Answer the next three questions based on the following information. The mean speed of the internet in your apartment is usually $35MBps.$ The internet provider, Optimum, charged you more for the last month. They claimed the mean speed of your internet connection to be more than $35$ MBps$.$ You are skeptical. So you tracked your daily internet speed for the last $30$ days. Your speed data yields the sample mean of $36\mathrm{.}2$ and the sample standard deviation of $4\mathrm{.}32.$

$7.$ Identify the appropriate null and alternative hypotheses.

a$)H_0$:$\stackrel{}{x}=35$ vs $H_A$:$\stackrel{}{x}>35$

b$)H_0$:$\stackrel{}{x}<mn\; id="EditorElement\_312931">3</mn><mn\; id="EditorElement\_312932">5</mn>$ vs $H_A$:$\stackrel{}{x}=35$

c$)H_0$:$\stackrel{}{x}=36\mathrm{.}2$ vs $H_A$:$\stackrel{}{x}>36\mathrm{.}2$

d$){?}^{********}{H}_0$:$=35$ vs $H_A$:$>35$

e$)H_0$:$<mn\; id="EditorElement\_312648">3</mn><mn\; id="EditorElement\_312649">5</mn>$ vs $H_A$:$=35$

$8.$ After choosing the appropriate test $($think of the appropriate null distribution carefully$),$ what would be your conclusion at the $=0\mathrm{.}05$ level of significance?

a$)$ We reject the null hypothesis at the $5\%$ level and conclude that Optimum was correct.

b$)$ We reject the null hypothesis at the $5\%$ level and conclude that Optimum was wrong.

c$)$ We fail to reject the null hypothesis at the $5\%$ level and conclude that Optimum was correct.

d$)********$ We fail to reject the null hypothesis at the $5\%$ level and conclude that Optimum was wrong.

$9.$ Dave is feeling sick, so he goes to the doctor and gets a chest X$-$ray to determine whether or not he has pneumonia. Let the null hypothesis be that Dave doesn't have pneumonia and the alternative hypothesis be that Dave has pneumonia. What would be a Type II error in this case?

a$)$ Dave has pneumonia and the $x-$ray shows he has pneumonia.

b$)$ Dave doesn't have pneumonia and the X$-$ray shows he has pneumonia.

c$){?}^{********}$ Dave has pneumonia and the X$-$ray shows he doesn't have pneumonia.

d$)$ Dave doesn't have pneumonia and the X$-$ray shows he doesn't have pneumonia.