$20\mathrm{.}48$ Mutual Funds Performance. Mutual funds often compare their performance with a benchmark provided by an "index" that describes the performance of the class of assets in which the fund invests. For example, the Vanguard International Growth Fund benchmarks its performance against the Spliced International Index. Table $20\mathrm{.}3$ gives annual returns $($percent$)$ for the fund and the index. Does the fund's mean annual percentage return differ significantly from that of its benchmark? Ilf MFUND

a$.$ Explain clearly why the matched pairs $t$ test is the proper choice to answer this question.

b$.$ Do a complete analysis that answers the question posed.

c$.$ The paired $t$ test uses the mean of the differences in the annual percentage returns to compare the fund's performance to that of its benchmark over time. What do you think of this as way to compare performance over time? $($Consider a simple example over two years. Fund $1$ has a $50\%$ annual return in year $1$ and a $-50\%$ return in year $2.$ The benchmark has a $0\%$ annual return both years. The mean of the differences in annual percentage returns is $0.$ However, what is an initial $$100$ investment in Fund $1$ worth after the two years? What is an initial $$100$ investment in the benchmark worth after two years?$)$

Table $20\mathrm{.}3$ A mutual fund versus its benchmark index

$\backslash$table$[[$Year$,\backslash$table$[[$Fund$],[$Return$]],\backslash$table$[[$Index$],[$Beturn $\%]],$Vear,Fund Return $\%,\backslash$table$[[$index$],[$Return $\%$