Question

Point(s) possible

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=1093 and x = 538 who
said "yes." Use a 90% confidence level.
Click the icon to view a table of z scores.
b) Identify the value of the margin of error E.
(Round to three decimal places as needed.)
c) Construct the confidence interval.

(Round to three decimal places as needed.)
d) Write a statement that correctly interprets the confidence interval. Choose the correct answer below.
OA. One has 90% confidence that the interval from the lower bound to the upper bound actually does contain the
true value of the population proportion.
OB. 90% of sample proportions will fall between the lower bound and the upper bound.
C. One has 90% confidence that the sample proportion is equal to the population proportion.
D. There is a 90% chance that the true value of the population proportion will fall between the lower bound and the
upper bound.

Answer 1

a) Since n = 1093, x = 538, and we are using a 90% confidence level, we can find the standard error of the proportion using the following formula:

standard error = sqrt((p-hat * (1 - p-hat)) / n)

where p-hat is the sample proportion.

Substituting the given values:

p-hat = x / n = 538 / 1093 = 0.4925

standard error = sqrt((0.4925 * (1 - 0.4925)) / 1093)

standard error ≈ 0.015

b) To find the margin of error, we can use the formula:

margin of error = z* * standard error

where z* is the z-score corresponding to the desired confidence level.

Since we are using a 90% confidence level, the z-score is 1.645 (see the table of z-scores).

Substituting the values:

margin of error = 1.645 * 0.015

margin of error ≈ 0.025

Therefore, the margin of error is approximately 0.025.

c) The confidence interval can be constructed using the formula:

confidence interval = p-hat ± margin of error

Substituting the given values:

confidence interval = 0.4925 ± 0.025

confidence interval = (0.4675, 0.5175)

Therefore, the 90% confidence interval for the proportion of people who feel vulnerable to identity theft is (0.4675, 0.5175).

d) The correct statement that interprets the confidence interval is:

OA. One has 90% confidence that the interval from the lower bound to the upper bound actually does contain the true value of the population proportion.

This statement means that if we repeated the sampling process many times and constructed a 90% confidence interval each time, approximately 90% of the intervals would contain the true proportion of people who feel vulnerable to identity theft.