Answer to Question #318080 in Statistics and Probability for Owen

Question

Based from the results of a survey conducted by a group of researchers, the proportion of feinales holding executive positions in the government agencies in a certain populated city in the Philippines is normally distributed, in a random sample of 300 female employees in government agencies, 75 female employees hold executive positions.

1. Compute the proportion estimate of the parameter.

2. Compute the standard error of the proportion.

3. Compute the margin of error of the proportion at 90% confidence level.

4. Find 90% and 95% confidence interval for all the female employees in the government agencies holding executive positions.

Answer 1

$1:\hat{p}=\frac{75}{300}=0.25\\2:\varepsilon =\sqrt{\frac{\hat{p}\left( 1-\hat{p} \right)}{n}}=\sqrt{\frac{0.25\cdot 0.75}{300}}=0.025\\3:E=\varepsilon z_{\frac{1+\gamma}{2}}=0.025\cdot 1.6449=0.0411225\\4:90\%\left( \hat{p}-E,\hat{p}+E \right) =\left( 0.25-0.0411,0.25+0.0411 \right) =\\=\left( 0.2089,0.2911 \right) \\95\%\left( \hat{p}-\sqrt{\frac{\hat{p}\left( 1-\hat{p} \right)}{n}}z_{\frac{1+0.95}{2}},\hat{p}+\sqrt{\frac{\hat{p}\left( 1-\hat{p} \right)}{n}}z_{\frac{1+0.95}{2}} \right) =\\=\left( 0.25-0.025\cdot 1.960,0.25+0.025\cdot 1.960 \right) =\\=\left( 0.201,0.299 \right)$