Question

using a random sample of 3648 tv households, acme media statistics found that 37.3% watched the final episode of \it aint over yet.\
a. find the margin of error in this percent
b. write a statement about the percentage of tv households in the population who tuned into the final episode of \it aint over yet.\
a. the margin of error is ± 1.66%.
(do not round until the final answer. then round to the nearest hundredth as needed.)
b. complete the following statement.
we can be 95% confident that between □% and □% of all households watched the final episode of \it aint over yet.\
(use ascending order. round to the nearest hundredth as needed.)

Explanation:

Step1: Calculate the lower and upper bounds for part b

The formula for a confidence interval for a proportion is \(\hat{p}\pm E\), where \(\hat{p}\) is the sample proportion and \(E\) is the margin of error.
Given \(\hat{p} = 37.3\%\) or \(0.373\) in decimal form and \(E=1.66\%\) or \(0.0166\) in decimal form.
The lower bound \(L=\hat{p}-E\).
\(L = 0.373-0.0166=0.3564\) (in decimal). Converting back to percentage: \(0.3564\times100 = 35.64\%\)
The upper bound \(U=\hat{p}+E\).
\(U=0.373 + 0.0166=0.3896\) (in decimal). Converting back to percentage: \(0.3896\times 100=38.96\%\)

Answer:

a. The margin of error is \(1.66\%\) (already given).
b. We can be \(95\%\) confident that between \(35.64\%\) and \(38.96\%\) of all households watched the final episode of “It Ain't Over Yet.”