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:

Step1: Recall confidence interval concept

A confidence interval for a proportion \(p\) (in this case, the proportion of adults with credit - card debt in the population) is constructed as \(\hat{p}\pm z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\), where \(\hat{p}\) is the sample proportion, \(n\) is the sample size, and \(z\) is the critical value. For a 95% confidence interval, \(z = 1.96\). Given \(\hat{p}=0.41\) and \(n = 1000\), \(\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=\sqrt{\frac{0.41\times(1 - 0.41)}{1000}}\approx\sqrt{\frac{0.41\times0.59}{1000}}\approx\sqrt{\frac{0.2419}{1000}}\approx0.0156\). Then \(\hat{p}-z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=0.41-1.96\times0.0156\approx0.41 - 0.0306=0.3794\approx0.38\) and \(\hat{p}+z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=0.41 + 1.96\times0.0156\approx0.41+0.0306 = 0.4406\approx0.44\). But the interval given is \(0.369\) to \(0.451\).

Step2: Analyze option B

There is no information given in the problem about non - response bias (people being ashamed to admit credit - card debt). We assume the sample is a simple random sample (SRS) as stated in the problem ("a random sample of 1000 adults").

Step3: Analyze option C

The formula for the confidence interval for a proportion is \(\hat{p}\pm z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\). The given interval \(0.369\) to \(0.451\) is incorrect. Using the formula \(\hat{p}=0.41\), \(n = 1000\), and \(z = 1.96\) (for 95% confidence), the correct interval is approximately \((0.38,0.44)\)

Step4: Analyze option D

There is no general rule that financial data requires a 99% confidence interval. The choice of confidence level (\(95\%\) or \(99\%\)) depends on the balance between the margin of error and the level of confidence, not the type of data (financial in this case)

Step5: Analyze option E

A confidence interval is about estimating the population parameter (\(p\), the proportion of all adults in the population with credit - card debt), not the sample proportion (\(\hat{p}\)). The sample proportion \(\hat{p}=0.41\) is known (it is given as the result of the sample survey)

Answer:

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