Q. 74 - The Practice of Statistics for AP

Question

Taking stock 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. Yes.

b. $p-value = 0.003$

c. It appears that these data give convincing evidence to support the lawyer's case.

Step 1: Given information

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.

Step 2: Explanation (part a)

The first quartile $Q1$, which represents a quarter of the way through the list of all data. $Q_1=-3.418$

The third quartile $Q3$, which represents three-quarters of the way through the list of all data. There is no third quartile. $Q_3=1.543$

Therefore, $IQR=4.961$ which is LESS than $MAX -Q_3$ and $Q_1- MIN$ . So, there are outliers.

Step 3: Explanation (part b)

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.  

$p-value = 0.00521286$

Decision: You can reject $H₀$ at the significance level $0.05$, because your p-value does not exceed $0.05$.

Step 4: Explanation (part c)

State: $H_0 : \mu = 0.95 versus H_a : \mu < 0.95$, where $\mu$ is the actual mean broker's returns for a random sample of $36$ weeks.

Plan: One-sample t test for $\mu$.

Random: The sample was randomly selected. 

Normal: The sample size was $36$, which is at least $30$.

Independent: There are clearly many more than $500$ index gained on an average of $0.95\%$ per month.

Do:$t =-2.98$, P-value is approximately $0$.

Conclude: Since our P-value is less than $0.05$, we reject $H_0$.

It appears that these data give convincing evidence to support the lawyer's case.