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 equals 1094 and x equals 592 who said​ "yes." Use a 99 % confidence level.

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

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

c ) Construct the confidence interval.

d)Write a statement that correctly interprets the confidence interval.

Choose the correct answer below.

A. 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.

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. There is a 99​% chance that the true value of the population proportion will fall between the lower bound and the upper bound.

Answer 1

Solution:

a) Sample proportion is the unbiased point estimate of the population proportion.

Sample proportion of the respondent who felt vulnerable to identity theft is,

Hence, point estimate of population proportion P is 0.5411.

b) The margin of error at 99% confidence level is given as follows:

  

Where, E is margin of error, n is sample size, p̂ is point estimate of population proportion and q̂ = 1 - p̂.

Z_(0.01/2) is critical z-value to construct 99% confidence interval.

We have, p̂ = 0.5411, q̂ = (1 - 0.5411) = 0.4589 and n = 1094

Using Z-table we get,

Margin of error is 0.0389.

c) The 99% confidence interval for population proportion is given as follows:

(Sample proportion ± margin of error)

From part (a) we have, sample proportion = 0.5411

From part (b) we have, margin of error = 0.0389

Hence, 99% confidence interval for population proportion is,

The 99% confidence interval for population proportion is,

(0.5022, 0.5800)

d) Interpretation of confidence interval:

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.

Hence, option (A) is correct.

Please rate the answer. Thank you.