Table below gives summary statistics on annual returns of two portfolios. A portfolio of small

stocks, and a portfolio of big stocks. The last column shows data for a portfolio that buys the

small portfolio and shorts the big portfolio. The sample has N $=100$ observations.

Measure Small Big Difference

Mean $15\mathrm{.}50\%11\mathrm{.}00\%4\mathrm{.}50\%$

Standard Deviation $35\mathrm{.}00\%20\mathrm{.}00\%15\mathrm{.}00\%$

$($a$)$ Construct a $95\%$ confidence interval for the next year$$s annual return on Small and Big

portfolios. What is the probability of next year return on Small stocks to be less than zero?

What is the probability of the average of the next four$-$years of return on Small stocks to be

less than zero?

$($b$)$ Formulate the null and alternative hypothesis consistent with testing whether the average

return on Small and Big portfolios are different from zero. Use a significance level of $5\%.$

$($H$0$ : $\backslash$theta $=0$ versus Ha : $\backslash$theta $=0,$ two$-$sided hypothesis test$)$

$($c$)$ Test the null hypothesis of equality of mean returns between Small and Big portfolios. The

data for a portfolio that buys small and shorts the big portfolio is provided in the column

titled $$Difference$.$ Use a significance level of $5\%.$