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 2415 subjects randomly selected from an online group involved with ears. 1176 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. 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. 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.
C. 95​% of sample proportions will fall between the lower bound and the upper bound.
D. One has 95​% confidence that the sample proportion is equal to the population proportion.

Answer 1

Solution :

Given that,

n = 2415

x = 1176

a)Point estimate = sample proportion = = x / n = 1176 / 2415 = 0.487

1 - = 1 - 0.487 = 0.513

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 (((0.487 * 0.513) / 2415)

= 0.020

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

- E < p < + E

0.487 - 0.020 < p < 0.487 + 0.020

0.467 < p < 0.507

B. 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.