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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.

Calculating the lower and upper boundaries,

$Q_1=−3.418and{Q}_3=1.543$

$IQR=Q_3−Q_1=1.543−(−3.418)=4.961$

Lower boundary,

$=Q_1−1.5×IQR=-3.418-1.5×4.961=-10.8595$

Upper boundary,

$=Q_3+1.5×IQR=1.543+1.5×4.961=8.9845$

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

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.

Calculating the null and alternative hypotheses,

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

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.