Using a random sample of 5284 TV households, Acme Media Statistics found that $31.1 \%$ watched the final episode of "When Will it End?"
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 "When Will it End?"
a. The margin of error is $\pm \square \%$.
(Do not round until the final answer. Then round to the nearest hundredth as needed.)
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Answer
We can say that we are 95% confident that the true proportion of TV households who watched the final episode of "When Will it End?" is between \(\boxed{29.69\%}\) and \(\boxed{32.51\%}\) (31.1% ± 1.41%)
Step 1 ：Calculate the standard deviation using the formula for a binomial distribution: \(\sigma = \sqrt{p(1-p)/n}\)
Step 2 ：Substitute the given values into the formula: \(\sigma = \sqrt{0.311(1-0.311)/5284}\)
Step 3 ：Calculate the standard deviation: \(\sigma = 0.0072\)
Step 4 ：Calculate the margin of error for a 95% confidence level by multiplying the standard deviation by 1.96: \(\text{Margin of error} = 1.96 * 0.0072\)
Step 5 ：Calculate the margin of error: \(\text{Margin of error} = 0.0141\) or 1.41% when expressed as a percentage
Step 6 ：The confidence interval is calculated by adding and subtracting the margin of error from the sample proportion: \(\text{Confidence interval} = 0.311 \pm 0.0141\)
Step 7 ：We can say that we are 95% confident that the true proportion of TV households who watched the final episode of "When Will it End?" is between \(\boxed{29.69\%}\) and \(\boxed{32.51\%}\) (31.1% ± 1.41%)
