with 14 - 1 = 13 degrees of freedom. From the t-table, 2.160 is the critical value to which the analyst should compare the calculated t-statistic.

Consider the following analysis of variance (ANOVA) table: Source | Sum of squares | Degrees of freedom | Mean sum of squares Regression | 550 | 1 | 550.000 Error | 750 | 38 | 19.737 Total | 1,300 | 39 The F-statistic for the test of the fit of the model is closest to: A) 0.42. B) 27.87. C) 0.97.

B F = sum of squares regression / mean squared error = 550 / 19.737 = 27.867.

If the probability of a Type I error decreases, then the probability of: A) incorrectly rejecting the null increases. B) a Type II error increases. C) incorrectly accepting the null decreases.

B If P(Type I error) decreases, then P(Type II error) increases. A null hypothesis is never accepted. We can only fail to reject the null.

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.

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.

Results from a regression analysis based on 36 observations are presented in the following figures. Estimated Coefficients Coefficient | Coefficient Estimate | Standard Error b0 | 0.0023 | 0.0022 b1 | 1.1163 | 0.0624 Partial ANOVA Table Source of Variation | Sum of Squares Regression (explained) | 0.0228 Error (unexplained) | 0.0024 To test the following hypothesis: H0: b1 ≤ 1 versus H1: b1 > 1, at the 1% significance level, the calculated t-statistic and the appropriate conclusion are: Calculated t-statistic | Appropriate conclusion A) 1.86 | Reject H0 B) 1.86 | Fail to reject H0 C) 2.44 | Reject H0

B Note that this is a one-tailed test. The critical one-tailed 1% t-value with 34 degrees of freedom is approximately 2.44. The calculated t-statistic for the slope coefficient is (1.1163 − 1) / 0.0624 = 1.86. Therefore, the slope coefficient is not statistically different from one at the 1% significance level and the analyst should fail to reject the null hypothesis.

Consider the following analysis of variance (ANOVA) table: Source | Sum of squares | Degrees of freedom | Mean sum of squares Regression | 556 | 1 | 556 Error | 679 | 50 | 13.5 Total | 1,235 | 51 The R2 for this regression is closest to: A) 0.55. B) 0.45. C) 0.82.

B R2 = sum of squares regression / sum of squares total = 556 / 1,235 = 0.45.

An analyst who wants to display the relationship between two variables graphically is most likely to use: A) a histogram. B) a scatterplot. C) a frequency polygon.

B Scatterplots illustrate the relationship between two variables. Histograms and frequency polygons show the distribution of observations for a single variable.

A client will move his investment account unless the portfolio manager earns at least a 10% rate of return on his account. The rate of return for the portfolio that the portfolio manager has chosen has a normal probability distribution with an expected return of 19% and a standard deviation of 4.5%. What is the probability that the portfolio manager will keep this account? A) 0.750. B) 0.950. C) 0.977.

B Since we are only concerned with values that are below a 10% return this is a 1 tailed test to the left of the mean on the normal curve. With μ = 19 and σ = 4.5, P(X ≥ 10) = P(X ≥ μ - 2σ) therefore looking up -2 on the cumulative Z table gives us a value of 0.0228, meaning that (1 - 0.0228) = 97.72% of the area under the normal curve is above a Z score of -2. Since the Z score of -2 corresponds with the lower level 10% rate of return of the portfolio this means that there is a 97.72% probability that the portfolio will earn at least a 10% rate of return.

Ralph will retire 15 years from today and has saved $121,000 in his investment account for retirement. He believes he will need $37,000 at the beginning of each year for 25 years of retirement, with the first withdrawal on the day he retires. Ralph assumes that his investment account will return 8%. The amount he needs to deposit at the beginning of this year and each of the following 14 years (15 deposits in all) is closest to: A) $1,350. B) $1,450. C) $1,550.

B Step 1: Calculate the amount needed at retirement at t = 15, with your calculator in BGN mode. N = 25, FV = 0, I/Y = 8, PMT = 37,000, CPT PV = -426,564 Step 2: Calculate the required deposits at t = 0,1,....,14 to result in a time 15 value of 426,564, with your calculator still in BGN mode. PV = -121,000, N = 15, I/Y = 8, FV = 426,564, CPT PMT = -$1,457.21

An analyst is interested in predicting annual sales for XYZ Company, a maker of paper products. The following table reports a regression of the annual sales for XYZ against paper product industry sales. Regression Output: Parameters | Coefficient | Standard Error of the Coefficient Intercept | -94.88 | 32.97 Slope (industry sales) | 0.2796 | 0.0363 The correlation between company and industry sales is 0.9757. The regression was based on five observations. Which of the following is closest to the value and reports the most likely interpretation of the R2 for this regression? The R2 is:

B The R2 is computed as the correlation squared: (0.9757)2 = 0.952. The interpretation of this R2 is that 95.2% of the variation in Company XYZ's sales is explained by the variation in industry sales. The independent variable (industry sales) explains the variation in the dependent variable (company sales). This interpretation is based on the economic reasoning used in constructing the regression model.

If the probability of an event is 0.20, what are the odds against the event occurring? A) Five to one. B) Four to one. C) One to four.

B The answer can be determined by dividing the probability of the event by the probability that it will not occur: (1/5) / (4/5) = 1 to 4. The odds against the event occurring is four to one, i.e. in five occurrences of the event, it is expected that it will occur once and not occur four times.

The appropriate regression model for a linear relationship between the relative change in an independent variable and the absolute change in the dependent variable is a: A) log-lin model. B) lin-log model. C) lin-lin model.

B The appropriate model would be a lin-log model, in which the values of the dependent variable (Y) are regressed on the natural logarithms of the independent variable (X), Y = b0 + b1 ln X

Allan Jabber invested $400 at the beginning of each of the last 12 months in the shares of a mutual fund that paid no dividends. Which method will he correctly choose to calculate his average price per share from the monthly share prices? A) Arithmetic mean. B) Harmonic mean. C) Geometric mean.

B The harmonic mean of the 12 purchase prices will be his average price paid per share.

Nikki Ali and Donald Ankard borrowed $15,000 to finance their wedding and reception. The fully amortizing loan at 11% requires equal payments at the end of each of the next seven years. The principal portion of the first payment is closest to: A) $1,500. B) $1,530. C) $1,560.

B The interest portion of the first payment is simply principal × interest rate = (15,000 × 0.11) = 1,650. Using a financial calculator: PV = 15,000, FV = 0, I/Y = 11, N = 7, CPT PMT= $3,183 Principal = payment − interest = 3,183 − 1,650 = 1,533

An analyst wants to construct a hypothesis test to determine whether the mean weekly return on a stock is positive. The null hypothesis for this test should be that the mean return is: A) greater than zero. B) less than or equal to zero. C) greater than or equal to zero.

B The null hypothesis should state the condition which, if rejected, would lend statistical support to the alternative hypothesis. Here, the null hypothesis should be that the mean return is less than or equal to zero, and the alternative hypothesis should be that the mean return is greater than zero.

X and Y are independently distributed random variables. The probability of X is 30% and the probability of Y is 40%. Which of the following is closest to the probability that either X or Y will occur? A) 70%. B) 58%. C) 12%.

B The probability of X or Y is P(X) + P(Y) − P(XY). 0.3 + 0.4 − (0.3)(0.4) = 58%

The mean return of a portfolio is 20% and its standard deviation is 4%. The returns are normally distributed. Which of the following statements about this distribution are least accurate? The probability of receiving a return: A) between 12% and 28% is 0.95. B) in excess of 16% is 0.16. C) of less than 12% is 0.025.

B The probability of receiving a return greater than 16% is calculated by adding the probability of a return between 16% and 20% (given a mean of 20% and a standard deviation of 4%, this interval is the left tail of one standard deviation from the mean, which includes 34% of the observations.) to the area from 20% and higher (which starts at the mean and increases to infinity and includes 50% of the observations.) The probability of a return greater than 16% is 34 + 50 = 84%. Note: 0.16 is the probability of receiving a return less than 16%.

A study was conducted by the British Department of Transportation to estimate urban travel time between locations in London, England. Data was collected for motorcycles and passenger cars. Simple linear regression was conducted using data sets for both types of vehicles, where Y = urban travel time in minutes and X = distance between locations in kilometers. The following results were obtained: Regression Results for Travel Times Between Distances in London Passenger cars: Y_hat = 1.85 + 3.86X | R2 = 0.758 Motorcycles: Y_hat = 2.50 + 1.93X | R2 = 0.676 The estimated increase in travel time for a motorcycle commuter planning to move 8 km farther from his workplace in London is closest to: A) 31 minutes. B) 15 minutes. C) 0.154 hours.

B The slope coefficient is 1.93, indicating that each additional kilometer increases travel time by 1.93 minutes: 1.93 × 8 = 15.44

What is the most appropriate interpretation of a slope coefficient estimate equal to 10.0? A) The predicted value of the dependent variable when the independent variable is zero is 10.0. B) For every one unit change in the independent variable, the model predicts that the dependent variable will change by 10 units. C) For every 1-unit change in the independent variable, the model predicts that the dependent variable will change by 0.1 units.

B The slope coefficient is best interpreted as the predicted change in the dependent variable for a 1-unit change in the independent variable. If the slope coefficient estimate is 10.0 and the independent variable changes by 1 unit, the dependent variable is expected to change by 10 units. The intercept term is best interpreted as the value of the dependent variable when the independent variable is equal to zero.

An analyst is interested in predicting annual sales for XYZ Company, a maker of paper products. The following table reports a regression of the annual sales for XYZ against paper product industry sales. Regression Output: Parameters | Coefficient | Standard Error of the Coefficient Intercept | -94.88 | 32.97 Slope (industry sales) | 0.2796 | 0.0363 The correlation between company and industry sales is 0.9757. The regression was based on five observations. Based on the regression results, XYZ Company's market share of any increase in industry sales is expected to be closest to: A) 4%. B) 28%. C) 45%.

B The slope coefficient of 0.2796 indicates that a $1 million increase in industry sales will result in an increase in firm sales of approximately 28% of that amount ($279,600).

Joe Sutton is evaluating the effects of the 1987 market decline on the volume of trading. Specifically, he wants to test whether the decline affected trading volume. He selected a sample of 500 companies and collected data on the total annual volume for one year prior to the decline and for one year following the decline. What is the set of hypotheses that Sutton is testing? A) H0: µd = µd0 versus Ha: µd > µd0. B) H0: µd = µd0 versus Ha: µd ≠ µd0. C) H0: µd ≠ µd0 versus Ha: µd = µd0.

B 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.

The power of the test is: A) equal to the level of confidence. B) the probability of rejecting a false null hypothesis. C) the probability of rejecting a true null hypothesis.

B This is the definition of the power of the test: the probability of correctly rejecting the null hypothesis (rejecting the null hypothesis when it is false).

An analyst plans to use the following test statistic: t_(n−1) = (x − μ) / (s / √n) This test statistic is appropriate for a hypothesis about: A) the equality of two population means of two normally distributed populations based on independent samples. B) the population mean of a normal distribution with unknown variance. C) the mean difference of two normal populations.

B When testing hypotheses about the population mean, the sample standard deviation must be used in the denominator of the test statistic when the population standard deviation is unknown, the population is normal, and/or the sample is large. The statistic is a t-stat with n − 1 degrees of freedom. The numerator is the sampling error for the population mean if the true mean is µ0 and the denominator is the standard error of the sample mean around the true mean.

Student's t-distribution, level of significance 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 coefficient of −0.525 from a sample size of 19, an analyst calculates a t-statistic of [−0.525 √(19−2)] / [√(1 − (−0.525)^2)] = −2.5433. The analyst can reject the hypothesis that the population correlation coefficient equals zero: A) at a 2% significance level, but not at a 1% significance level. B) at a 5% significance level, but not at a 2% significance level. C) at a 1% significance level.

B With 19 − 2 = 17 degrees of freedom, the critical values are plus-or-minus 2.110 at a 5% significance level, 2.567 at a 2% significance level, and 2.898 at a 1% significance level. Because the t-statistic of −2.5433 is less than −2.110, the hypothesis can be rejected at a 5% significance level. Because the t-statistic is greater than −2.567, the hypothesis cannot be rejected at a 2% significance level (or any smaller significance level).

The average annual return over 20 years for a sector of mutual funds, calculated for the population of funds in that sector that have 20 years of performance history, is most likely to: A) understate returns for the fund sector. B) fairly state returns for the fund sector. C) overstate returns for the fund sector.

C A sample suffers from survivorship bias if only surviving funds are captured in the data. Funds that cease to exist (due to poor performance) are excluded. This results in an average that overstates the annual return an investor in the sector could actually have expected to earn over the period.

To test a hypothesis that the population correlation coefficient of two variables is equal to zero, an analyst collects a sample of 24 observations and calculates a sample correlation coefficient of 0.37. Can the analyst test this hypothesis using only these two inputs? A) No, because the sample means of the two variables are also required. B) No, because the sample standard deviations of the two variables are also required. C) Yes.

C The t-statistic for a test of the population correlation coefficient is [r√(n−2)] / [√(1−r2)], where r is the sample correlation coefficient and n is the sample size.

An investment product promises to pay a lump sum of $25,458 at the end of 9 years. If an investor feels this investment should produce a rate of return of 14%, compounded annually, the present value is closest to: A) $9,426.00. B) $7,618.00. C) $7,829.00.

C 25,458 / 1.149 = 7,828.54 Alternatively, N = 9; I/Y = 14; FV = -25,458; PMT = 0; CPT → PV = $7,828.54.

For a hypothesis test with a probability of a Type II error of 60% and a probability of a Type I error of 5%, which of the following statements is most accurate? A) The power of the test is 40%, and there is a 5% probability that the test statistic will exceed the critical value(s). B) There is a 95% probability that the test statistic will be between the critical values if this is a two-tailed test. C) There is a 5% probability that the null hypothesis will be rejected when actually true, and the probability of rejecting the null when it is false is 40%.

C A Type I error is rejecting the null hypothesis when it's true. The probability of rejecting a false null is [1 - Prob Type II] = [1 - 0.60] = 40%, which is called the power of the test. A and B are not necessarily true, since the null may be false and the probability of rejection unknown

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) A Type I error represents the failure to reject the null hypothesis when it is, in fact, false. B) One minus the confidence level of the test represents the probability of making a Type II error. C) The significance level of the test represents the probability of making a Type I error.

C 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.

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.

C This is a one-sided alternative because of the "greater than" belief. We expect to reject the null.

Which of the following statements about probability distributions is least accurate? A) Continuous uniform distributions have cumulative distribution functions that are straight lines from zero to one. B) The probability that a continuously distributed random variable will take on a specific value is always zero. C) A normally distributed random variable divided by its standard deviation will follow a standard normal probability distribution.

C A standard normal probability distribution has a mean of zero, so subtracting the mean from a normal random variable before dividing by its standard deviation is necessary to produce a standard normal probability distribution

Joe Bay, CFA, wants to test the hypothesis that the variance of returns on energy stocks is equal to the variance of returns on transportation stocks. Bay assumes the samples are independent and the returns are normally distributed. The appropriate test statistic for this hypothesis is: A) a t-statistic. B) a Chi-square statistic. C) an F-statistic.

C Bay is testing a hypothesis about the equality of variances of two normally distributed populations. The test statistic used to test this hypothesis is an F-statistic. A chi-square statistic is used to test a hypothesis about the variance of a single population. A t-statistic is used to test hypotheses concerning a population mean, the differences between means of two populations, or the mean of differences between paired observations from two populations.

The appropriate test statistic to test the hypothesis that the variance of a normally distributed population is equal to 13 is: A) the t-test. B) the F-test. C) the χ2 test.

C Explanation A test of the population variance is a chi-square test.

Which of the following assumptions is least likely required for the difference in means test based on two samples? A) The two samples are independent. B) The two populations are normally distributed. C) The two populations have known variances.

C Explanation The difference-in-means test does not require the two population variances to be known.

In a simple regression model, the least squares criterion is to minimize the sum of squared differences between: A) the estimated and actual slope coefficient. B) the intercept term and the residual term. C) the predicted and actual values of the dependent variable.

C Explanation The least squares criterion defines the best-fitting linear relationship as the one that minimizes the sum of squared errors, the squared vertical distances between the predicted and actual values of the dependent variable.

To test whether the mean of a population is greater than 20, the appropriate null hypothesis is that the population mean is: A) less than 20. B) greater than 20. C) less than or equal to 20.

C Explanation To test whether the population mean is greater than 20, the test would attempt to reject the null hypothesis that the mean is less than or equal to 20. The null hypothesis must always include the "equal to" condition.

Given the relationship: Y = 2.83 + 1.5X What is the predicted value of the dependent variable when the value of the independent variable equals 2? A) 2.83. B) -0.55. C) 5.83.

C Explanation Y = 2.83 + (1.5)(2) = 2.83 + 3 = 5.83.

Which of the following statements about parametric and nonparametric tests is least accurate? A) The test of the mean of the differences is used when performing a paired comparison. B) The test of the difference in means is used when you are comparing means from two independent samples. C) Nonparametric tests rely on population parameters.

C 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.

Which of the following statements about hypothesis testing is most accurate? A) A Type II error is rejecting the null when it is actually true. B) The significance level equals one minus the probability of a Type I error. C) A two-tailed test with a significance level of 5% has z-critical values of ±1.96.

C Rejecting the null when it is actually true is a Type I error. A Type II error is failing to reject the null hypothesis when it is false. The significance level equals the probability of a Type I error

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. He takes a sample to determine whether the mean length of Northern Pike in the lake exceeds 18 inches. If the sample t-statistic indicates that the mean length of the fish is significantly greater than 18 inches, when the population mean is actually 17.8 inches, the t-test resulted in: A) a Type II error only. B) both a Type I and a Type II error. C) a Type I error only.

C Rejection of a null hypothesis when it is actually true is a Type I error. Here, Ho: μ ≤ 18 inches and Ha: μ > 18 inches. Type II error is failing to reject a null hypothesis when it is actually false. Because a Type I error can only occur if the null hypothesis is true, and a Type II error can only occur if the null hypothesis is false, it is logically impossible for a test to result in both types of error at the same time.

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.

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.

Results from a regression analysis based on 36 observations are presented in the following figures. Estimated Coefficients Coefficient | Coefficient Estimate | Standard Error b0 | 0.0023 | 0.0022 b1 | 1.1163 | 0.0624 Partial ANOVA Table Source of Variation | Sum of Squares Regression (explained) | 0.0228 Error (unexplained) | 0.0024 Are the intercept term and the slope coefficient statistically significantly different from zero at the 5% significance level? Intercept term significant? | Slope coefficient significant? A) Yes | Yes B) Yes | No C) No | Yes

C The critical two-tailed 5% t-value with 34 degrees of freedom is approximately 2.03. The calculated t-statistics for the intercept term and slope coefficient are, respectively, 0.0023 / 0.0022 = 1.05 and 1.1163 / 0.0624 = 17.9. Therefore, the intercept term is not statistically different from zero at the 5% significance level, while the slope coefficient is.

For a regression model of Y = 5 + 3.5X, the analysis (based on a large data sample) provides the standard error of the forecast as 2.5 and the standard error of the slope coefficient as 0.8. A 90% confidence interval for the estimate of Y when the value of the independent variable is 10 is closest to: A) 35.1 to 44.9. B) 35.6 to 44.4. C) 35.9 to 44.1.

C The estimate of Y, given X = 10 is: Y = 5 + 3.5(10) = 40. The critical value for a 90% confidence interval with a large sample size (z-statistic) is approximately 1.65. Given the standard error of the forecast of 2.5, the confidence interval for the estimated value of Y is 40 +/- 1.65(2.5) = 35.875 to 44.125

A study was conducted by the British Department of Transportation to estimate urban travel time between locations in London, England. Data was collected for motorcycles and passenger cars. Simple linear regression was conducted using data sets for both types of vehicles, where Y = urban travel time in minutes and X = distance between locations in kilometers. The following results were obtained: Regression Results for Travel Times Between Distances in London Passenger cars: Y_hat = 1.85 + 3.86X | R2 = 0.758 Motorcycles: Y_hat = 2.50 + 1.93X | R2 = 0.676 Based on the regression results, which model is more reliable? A) The passenger car model because 3.86 > 1.93. B) The motorcycle model because 1.93 < 3.86. C) The passenger car model because 0.758 > 0.676.

C The higher R2 for the passenger car model indicates that regression results are more reliable. Distance is a better predictor of travel time for cars. Perhaps the aggressiveness of the driver is a bigger factor in travel time for motorcycles than it is for autos.

In the process of hypothesis testing, what is the proper order for these steps? A) Collect the sample and calculate the sample statistics. State the hypotheses. Specify the level of significance. Make a decision. B) Specify the level of significance. State the hypotheses. Make a decision. Collect the sample and calculate the sample statistics. C) State the hypotheses. Specify the level of significance. Collect the sample and calculate the test statistics. Make a decision.

C 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.

Which of the following is least likely a necessary assumption of simple linear regression analysis? A) The residuals are normally distributed. B) There is a constant variance of the error term. C) The dependent variable is uncorrelated with the residuals.

C The model does not assume that the dependent variable is uncorrelated with the residuals. It does assume that the independent variable is uncorrelated with the residuals.

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) To test the null hypothesis, the analyst must determine the exact sample size and calculate the degrees of freedom for the test. B) The analyst should reject the null hypothesis and conclude that the earnings estimates are significantly different from reported earnings. C) The analyst should fail to reject the null hypothesis and conclude that the earnings estimates are not significantly different from reported earnings.

C 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 critical 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.

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.

C The null must be either equal to, less than or equal to, or greater than or equal to.

The variation in the dependent variable explained by the independent variable is measured by: A) the mean squared error. B) the sum of squared errors. C) the regression sum of squares.

C The regression sum of squares measures the amount of variation in the dependent variable explained by the independent variable (i.e., the explained variation). The sum of squared errors measures the variation in the dependent variable not explained by the independent variable. The mean squared error is equal to the sum of squared errors divided by its degrees of freedom.

Which of the following statements about testing a hypothesis using a Z-test is least accurate? A) 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. B) If the calculated Z-statistic lies outside the critical Z-statistic range, the null hypothesis can be rejected. C) The calculated Z-statistic determines the appropriate significance level to use.

C The significance level is chosen before the test so the calculated Z-statistic can be compared to an appropriate critical value.

A researcher has 28 quarterly excess returns to an investment strategy and believes these returns are approximately normally distributed. The mean return on this sample is 1.645% and the standard deviation is 5.29%. For a test with a 5% significance level of the hypothesis that excess returns are less than or equal to zero, the researcher should: A) reject the null hypothesis because the critical value for the test is 1.645. B) not draw any conclusion because the sample size is less than 30. C) fail to reject the null because the critical value is greater than 1.645.

C The standard error is (5.29/√28) = 1.0. Test statistic = 1.645/1.0 = 1.645. The critical value for a t-test is greater than the critical value for a z-test at a 5% significance level (which is 1.645 for a one-tailed test), so the calculated test statistic of 1.645 must be less than the critical value for a t-test (which is 1.703 for a one-tailed test with 27 degrees of freedom) and we cannot reject the null hypothesis that mean excess return is less than or equal to zero

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.

C This is a one-sided alternative because of the "less than" belief.

An economist estimates a 60% probability that the economy will expand next year. The technology sector has a 70% probability of outperforming the market if the economy expands and a 10% probability of outperforming the market if the economy does not expand. Given the new information that the technology sector will not outperform the market, the probability that the economy will not expand is closest to: A) 33%. B) 54%. C) 67%.

C Using the new information we can use Bayes' formula to update the probability. P(economy does not expand | tech does not outperform) = P(economy does not expand and tech does not outperform) / P(tech does not outperform). P(economy does not expand and tech does not outperform) = P(tech does not outperform | economy does not expand) × P(economy does not expand) = 0.90 × 0.40 = 0.36. P(economy does expand and tech does not outperform) = P(tech does not outperform | economy does expand) × P(economy does expand) = 0.30 × 0.60 = 0.18. P(economy does not expand) = 1.00 - P(economy does expand) = 1.00 - 0.60 = 0.40. P(tech does not outperform | economy does not expand) = 1.00 - P(tech does outperform | economy does not expand) = 1.00 - 0.10 = 0.90. P(tech does not outperform) = P(tech does not outperform and economy does not expand) + P(tech does not outperform and economy does expand) = 0.36 + 0.18 = 0.54. P(economy does not expand | tech does not outperform) = P(economy does not expand and tech does not outperform) / P(tech does not outperform) = 0.36 / 0.54 = 0.67.

Which of the following statements about hypothesis testing is most accurate? A) A Type I error is rejecting the null hypothesis when it is true, and a Type II error is rejecting the alternative hypothesis when it is true. B) A hypothesis that the population mean is less than or equal to 5 should be rejected when the critical Z-statistic is greater than the sample Z-statistic. C) A hypothesized mean of 3, a sample mean of 6, and a standard error of the sampling means of 2 give a sample Z-statistic of 1.5.

C Z = (6 - 3)/2 = 1.5. A Type II error is failing to reject the null hypothesis when it is false. The null hypothesis that the population mean is less than or equal to 5 should be rejected when the sample Z-statistic is greater than the critical Z-statistic.