Answer to Question #269471 in Statistics and Probability for Ching

Question

question 1:Work Interruptions A survey found that out of 200 workers, 168 said they were interrupted three or more times an hour by phone messages, faxes, etc. Find the 90% confidence interval of the population proportion of workers who are interrupted three or more times an hour

question 2:High School Graduates Who Take the SAT The national average for the percentage of high school graduates taking the SAT is 49%, but the state averages vary from a low of 4% to a high of 92%. A random sample of 300 graduating high school seniors was polled across a particular tristate area, and it was found that 195 of them had taken the SAT. Estimate the true proportion of high school graduates in this region who take the SAT with 95% confidence.

Answer 1

1.

We first need to find the proportion estimate given as,

$\hat{p}=x/n$ where the number of success $x=168$ and the sample size is $n=200.$ Therefore, $\hat{p}=168/200=0.84$ Let $\hat{q}=(1-\hat{p})=1-0.84=0.16.$.

A 90% confidence interval for the proportion of workers is given as,

$C.I= \hat{p}\pm Z_{\alpha/2}*\sqrt{(\hat{p}\hat{q})/n}$ where $\alpha=0.1$ and $Z_{\alpha/2}=Z_{0.1/2}=Z_{0.05}=1.645$

Thus,

$C.I=0.84\pm1.645\sqrt{(0.84*0.16)/200}=0.84\pm1.645*0.02592296=0.84\pm0.04264327=(0.7974,\space 0.8826)$

Therefore, a 90% confidence interval of the population proportion of workers who are interrupted three or more times an hour rounded to 4 decimal places is (0.7974,0.8826),

2.

We are given the following from the information above,

$x=195, \space n=300$

$\hat{p}=x/n=195/300=0.65$. Let $\hat{q}=1-\hat{p}=1-0.65=0.35$

$\alpha=0.05$

We are required to find the 95% confidence interval fort the population proportion $p$.

Now, the 95% confidence interval is given as,

$C.I= \hat{p}\pm Z_{\alpha/2}*\sqrt{(\hat{p}\hat{q})/n}$ . $Z_{\alpha/2}=Z_{0.05/2}=Z_{0.025}=1.96$

$C.I=0.65\pm1.96\sqrt{(0.65*0.35)/300}=0.65\pm0.05397419=(0.5960,0.7040)$

Therefore, the true proportion, $p$ of high school graduates in this region who take the SAT lies in the interval (0.5960,0.7040) with 95% confidence.