Question

1. what proportion of adults have credit card debt? a financial institution conducts a survey of a random sample of 1000 adults and finds that the proportion of the adults in the sample who have credit card debt is 0.41. when a 95% confidence interval is constructed based on this information, it is found to be from 0.369 to 0.451. the financial institution interprets the interval as follows: “we are 95% confident the interval from 0.369 to 0.451 includes the proportion of all adults in the population who have credit card debt.” is anything wrong with this interval or the interpretation of the interval?

a. no, nothing is wrong.
b. yes, we shouldn’t trust that the sample proportion is 0.41 because most people who have credit card debt are ashamed to admit it.
c. yes, the lower and upper bounds of the confidence interval are not correct.
d. yes, because we are dealing with financial data, a 99% confidence interval should have been constructed instead of a 95% confidence interval.
e. yes, a correct interpretation of the interval is that we are 95% confident the interval from 0.369 to 0.451 includes the proportion of adults in the sample who have credit card debt.

Explanation:

A confidence interval is constructed to estimate the population proportion. The interpretation of a 95% confidence interval is that we are 95% confident that the interval contains the True population proportion. Here, the sample proportion \( \hat{p}=0.41\), sample size \(n = 1000\). The formula for the confidence interval for a proportion is \(\hat{p}\pm z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\), where for a 95% confidence interval \(z = 1.96\).

Calculating the margin of error \(E=z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=1.96\sqrt{\frac{0.41\times(1 - 0.41)}{1000}}\approx1.96\times0.0155\approx0.0304\)

The confidence interval is \(\hat{p}-E=0.41- 0.0304=0.3796\approx0.38\) and \(\hat{p}+E=0.41 + 0.0304=0.4404\approx0.44\). But the given confidence interval is \(0.369\) to \(0.451\).

Answer:

C. Yes, the lower and upper bounds of the confidence interval are not correct.