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%)