Question

A magazine provided results from a poll of 1000 adults who were asked to identify their favorite pie. Among the

1000 ​respondents,11​% chose chocolate​ pie, and the margin of error was given as plus or minus±5 percentage points. Given specific sample​ data, which confidence interval is​ wider: the 90​% confidence interval or the 80​% confidence​ interval? Why is it​ wider?

Choose the correct answer below.

A. A 90​% confidence interval must be wider than an 80​% confidence interval in order to be more confident that it captures the true value of the population proportion.

B. An 80​% confidence interval must be wider than a 90​%confidence interval because it contains​100%minus−8080​%equals=20​% of the true population​ parameters, while the 90​% confidence interval only contains ​100%minus−9090​%equals=1010​% of the true population parameters.

C.An 80​% confidence interval must be wider than a 90​% confidence interval in order to be more confident that it captures the true value of the population proportion.

D. A 90​% confidence interval must be wider than an 80​%confidence interval because it contains 90​% of the true population​ parameters, while the 80​%confidence interval only contains 80​%of the true population parameters.

Answer 1

Solution

Option A Answer 1

Explanation

Theoretically, 100(1 - α) % Confidence Interval for the population proportion, p is:

p_hat ± MoE, where MoE = Z_α/2[√{p_hat (1 – p_hat)/n}]

with

Z_α/2 is the upper (α/2)% point of N(0, 1),

p_hat = sample proportion, and

n = sample size.

The width of the confidence interval depends only on MoE which in turn depends on Z_α/2 only when other things remain the same.

As confidence level increases, α decreases and as α decreases, Z_α/2 increases resulting in wider interval.

Physically, more confidence can be gained only by widening the interval.

DONE