Question

in a survey of 2267 adults in a recent year, 1419 say they have made a new years resolution.
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 (0.609, 0.643).
(round to three decimal places as needed.)
the 95% confidence interval for the population proportion p is (0.606, 0.646).
(round to three decimal places as needed.)
with the given confidence, it can be said that the of adults who say they have made a new years resolution is of the given confidence interval.

Explanation:

Step1: Understand confidence interval interpretation

A confidence interval gives a range of values within which the population parameter (here, the proportion of adults who made a New - Year's resolution) is likely to lie.

Step2: Interpret the 90% and 95% confidence intervals

For a 90% confidence interval \((0.609,0.643)\), we can say with 90% confidence that the proportion of adults who made a New - Year's resolution is within this interval. For a 95% confidence interval \((0.606,0.646)\), we can say with 95% confidence that the proportion of adults who made a New - Year's resolution is within this interval.

Step3: Compare the widths of the confidence intervals

The width of a confidence interval is calculated as \(Upper - bound - Lower - bound\).
For the 90% confidence interval: \(0.643 - 0.609=0.034\)
For the 95% confidence interval: \(0.646 - 0.606 = 0.04\)

Answer:

With the given confidence, it can be said that the proportion of adults who say they have made a New Year's resolution is within the range of the given confidence interval. The 95% confidence interval \((0.606,0.646)\) is wider than the 90% confidence interval \((0.609,0.643)\) because a higher confidence level requires a larger range of values to be more certain of capturing the population proportion.