Question

11. [6 points] Using a sample of 100 children, a 95% confidence interval for the proportion of children under the age of 18 who had asthma was constructed to be (0.085,6.23) a) Which of the following would produce a confidence interval which is narrower than the constructed 95% confidence interval? Sample 150 children rather than 100, (Maintain the confidence level at 95%) Sample only 80 children rather than 100, (Maintain the confidence level at 95%) . (b) Which of the following would produce a confidence interval which is narrower than the constructed 95% confidence interval? Construct a 99% confidence interval rather than the 95% confidence interval. (Maintain the sample size of 100.) Construct a 91% confidence interval rather than the 95% confidence interval. the sample size of 100.) (Maintain .

Answer 1

Solution:

Given:

Sample size = n = 100
95% confidence interval for the proportion of children under the age of 18 who had asthma was (0.085 , 6.23)

Part a) We have to find which of the following would produce a confidence interval which is narrower than the constructed 95% confidence level.
Confidence interval for proportion is given by:

$( \hat{p} - E \: \: ,\: \: \hat{p} + E )$

where

$E = Z_{c}\times \sqrt{\frac{\hat{p}\times (1-\hat{p})}{n}}$

Margin of Error E is inversely proportional to sample size. That is as n increases, E decreases and hence length of confidence interval decreases, that is it becomes narrower when sample size increases.

Thus if we increase n from 100 to 150, then 95% confidence interval will becomes narrower.

In the second case we are given sample size = n = 80 , so as n decreases , then E increases and hence confidence interval becomes wider.

Thus correct answer is:

Sample of 150 children rather than 100. ( maintain confidence level 95%)

Part b) In part b) we have to check with different confidence levels

Margin of Error is:

$E = Z_{c}\times \sqrt{\frac{\hat{p}\times (1-\hat{p})}{n}}$

we can see E= Margin of error is directly proportional to Z_c .

Where Z_c is z critical value which depends on confidence level.

As confidence level increases , Zc value increases.

Zc value for 95% confidence level is 1.96 and for 99% confidence level is 2.575

Thus as confidence level increases, Zc value increases and hence E = Margin of Error is also increases and hence confidence interval becomes wider.

Thus as to get narrower confidence interval , we should decrease the confidence level.

Thus correct option is:

Construct 91% confidence interval rather than 95% confidence interval. ( maintain sample size of 100)