For Questions $23$ to $26,$ refer to the following information

An investment specialist claims that if one holds a portfolio that moves in the opposite direction to the market index like the S&P $500,$ then it is possible to reduce the variability of the portfolio's return. In other words, one can create a portfolio with positive returns but less exposure to risk.

A sample of $26$ years of S&P $500$ index and a portfolio consisting of stocks of private prisons, which are believed to be negatively related to the S&P $500$ index, is collected. A regression analysis was performed by regressing the returns of the prison stocks portfolio $(Y)$ on the returns of S&P $500$ index $(x)$ to prove that the prison stocks portfolio is negatively related to the S&P $500$ index at a $5\%$ level of significance. The results are given in the following EXCEL output.

$\backslash$table$[[,$Coefficients,Standard Error,T Stat,$P-$value$],[$Intercept$,4\mathrm{.}8660,0\mathrm{.}3574,13\mathrm{.}6136,0\mathrm{.}0000],[S$&$P,-0\mathrm{.}5025,0\mathrm{.}0716,-7\mathrm{.}0186,0\mathrm{.}0000]]$

Question $23$

To test whether the prison stocks portfolio is negatively related to the S&P $500$ index, the appropriate null and alternative hypotheses are, respectively,

a$)H_0$:$=0$ vs$.H_1$:$0$

b$.H_0$:$0$ vs$.H_1$:$<mn\; id="EditorElement\_181334">0</mn>$

c$.H_0$:$0$ vs$.H_1$:$>0$

d$.,H_0$:$0$ vs$.H_1$:$=0$

e$.$ None of the above

$6$

Question $24$

To test whether the prison stocks portfolio is negatively related to the S&P $500$ index, the measured value of the test statistic is

a$.-7\mathrm{.}019$

b$.-0\mathrm{.}503$

c$.0\mathrm{.}072$

d$.0\mathrm{.}357$

e$.$ none of the above

Question $25$

To test whether the prison stocks portfolio is negatively related to the S&P $500$ index, the $p-$value of the associated test statistic is

A$.0\mathrm{.}0000$

B$.0\mathrm{.}1234$

C$.0\mathrm{.}2315$

D$.0\mathrm{.}0026$

E$.0\mathrm{.}0017$

F$.$ None of the above

Question $26.$

which of the following will be a correct conclusion?

a$.$ You cannot reject the null hypothesis and, therefore, conclude that there is sufficient evidence to show that the prisons stock portfolio and S&P $500$ index are negatively related.

b$.$ You can reject the null hypothesis and, therefore, conclude that there is sufficient evidence to show that the prisons stock portfolio and S&P $500$ index are negatively related.

c$.$ You cannot reject the null hypothesis and, therefore, conclude that there is insufficient evidence to show that the prisons stock portfolio and S&P $500$ index are negatively related.

d$.$ You can reject the null hypothesis and conclude that there is insufficient evidence to show that the prisons stock portfolio and S&P $500$ index are negatively related.

e$.$ None of the above