you may need to use the appropriate appendix ...
you may need to use the appropriate appendix table or technology to answer this question. a magazine reports that women trust recommendations from a particular social networking site more than recommendations from any other s following sample data show the number of women and men who stated in a recent sample that they trust recommendations made on this parti\n\n| | women | men |\n|--|--|--|\n| sample | 150 | 170 |\n| trust recommendations made on the social networking site | 123 | 102 |\n\n(a) what is the point estimate of the proportion of women who trust recommendations made on this particular social networking site? 0.82 \n\n(b) what is the point estimate of the proportion of men who trust recommendations made on this particular social networking site? 0.6 \n\n(c) provide a 95% confidence interval estimate of the difference between the proportion of women and men who trust recommendations ma 0.1240 × 1 to 0.3160 × 1 \n\nneed help? read it
Answer
# Explanation:
## Step1: Identify sample - proportions
Let $\hat{p}_1$ be the proportion of women and $\hat{p}_2$ be the proportion of men. We know from (a) $\hat{p}_1=\frac{123}{150} = 0.82$ and from (b) $\hat{p}_2=\frac{102}{170}=0.6$. The sample sizes are $n_1 = 150$ and $n_2=170$.
## Step2: Find critical - value
For a 95% confidence interval, the significance level $\alpha=1 - 0.95 = 0.05$, and $\alpha/2=0.025$. The critical - value $z_{\alpha/2}=z_{0.025}=1.96$.
## Step3: Calculate the standard error
The formula for the standard error $SE$ of the difference in proportions is $SE=\sqrt{\frac{\hat{p}_1(1 - \hat{p}_1)}{n_1}+\frac{\hat{p}_2(1 - \hat{p}_2)}{n_2}}$.
Substitute $\hat{p}_1 = 0.82$, $n_1 = 150$, $\hat{p}_2 = 0.6$, and $n_2 = 170$ into the formula:
\[
\begin{align*}
SE&=\sqrt{\frac{0.82\times(1 - 0.82)}{150}+\frac{0.6\times(1 - 0.6)}{170}}\\
&=\sqrt{\frac{0.82\times0.18}{150}+\frac{0.6\times0.4}{170}}\\
&=\sqrt{\frac{0.1476}{150}+\frac{0.24}{170}}\\
&=\sqrt{0.000984 + 0.001412}\\
&=\sqrt{0.002396}\\
&\approx0.0489
\end{align*}
\]
## Step4: Calculate the confidence interval
The formula for the 95% confidence interval for the difference in proportions $(\hat{p}_1-\hat{p}_2)$ is $(\hat{p}_1 - \hat{p}_2)\pm z_{\alpha/2}\times SE$.
Substitute $\hat{p}_1 - \hat{p}_2=0.82 - 0.6 = 0.22$, $z_{\alpha/2}=1.96$, and $SE\approx0.0489$ into the formula:
Lower limit: $0.22-1.96\times0.0489=0.22 - 0.095 = 0.125$
Upper limit: $0.22 + 1.96\times0.0489=0.22+0.095 = 0.315$
# Answer:
$0.125$ to $0.315$