Chapter 9: Q.74 (page 589)

An investor with a stock portfolio worth several hundred thousand dollars sued his broker due to the low returns he got from the portfolio at a time when the stock market did well overall. The investor’s lawyer wants to compare the broker’s performance against the market as a whole. He collects data on the broker’s returns for a
random sample of 36 weeks. Over the 10-year period that the broker has managed portfolios, stocks in the Standard & Poor’s 500 index gained an average of 0.95% per month. The Minitab output below displays descriptive statistics for these data, along with the results of a significance test.
(a) Determine whether there are any outliers. Show your work.
(b) Interpret the P-value in context.
(c) Do these data give convincing evidence to support the lawyer’s case? Carry out a test to help you answer this question.

Short Answer

Expert verified

a. There are no outliers

b. $0.003$

c. There is convincing evidence to support the lawyer’s case

Step by step solution

Introduction

An investor with a stock portfolio worth several hundred thousand dollars sued his broker due to the low returns he got from the portfolio at a time when the stock market did well overall. The investor’s lawyer wants to compare the broker’s performance against the market as a whole. Over the $10$ -year period that the broker has managed portfolios, stocks in the Standard & Poor’s $500$ index gained an average of $0.95 %$ per month.

Explanation Part (a)

Calculating the lower and upper boundaries,

$Q_{1} = - 3.418 a n d Q_{3} = 1.543$

$I Q R = Q_{3} - Q_{1} = 1.543 - (- 3.418) = 4.961$

Lower boundary,

$= Q_{1} - 1.5 \times I Q R = - 3.418 - 1.5 \times 4.961 = - 10.8595$

Upper boundary,

$= Q_{3} + 1.5 \times I Q R = 1.543 + 1.5 \times 4.961 = 8.9845$

Hence the values lie between the boundaries and there are no outliers.

Explanation Part (b)

Probability of getting a random sample of $36$ weeks with an average return of $- 1.441$ or less is approximately $0.003$ if the average percentage of return is $0.95$ per month.

Explanation Part (c)

Calculating the null and alternative hypotheses,

$H_{0} : μ = 0.95 H_{0} : μ < 0.95$

The population is of enormous size, so we can approximately expect it to be a typical distribution. There will be over $360$ weeks where the securities exchange did well, so $10 %$ condition is likewise satisfied.

A one example t test is run and we get test statistic of t = $- 2.98$ and the corresponding p value as $0.003$ . As the p value is not exactly the degree of significance, we have sufficient evidence at $5 %$ level of significance to dismiss the null hypothesis.