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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 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. localid="1650714832411" $Q_3=1.543$

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

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.

Decision:You can reject$Hâ‚\P Ä$ at the significance level $0.05$, because your p-value does not exceed $0.05$.

State: $H_0:\mathrm{μ}=0.95versus{H}_a:\mathrm{μ}<0.95$, where $\mathrm{μ}$is the actual mean broker's returns for a random sample of $36$weeks.

Plan: One-sample t test for $\mathrm{μ}$.

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.