Question

Use the sample data and confidence level given below to complete parts​ (a) through​ (d). In a study of cell phone use and brain hemispheric​ dominance, an Internet survey was​ e-mailed to 2397 subjects randomly selected from an online group involved with ears. 1025 surveys were returned. Construct a 99​% confidence interval for the proportion of returned surveys. LOADING... Click the icon to view a table of z scores. ​a) Find the best point estimate of the population proportion p. nothing ​(Round to three decimal places as​ needed.) ​b) Identify the value of the margin of error E. Eequals nothing ​(Round to three decimal places as​ needed.) ​c) Construct the confidence interval. nothingless than p less than nothing ​(Round to three decimal places as​ needed.) ​d) Write a statement that correctly interprets the confidence interval. Choose the correct answer below. A. There is a 99​% chance that the true value of the population proportion will fall between the lower bound and the upper bound. B. 99​% of sample proportions will fall between the lower bound and the upper bound. C. One has 99​% confidence that the sample proportion is equal to the population proportion. D. One has 99​% confidence that the interval from the lower bound to the upper bound actually does contain the true value of the population proportion.

Answer 1

Answer:

a) p-hat = 0.428

b) Error = 0.026

c) 0.402 ≤ p ≤ 0.454

Step-by-step explanation:

We have a survey that was emailed to 2397 subjects and only 1025 were returned.

a) The best point estimate for p is p-hat=x/n:

$\hat{p}=X/n=1025/2397=0.428$

b) The margin of error is the product of the z-score for a 99% interval and the standard deviation of the proportion.

The z-score for a 99% CI is z=2.576.

The standard deviation is:

$\sigma=\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}=\sqrt{\frac{0.428*0.572}{2397}}=0.010$

Then, the margin of error is:

$E=z*\sigma=2.576*0.01=0.02576$

c) the confidence interval can be constructed as:

$\hat{p}-z\sigma\leq p \leq \hat{p}-z\sigma\\\\0.428-0.026\leq p \leq 0.428+0.026\\\\ 0.402\leq p \leq0.454$