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 2510 subjects randomly selected from an online group involved with ears. 1092 surveys were returned. Construct a 95​% confidence interval for the proportion of returned surveys.

a) Find the best point estimate of the population proportion p.

___ (Round to three decimal places as​ needed.)

​b) Identify the value of the margin of error E.

___ ​(Round to three decimal places as​ needed.)

​c) Construct the confidence interval.

___ < p < ___​ (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 95​% chance that the true value of the population proportion will fall between the lower bound and the upper bound.

B. 95% of sample proportions will fall between the lower bound and the upper bound.

C. One has 95​% confidence that the interval from the lower bound to the upper bound actually does contain the true value of the population proportion.

D. One has 95​% confidence that the sample proportion is equal to the population proportion.

Answer 1

Solution :

Given that,

n = 2510

x = 1092

a) Point estimate = sample proportion = = x / n = 1092 / 2510 = 0.435

1 - = 1 - 0.435 = 0.565

At 95% confidence level

= 1 - 95%

=1 - 0.95 =0.05

/2 = 0.025

Z/2 = Z0.025 = 1.960

b) Margin of error = E = Z / 2 * (( * (1 - )) / n)

= 1.96 (((0.435 * 0.565) / 2510)

= 0.019

c) A 95% confidence interval for population proportion p is ,

- E < p < + E

0.435 - 0.019 < p < 0.435 + 0.019

( 0.416 < p < 0.454 )

d) C. One has 95​% confidence that the interval from the lower bound to the upper bound actually does contain the true value of the population proportion.