Refer to Exercise $15$.

(a) Carry out a significance test at the $\alpha =0.05$level.

(b) Construct and interpret a $95\%$confidence interval for the difference between the population proportions. Explain how the confidence interval is consistent with the results of the test in part (a).

Given

${\hat{p}}_1=79\%=0.79$

$n_1=800$

${\hat{p}}_2=67\%=0.67$

$n_2=400$

Determine the hypothesis

$H_0:p_1-p_2=0$

$H_a:p_1-p_2\ne 0$

The sample proportion is the number of successes divided by the sample size:

$$\begin{gathered} {\hat{p}}_1=\frac{x_1}{n_1} \\ =\frac{632}{800} \\ =0.79 \end{gathered}$$

$$\begin{gathered} {\hat{p}}_2=\frac{x_2}{n_2} \\ =\frac{268}{400} \\ =0.67 \end{gathered}$$

$$\begin{gathered} {\hat{p}}_p=\frac{x_1+x_2}{n_1+n_2} \\ =\frac{632+268}{800+400} \\ =\frac{900}{1200} \\ =0.75 \end{gathered}$$

Determine the value of the test statistic:

$$\begin{gathered} z=\frac{{\hat{p}}_1-{\hat{p}}_2}{\sqrt{{\hat{p}}_p\left(1-{\hat{p}}_p\right)}\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}} \\ =\frac{0.79-0.67}{\sqrt{0.75(1-0.75)}\sqrt{\frac{1}{800}+\frac{1}{400}}} \\ \approx 4.53 \end{gathered}$$

The $p$-value is the probability of obtaining the value of the test statistic, or a value more extreme. Determine the$p$-value using table A:
localid="1650450174625" $$\begin{gathered} P=P(Z<-4.53\text{or}Z>4.53) \\ =2\times P(Z<-4.53) \\ =2\times 0.0001 \\ =0.0002 \end{gathered}$$

If the P-value is smaller than the significance level, reject the null hypothesis:

$P<0.05\Rightarrow \text{Reject}{H}_0$

Given

${\hat{p}}_1=79\%=0.79$

$n_1=800$

${\hat{p}}_2=67\%=0.67$

$n_2=400$

Determine the hypothesis

$H_0:p_1-p_2=0$

$H_a:p_1-p_2\ne 0$

The sample proportion is the number of successes divided by the sample size:

$$\begin{gathered} {\hat{p}}_1=\frac{x_1}{n_1} \\ =\frac{632}{800} \\ =0.79 \end{gathered}$$

$$\begin{gathered} {\hat{p}}_2=\frac{x_2}{n_2} \\ =\frac{268}{400} \\ =0.67 \end{gathered}$$

For confidence level $1-\alpha =0.95$, determine $z_{\alpha /2}=z_{0.025}$Using table II (look up x$z_{\alpha /2}=z_{0.025}$in the table, the $z$-score is then the found $z$-score with opposite sign):

$z_{\alpha /2}=1.96$

The endpoints of the confidence interval for $p_{1}-p_{2}$ are then:

localid="1650450195998" $$\begin{gathered} \left({\hat{p}}_1-{\hat{p}}_2\right)-z_{\alpha /2}·\sqrt{\frac{{\hat{p}}_1\left(1-{\hat{p}}_1\right)}{n_1}+\frac{{\hat{p}}_2\left(1-{\hat{p}}_2\right)}{n_2}}=(0.79-0.67)-1.96\sqrt{\frac{0.79(1-0.79)}{800}+\frac{0.67(1-0.67)}{400}} \\ \approx 0.066 \end{gathered}$$

localid="1650450210366" $$\begin{gathered} \left({\hat{p}}_1-{\hat{p}}_2\right)+z_{\alpha /2}·\sqrt{\frac{{\hat{p}}_1\left(1-{\hat{p}}_1\right)}{n_1}+\frac{{\hat{p}}_2\left(1-{\hat{p}}_2\right)}{n_2}}=(0.79-0.67)+1.96\sqrt{\frac{0.79(1-0.79)}{800}+\frac{0.67(1-0.67)}{400}} \\ \approx 0.174 \end{gathered}$$