Question

select the correct answer.
the city transit authority surveyed 1,000 citizens who regularly use public transportation. of those surveyed, 658 users said they paid using a monthly subscription as opposed to paying for each ride individually. the transit authoritys report claims, we can be 95% confident that the percentage of citizens using public transportation with a monthly subscription is between 63.33% and 68.27%.
which statement best describes this claim?

a. the reports claim is false because the margin of error is 3.87%.
b. the reports claim is false because the margin of error is 2.94%.
c. the reports claim is true.
d. there is not enough information to evaluate the reports claim.

Explanation:

Step1: Calculate the sample proportion

The sample proportion $\hat{p}=\frac{658}{1000} = 0.658$

Step2: Calculate the margin of error for a 95% confidence interval

For a large - sample proportion confidence interval, the margin of error $E$ for a 95% confidence interval is calculated using the formula $E = z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$, where $z\approx1.96$ for a 95% confidence interval, $\hat{p}=0.658$, and $n = 1000$.

First, calculate $\hat{p}(1 - \hat{p})=0.658\times(1 - 0.658)=0.658\times0.342 = 0.225036$

Then, $\frac{\hat{p}(1 - \hat{p})}{n}=\frac{0.225036}{1000}=0.000225036$

$\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=\sqrt{0.000225036}\approx0.015$

$E = 1.96\times0.015=0.0294 = 2.94\%$

The confidence interval is $\hat{p}\pm E=0.658\pm0.0294=(0.6286,0.6874)$ or $(62.86\%,68.74\%)$

The given confidence interval in the report is $(63.33\%,68.27\%)$ which is within the calculated confidence - interval range.

Answer:

C. The report's claim is true.