Q.74 - The Practice of Statistics for AP

Question

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.

Step-by-Step Solution

Answer

a. There are no outliers 

b.$0.003$

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

Step 1: 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.

Step 2: Explanation Part (a)

Calculating the lower and upper boundaries,

$Q_1=-3.418 and Q_3=1.543$

$IQR=Q_3-Q_1=1.543-(-3.418)=4.961$

Lower boundary,

$=Q_1-1.5\times IQR=-3.418-1.5\times 4.961=-10.8595$

Upper boundary,

$=Q_3+1.5\times IQR=1.543+1.5\times 4.961=8.9845$

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

Step 3: 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.

Step 4: Explanation Part (c)

Calculating the null and alternative hypotheses,

$$\begin{gathered} H_0:\mu =0.95 \\ {H}_0:\mu <0.95 \end{gathered}$$

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.