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$.H_1$:$<mn\; id="EditorElement\_8041132">0</mn>$

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

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

e$.$ None of the above

$6$