Question

Use the sample data and confidence level given below to complete parts​ (a) through​ (d).

A research institute poll asked respondents if they felt vulnerable to identity theft. In the​ poll, n=1075 and x=514 who said​ "yes." Use a 95 % confidence level.

​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. 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.

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

C. There is a 95​% chance that the true value of the population proportion will fall between the lower bound and the upper bound.

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

Answer 1

Solution :

Given that,

n = 1075

x = 514

Point estimate = sample proportion = = x / n = 514/1075=0.478

1 -   = 1- 0.478 =0.522

At 95% confidence level the z is ,

= 1 - 95% = 1 - 0.95 = 0.05

/ 2 = 0.05 / 2 = 0.025

Z/2 = Z0.025 = 1.96   ( Using z table )

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

= 1.96 (((0.478*0.522) / 1075)

E = 0.030

A 95% confidence interval for population proportion p is ,

- E < p < + E

0.478- 0.030< p <0.478+ 0.030

0.448< p < 0.508

The 95% confidence interval for the population proportion p is : 0.448< p < 0.508

correct option

A. 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.