quant quiz

A chi-square test is most appropriate for tests concerning: a single variance. differences between two population means with variances assumed to be equal. differences between two population means with variances assumed to not be equal.

A is correct. A chi-square test is used for tests concerning the variance of a single normally distributed population.

An increase in which of the following items will most likely result in a wider confidence interval for the population mean? Reliability factor Sample size Degrees of freedom

A is correct. An increase in the reliability factor (the degree of confidence) increases the width of the confidence interval. Increasing the sample size and increasing the degrees of freedom both shrink the confidence interval.

A distribution with excess kurtosis less than zero is termed: mesokurtic. platykurtic. leptokurtic.

B is correct. A platykurtic distribution has excess kurtosis less than zero.

Which of the following should be used to test the difference between the variances of two normally distributed populations? t-test F-test Paired comparisons test

B is correct. An F-test is used to conduct tests concerning the difference between the variances of two normally distributed populations with random independent samples.

An estimator with an expected value equal to the parameter that it is intended to estimate is described as: efficient. unbiased. consistent.

B is correct. An unbiased estimator is one for which the expected value equals the parameter it is intended to estimate.

The total number of parameters that fully characterizes a multivariate normal distribution for the returns on two stocks is: 3. 4. 5.

C is correct. A bivariate normal distribution (two stocks) will have two means, two variances and one correlation. A multivariate normal distribution for the returns on n stocks will have n means, n variances and n(n - 1)/2 distinct correlations.

The confidence interval is most likely to be: wider as the sample size increases. wider as the point estimate increases. narrower as the reliability factor decreases.

C is correct. A confidence interval for a parameter = Point estimate ± Reliability factor × Standard error. For example, the reliability factors for confidence intervals based on the standard normal distribution are 1.65 for 90% confidence intervals and 1.96 for 95% confidence intervals. For a given point estimate and standard error, the confidence interval will be narrower with a lower reliability factor.

Compared with analytical methods, what are the strengths and weaknesses of Monte Carlo simulation for use in valuing securities?

Strengths. Monte Carlo simulation can be used to price complex securities for which no analytic expression is available, particularly European-style options. Weaknesses. Monte Carlo simulation provides only statistical estimates, not exact results. Analytic methods, when available, provide more insight into cause-and-effect relationships than does Monte Carlo simulation.

As the t-distribution's degrees of freedom decrease, the t-distribution most likely: exhibits tails that become fatter. approaches a standard normal distribution. becomes asymmetrically distributed around its mean value.

A is correct. A standard normal distribution has tails that approach zero faster than the t-distribution. As degrees of freedom increase, the tails of the t-distribution become less fat and the t-distribution begins to look more like a standard normal distribution. But as degrees of freedom decrease, the tails of the t-distribution become fatter.

In descriptive statistics, an example of a parameter is the: median of a population. mean of a sample of observations. standard deviation of a sample of observations.

A is correct. Any descriptive measure of a population characteristic is referred to as a parameter.

For a two-sided confidence interval, an increase in the degree of confidence will result in: a wider confidence interval. a narrower confidence interval. no change in the width of the confidence interval.

A is correct. As the degree of confidence increases (e.g., from 95% to 99%), a given confidence interval will become wider. A confidence interval is a range for which one can assert with a given probability 1 - α, called the degree of confidence, that it will contain the parameter it is intended to estimate.

All else held constant, the width of a confidence interval for a population mean is most likely to be smaller if the sample size is: larger and the degree of confidence is lower. larger and the degree of confidence is higher. smaller and the degree of confidence is lower.

A is correct. As the degree of confidence is increased, the confidence interval becomes wider. A larger sample size decreases the width of a confidence interval.

The value of a test statistic is best described as the basis for deciding whether to: reject the null hypothesis. accept the null hypothesis. reject the alternative hypothesis.

A is correct. Calculated using a sample, a test statistic is a quantity whose value is the basis for deciding whether to reject the null hypothesis.

Which of the following is a Type I error? Rejecting a true null hypothesis Rejecting a false null hypothesis Failing to reject a false null hypothesis

A is correct. The definition of a Type I error is when a true null hypothesis is rejected.

The total probability rule is used when an analyst is interested in: all potential outcomes. a set of events. a single outcome.

A is correct. When the scenarios (conditioning events) are mutually exclusive and exhaustive, no possible outcomes are left out, thereby covering all potential outcomes.

A Type II error is best described as: rejecting a true null hypothesis. failing to reject a false null hypothesis. failing to reject a false alternative hypothesis.

B is correct. A Type II error occurs when a false null hypothesis is not rejected.

When evaluating mean differences between two dependent samples, the most appropriate test is a: chi-square test. paired comparisons test. z-test.

B is correct. A paired comparisons test is appropriate to test the mean differences of two samples believed to be dependent.

Which sampling bias is most likely investigated with an out-of-sample test? Look-ahead bias Data-mining bias Sample selection bias

B is correct. An out-of-sample test is used to investigate the presence of data-mining bias. Such a test uses a sample that does not overlap the time period of the sample on which a variable, strategy, or model was developed.

After estimating the probability that an investment manager will exceed his benchmark return in each of the next two quarters, an analyst wants to forecast the probability that the investment manager will exceed his benchmark return over the two-quarter period in total. Assuming that each quarter's performance is independent of the other, which probability rule should the analyst select? Addition rule Multiplication rule Total probability rule

B is correct. Because the events are independent, the multiplication rule is most appropriate for forecasting their joint probability. The multiplication rule for independent events states that the joint probability of both A and B occurring is P(AB) = P(A)P(B).

For a positively skewed unimodal distribution, which of the following measures is most accurately described as the largest? Median Mean Mode

B is correct. For a positively skewed unimodal distribution, the mode is less than the median, which is less than the mean.

A pooled estimator is used when testing a hypothesis concerning the: equality of the variances of two normally distributed populations. difference between the means of two at least approximately normally distributed populations with unknown but assumed equal variances. difference between the means of two at least approximately normally distributed populations with unknown and assumed unequal variances.

B is correct. The assumption that the variances are equal allows for the combining of both samples to obtain a pooled estimate of the common variance.

The central limit theorem is best described as stating that the sampling distribution of the sample mean will be approximately normal for large-size samples: if the population distribution is normal. for populations described by any probability distribution. if the population distribution is symmetrical.

B is correct. The central limit theorem holds without regard for the distribution of the underlying population.

The arithmetic and geometric mean are calculated for the same data. If there is variability in the data, compared with the arithmetic mean, the geometric mean will most likely be: greater. smaller. equal.

B is correct. The geometric mean is always less than or equal to the arithmetic mean. The only time the two means will be equal is when there is no variability in the observations.

Common stock prices are approximately lognormally distributed. Therefore, it is most likely that conventional (discrete) common stock prices are: leptokurtic. skewed to the right. skewed to the left.

B is correct. The lognormal distribution is truncated at zero and skewed to the right (positively skewed).

Which of the following is characteristic of the normal distribution? Asymmetry Kurtosis of 3 Definitive limits or boundaries

B is correct. The normal distribution has a skewness of 0, a kurtosis of 3, and a mean, median and mode that are all equal.

The probability of correctly rejecting the null hypothesis is the: p-value. power of a test. level of significance.

B is correct. The power of a test is the probability of rejecting the null hypothesis when it is false.

An analysis of US share prices determines that there is consistent underpricing by $0.02 with a p-value of 0.0012. Assuming an average transaction cost of $0.05, which statement is most accurate? The underpricing result is: statistically significant and indicates a possible arbitrage opportunity. not economically meaningful. not statistically significant.

B is correct. The underpricing result is not economically meaningful when the average transaction cost is taken into consideration.

The value of the cumulative distribution function F(x), where x is a particular outcome, for a discrete uniform distribution: sums to 1. lies between 0 and 1. decreases as x increases.

B is correct. The value of the cumulative distribution function lies between 0 and 1 for any x: 0 ≤ F(x) ≤ 1.

An analyst is examining the monthly returns for two funds over one year. Both funds' returns are non-normally distributed. To test whether the mean return of one fund is greater than the mean return of the other fund, the analyst can use: a parametric test only. a nonparametric test only. both parametric and nonparametric tests.

B is correct. There are only 12 (monthly) observations over the one year of the sample and thus the samples are small. Additionally, the funds' returns are non-normally distributed. Therefore, the samples do not meet the distributional assumptions for a parametric test. The Mann-Whitney U test (a nonparametric test) could be used to test the differences between population means.

A return distribution with frequent small gains and a few extreme losses is most likely to be called: leptokurtic. positively skewed. negatively skewed.

C is correct. A return distribution with negative skew has frequent small gains and a few extreme losses.

Which sampling-related bias is most likely to result in finding apparent significance when none exists? Sample selection bias Look-ahead bias Data mining bias

C is correct. Data mining bias comes from overuse or misuse of the data and can result in finding models or patterns where none exist. Sample selection bias often results when data availability leads to certain data being excluded from the analysis. Look-ahead bias exists if the model uses data not available to the analyst at the time the analyst act on the model.

A hypothesis test for a normally-distributed population at a 0.05 significance level implies a: 95% probability of rejecting a true null hypothesis. 95% probability of a Type I error for a two-tailed test. 5% critical value rejection region in a tail of the distribution for a one-tailed test.

C is correct. For a one-tailed hypothesis test, there is a 5% critical value rejection region in one tail of the distribution.

If the distribution of the population from which samples of size n are drawn is positively skewed and given that the sample size, n, is large, the sampling distribution of the sample means is most likely to have a: mean smaller than the mean of the entire population. variance equal to that of the entire population. distribution that is approximately normal.

C is correct. Given a population that has a finite variance and a large sample size, the central limit theorem establishes that the sampling distribution of sample means will be approximately normal, will have a mean equal to the population mean, and will have a variance equal to the population variance divided by the sample size.

When working backward from the nodes on a binomial tree diagram, the analyst is attempting to calculate: the number of potential outcomes. the probability of a given scenario. an expected value as of today.

C is correct. In a tree diagram, a problem is worked backward to formulate an expected value as of today.

The best approach for creating a stratified random sample of a population involves: drawing an equal number of simple random samples from each subpopulation. selecting every kth member of the population until the desired sample size is reached. drawing simple random samples from each subpopulation in sizes proportional to the relative size of each subpopulation.

C is correct. Stratified random sampling involves dividing a population into subpopulations based on one or more classification criteria. Then, simple random samples are drawn from each subpopulation in sizes proportional to the relative size of each subpopulation. These samples are then pooled to form a stratified random sample.

For a binomial random variable with five trials, and a probability of success on each trial of 0.50, the distribution will be: skewed. uniform. symmetric.

C is correct. The binomial distribution is symmetric when the probability of success on a trial is 0.50, but it is asymmetric or skewed otherwise. Here it is given that p = 0.50

A manager will select 20 bonds out of his universe of 100 bonds to construct a portfolio. Which formula provides the number of possible portfolios? Permutation formula Multinomial formula Combination formula

C is correct. The combination formula provides the number of ways that r objects can be chosen from a total of n objects, when the order in which the r objects are listed does not matter. The order of the bonds within the portfolio does not matter.

The covariance of returns is positive when the returns on two assets tend to: have the same expected values. be above their expected value at different times. be on the same side of their expected value at the same time.

C is correct. The covariance of returns is positive when the returns on both assets tend to be on the same side (above or below) their expected values at the same time, indicating an average positive relationship between returns.

Which of the following represents a correct statement about the p-value? The p-value offers less precise information than does the rejection points approach. A larger p-value provides stronger evidence in support of the alternative hypothesis. A p-value less than the specified level of significance leads to rejection of the null hypothesis.

C is correct. The p-value is the smallest level of significance at which the null hypothesis can be rejected for a given value of the test statistic. The null hypothesis is rejected when the p-value is less than the specified significance level.

The distribution of all the distinct possible values for a statistic when calculated from samples of the same size randomly drawn from the same population is most accurately referred to as: a discrete uniform distribution. a multivariate normal distribution. the sampling distribution of a statistic.

C is correct. The sampling distribution of a statistic (like a sample mean) is defined as the probability distribution of a given sample statistic when samples of the same size are randomly drawn from the same population.

Which of the following best describes how an analyst would estimate the expected value of a firm under the scenarios of bankruptcy and survivorship? The analyst would use: the addition rule. conditional expected values. the total probability rule for expected value.

C is correct. The total probability rule for expected value is used to estimate an expected value based on mutually exclusive and exhaustive scenarios.

Which of the following statements is correct with respect to the p-value? It is a less precise measure of test evidence than rejection points. It is the largest level of significance at which the null hypothesis is rejected. It can be compared directly with the level of significance in reaching test conclusions.

C is correct. When directly comparing the p-value with the level of significance, it can be used as an alternative to using rejection points to reach conclusions on hypothesis tests. If the p-value is smaller than the specified level of significance, the null hypothesis is rejected. Otherwise, the null hypothesis is not rejected.

In setting the confidence interval for the population mean of a normal or approximately normal distribution, and given that the sample size is small, Student's t-distribution is the most appropriate approach when the variance is: known. large. unknown.

C is correct. When the sample size is small (and the population is normally or approximately normally distributed), the Student's t-distribution is preferred if the variance is unknown.