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 ±□%. (do not round until the final answer. then round to the nearest hundredth as needed.)

Explanation:

Step1: Recall the formula for margin of error for a proportion

The formula for the margin of error \(E\) for a proportion in a large - sample (when \(np\geq5\) and \(n(1 - p)\geq5\), where \(n\) is the sample size and \(p\) is the sample proportion) is \(E = z\sqrt{\frac{p(1 - p)}{n}}\). For a 95% confidence level, \(z\approx1.96\). Here, \(n = 3648\) and \(p=0.373\).

Step2: Calculate the value inside the square root

First, calculate \(p(1 - p)=0.373\times(1 - 0.373)=0.373\times0.627 = 0.233971\). Then \(\frac{p(1 - p)}{n}=\frac{0.233971}{3648}\approx0.0000641\).

Step3: Calculate the square root

\(\sqrt{\frac{p(1 - p)}{n}}=\sqrt{0.0000641}\approx0.008006\).

Step4: Calculate the margin of error

\(E = 1.96\times0.008006\approx0.0157\). To convert this to a percentage, multiply by 100. So \(E = 1.57\%\).

Answer:

a. The margin of error is \(\pm1.57\%\)
b. We can say with 95% confidence that the percentage of TV households in the population who tuned into the final episode of “It Ain't Over Yet” is between \((37.3 - 1.57)\%\) and \((37.3+1.57)\%\), that is between \(35.73\%\) and \(38.87\%\)