An investor with a stock portfolio worth several hundred thousand dollars sued his broker because he claimed that lack of diversification in his portfolio led to consistent poor performance over a period of more than $3$ years. The file below gives the monthly returns of the investor's account for the $39$ month period in question. A financial arbitration panel compared these returns with the benchmark S&P $500$ performance during the same period, and ended up awarding "substantial damages" to the investor.

Download the following data set and use JMP to test if the investor's monthly returns are significantly different than the S&P $500$: Monthly Returns.

$($a$)$ During these $39$ months, the S&P $500$ had an average monthly return of $1\mathrm{.}1\%.$ What was the average monthly return for the investor's brokerage account over this period?

$\%(4$ decimals$)$

$($b$)$ Which hypotheses should be used to test if the investor's mean monthly return was different than the S&P $500?$

H$0$:

p

$=$

$-1\mathrm{.}0997$

Ha:

p

$!=$

$-1\mathrm{.}0997$

$($c$)$ Use JMP to carry out the above hypothesis test:

Identify the test statistic:

$(4$ decimals$)$

Identify the degrees of freedom $($DF$)$:

Identify the correct p$-$value:

$(4$ decimals$)$

$($d$)$ For $=0\mathrm{.}05,$ should the arbitration panel conclude the mean return of the investor's brokerage account is significantly different from the S&P $500?$

$---$

$,$ there is

$---$

evidence these returns are

$---$

the S&P $500.$

$($e$)$ For $=0\mathrm{.}01,$ should the arbitration panel conclude the mean return of the investor's brokerage account is significantly different from the S&P $500?$

$---$

$,$ there is

$---$

evidence these returns are

$---$

the S&P $500.$

$($f$)$ The investor would like to claim the returns on his account are significantly different than the S&P $500.$ In contrast, the brokerage would like to claim these returns are not significantly different from the S&P $500,$ and any observed differences are merely due to ordinary random fluctuation. Which significance level should each party use?

The investor should use

$---$

; the brokerage should use

$---$

$.$