Question

1. researchers survey 1000 grocery store customers and find that, when given a choice, 68% of these customers prefer using the self - checkout. when a 95% confidence interval is constructed based on this sample of data, the interval is from 0.651 to 0.709. given what you know and understand about confidence intervals, which one of the following statements is most accurate?

a. with 95% confidence, approximately 68% of the sample prefer using the self - checkout option.
b. between 65.1% and 70.9% of all possible samples will produce sample proportions in this range.
c. between 65.1% and 70.9% of all grocery store customers prefer using the self - checkout.
d. if the sample size were increased, this would make the interval narrower, but only if the confidence level is also lowered.
e. none of the above answer options are correct.

Explanation:

• Option A: The sample proportion is \(68\%\) (given as \(0.68\)), and a confidence interval is not about the proportion of the sample itself in this way. The confidence interval is about estimating the population proportion.
• Option B: A confidence interval is not about the proportion of samples that will have sample - proportions in the interval. It is about estimating the population proportion.
• Option C: A \(95\%\) confidence interval does not mean that \(95\%\) of the population is in the interval. It means that if we were to take many samples and construct confidence intervals, about \(95\%\) of those intervals would contain the True population proportion.
• Option D: Increasing the sample size \(n\) will make the confidence interval narrower for a fixed confidence level. The formula for the margin of error for a proportion is \(E = z\sqrt{\frac{p(1 - p)}{n}}\), where \(z\) is related to the confidence level and \(n\) is the sample size. As \(n\) increases, \(E\) decreases (for a fixed \(z\) and \(p\)).

Answer:

E. None of the above answer options are correct.