Question

using a random sample of 3417 tv households, acme media statistics found that 48.9% watched the final episode of \still hanging on.\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 \still hanging on.\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

The formula for margin of error \(E\) for a proportion is \(E = z\sqrt{\frac{p(1 - p)}{n}}\). For a \(95\%\) confidence level (common in such media - statistics problems), \(z = 1.96\). Here, \(p=0.489\) and \(n = 3417\).

Step2: Calculate the value inside the square - root

First, calculate \(p(1 - p)=0.489\times(1 - 0.489)=0.489\times0.511 = 0.25\) (approx). Then \(\frac{p(1 - p)}{n}=\frac{0.25}{3417}\approx0.00007316\).

Step3: Calculate the square - root and multiply by \(z\)

\(\sqrt{\frac{p(1 - p)}{n}}=\sqrt{0.00007316}\approx0.00855\). Then \(E=1.96\times0.00855\approx0.01676\).

Step4: Convert to percentage

To convert to a percentage, multiply by \(100\). So \(E = 1.676\%\approx1.68\%\)

Answer:

a. The margin of error is \(\pm1.68\%\)
b. We are \(95\%\) confident that the true percentage of TV households in the population who tuned into the final episode of "Still Hanging On" is between \(48.9\% - 1.68\%=47.22\%\) and \(48.9\%+1.68\% = 50.58\%\)