Question

in a survey of 2007 adults in a recent year, 718 made a new years resolution to eat healthier.
construct 90% and 95% confidence intervals for the population proportion. interpret the results and compare the widths
of the confidence intervals.
the 90% confidence interval for the population proportion p is (0.340, 0.378)
(round to three decimal places as needed.)
the 95% confidence interval for the population proportion p is (0.337, 0.379)
(round to three decimal places as needed.)
with the given confidence, it can be said that the population proportion of adults who say they have made a
new years resolution to eat healthier is greater than the lower endpoint of the given confidence interval.

Explanation:

Step1: Confidence interval interpretation

A confidence interval for a population proportion gives a range of values within which the true population proportion is likely to lie. For example, a 90% confidence interval means that if we were to take many samples and construct confidence intervals, about 90% of them would contain the true population proportion.

Step2: Width comparison

The width of a confidence interval is calculated as \(upper - lower\) bound. For the 90% confidence interval: \(0.378 - 0.340=0.038\). For the 95% confidence interval: \(0.379 - 0.337 = 0.042\).

Answer:

Interpretation: With 90% confidence, we can say that the population proportion of adults who made a New Year's resolution to eat healthier is between \(0.340\) and \(0.378\). With 95% confidence, it is between \(0.337\) and \(0.379\).
Width comparison: The width of the 90% confidence interval is \(0.038\) and the width of the 95% confidence interval is \(0.042\). The 95% confidence interval is wider. This is because a higher confidence level (95% compared to 90%) requires a larger range of values to be more certain of capturing the true population proportion.