91影视

Null hypothesis: $H_0:渭=75$

Alternate hypothesis: $H_0:渭<75$

Thus, we have to sample finding the distribution of $\overline{x}$according to the central limit theorem. The sample mean follows the normal distribution with mean $渭$as well as standard deviation. $\frac{蟽}{\sqrt{n}}$

$$\begin{gathered} E\left(\overline{x}\right)=渭 \\ =75 \\ {蟽}_{\left(\overline{x}\right)}=\frac{蟽}{\sqrt{n}} \\ =\frac{15}{\sqrt{49}} \\ =\frac{15}{7} \end{gathered}$$

Thus, the sample distribution of $\left(\overline{x}\right)$is N$\left(75,\frac{15}{7}\right)$

They consider the rejection region corresponding to $a=0.10$. To calculate to the rejection region is $z<-1.28$then the statistics test is

$$\begin{gathered} {\overline{x}}_{oe}={渭}_0-1.28脳\frac{s}{\sqrt{n}} \\ =75-1.28脳2.142 \\ =75-2.741 \\ {\overline{x}}_{oe}=72.259 \end{gathered}$$

$$\begin{gathered} {\overline{x}}_{oe}=渭+1.28脳\frac{s}{\sqrt{n}} \\ =75+1.28脳2.142 \\ =75+2.741 \\ {\overline{x}}_{oe}=77.741 \end{gathered}$$

They necessary to obtain sampling distribution of $\overline{x}$under the assumption that the population mean is 70. Thus, we have to sample finding the distribution of $\overline{x}.$according to the central limit theorem. The sample mean follows a normal distribution with mean $渭$as well as standard deviation. $\frac{蟽}{\sqrt{n}}$

$$\begin{gathered} E\left(\overline{x}\right)=渭 \\ =70 \\ {蟽}_{\left(\overline{x}\right)}=\frac{蟽}{\sqrt{n}} \\ =\frac{15}{\sqrt{49}} \\ =\frac{15}{7} \end{gathered}$$

Thus, the sample distribution of $\left(\overline{x}\right)$is N $\left(70,\frac{15}{7}\right)$

Following values in part (a) to z-value in the alternative distribution with $渭=70$

$$\begin{gathered} zL=\frac{({\overline{)\overline{x}}}_{oe}-\overline{)渭})}{蟽\overline{x}} \\ =\frac{72.259-70}{2.142} \\ zL=1.054 \end{gathered}$$

Using Table IV in Appendix B, they see that the area among $z=0$as well as $z=1.054is0.3531,$while the area between $z=0$as well as $z=3.613$is about 0.5 (since $z=3.613$it is off the scale).

In this issue, we assume that $渭$it is truly equal to 70. Then there is a chance that the hypothesis test will cause the investigator to make a type - II error $\left(尾\right)$:

For this, we obtain the area between:

$$\begin{gathered} z=1.054 \\ z=3.613istreatedas \\ 尾=0.5-0.3531 \\ 尾=0.1469 \end{gathered}$$

The power is defined to be the probability of (correctly) the null hypothesis when the alternative is $H_a:渭=70true$

Power of the test: $(1-尾)$

$$\begin{gathered} Power=(1-尾) \\ =1-0.1469 \\ =0.8531 \end{gathered}$$