Question

99% confidence interval will be more accurate than a 95% confidence interval based on the same sample size, it will be less precise because it will be wider than a 95% confidence interval.

1. what is your favorite thing about the season of fall? a researcher surveys a random sample consisting of ( n = 725 ) adults and finds that 382 of these adults say the best thing about the fall season is cooler weather. from this information, calculate the sample proportion below and round your answer to exactly three decimal places.

( p=x / n )
( p = 382 / 725 = 0.527 )

1. using the information provided in question 1, construct a 99% confidence interval and show all work below. be sure to clearly label your lower and upper bounds.

Explanation:

Step1: Determine the values of \(n\), \(\hat{p}\), and \(z_{\alpha/2}\)

• \(n = 725\) (sample size)
• \(\hat{p}=\frac{382}{725}=0.527\) (sample proportion)
• For a \(99\%\) confidence interval, \(\alpha = 1 - 0.99=0.01\), and \(\alpha/2=0.005\). From the standard normal distribution table, \(z_{\alpha/2}=z_{0.005} = 2.576\)

Step2: Calculate the standard error \(SE\)

The formula for the standard error of a proportion is \(SE=\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\)
Substitute \(\hat{p}=0.527\) and \(n = 725\) into the formula:

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Step3: Calculate the margin of error \(ME\)

The formula for the margin of error for a proportion is \(ME = z_{\alpha/2}\times SE\)
Substitute \(z_{\alpha/2}=2.576\) and \(SE = 0.0185\) into the formula:
\(ME=2.576\times0.0185\approx0.0477\)

Step4: Calculate the lower and upper bounds of the confidence interval

• Lower bound \(=\hat{p}-ME\)

\(=0.527- 0.0477=0.479\)

• Upper bound \(=\hat{p}+ME\)

\(=0.527 + 0.0477=0.575\)

Answer:

The \(99\%\) confidence interval is \((0.479,0.575)\)