Question

question 8
1 pts
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?
no, nothing is wrong.
yes, we shouldnt trust that the sample proportion is 0.41 because most people who have credit card debt
are ashamed to admit it.
yes, the lower and upper bounds of the confidence interval are not correct.
yes, because we are dealing with financial data, a 99% confidence interval should have been constructed
instead of a 95% confidence interval.
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 is constructed to estimate the population proportion. The formula for a confidence interval for a proportion is $\hat{p}\pm z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$, where $\hat{p}$ is the sample proportion, $n$ is the sample size, and $z$ is the critical value. For a 95% confidence interval, the critical value $z\approx1.96$. Given $\hat{p}=0.41$ and $n = 1000$, 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 interval is $0.369$ to $0.451$.

Step2: Analyze each option

• Option 1: The calculated confidence interval (approx $0.38 - 0.44$) is different from the given ($0.369 - 0.451$), so this option is wrong.
• Option 2: There is no information given in the problem about people being ashamed to admit credit - card debt, so this is an unfounded assumption.
• Option 3: As shown in Step 1, the lower and upper bounds of the given confidence interval are not correct according to the formula.
• Option 4: There is no rule that financial data must use a 99% confidence interval. The choice of confidence level depends on the context (e.g., risk tolerance), and 95% is a common choice.
• Option 5: A confidence interval for a proportion is for the population proportion, not the sample proportion. The sample proportion is a single value ($0.41$ in this case).

Answer:

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