Question

why is it necessary to check that ( nhat{p}geq5 ) and ( nhat{q}geq5 )?

a. it is necessary to check that ( nhat{p}geq5 ) and ( nhat{q}geq5 ) because the confidence intervals estimating the population proportions will overlap if these values are less than 5.
b. it is necessary to check that ( nhat{p}geq5 ) and ( nhat{q}geq5 ) because, if either of the values are less than 5, the tails on either side of the left and right endpoints are not accounted for.
c. it is necessary to check that ( nhat{p}geq5 ) and ( nhat{q}geq5 ) to be certain that the minimum value of the sample size, ( n ), is met.
d. it is necessary to check that ( nhat{p}geq5 ) and ( nhat{q}geq5 ) because, if either of the values are less than 5, the distribution may not be normally distributed, thus ( z_{c} ) cannot be used to calculate the confidence interval.

Explanation:

When constructing a confidence interval for a population proportion, we use the normal approximation to the binomial distribution. The conditions \(n\hat{p}\geq5\) and \(n\hat{q}\geq5\) (where \(\hat{q} = 1-\hat{p}\)) ensure that the binomial distribution (which models the number of successes in \(n\) independent Bernoulli trials with probability of success \(\hat{p}\)) is approximately normal. If these conditions are not met (i.e., if either \(n\hat{p}<5\) or \(n\hat{q}<5\)), the binomial distribution is too skewed, and the normal approximation (which relies on the central limit theorem for sums of independent and identically - distributed random variables, in this case, Bernoulli random variables) does not hold well. When the normal approximation does not hold, the \(z -\)score (\(z_{c}\)) used in the formula for the confidence interval of a proportion \(\hat{p}\pm z_{c}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\) is not appropriate.

• Option A: The reason is not about overlapping confidence intervals. Confidence intervals for proportions are calculated based on the distribution of the sample proportion, not about overlapping in the way described.
• Option B: It is not about unaccounted - for tails in the sense of left and right endpoints not considering tails. The key is the distributional assumption.
• Option C: It is not just about the sample size \(n\). The relationship between \(n\), \(\hat{p}\), and \(\hat{q}\) is crucial for the distributional form.
• Option D: This option correctly states that if \(n\hat{p}<5\) or \(n\hat{q}<5\), the distribution (of the sample proportion \(\hat{p}\)) may not be normal, and thus the \(z -\)score (\(z_{c}\)) (which is based on the standard normal distribution) cannot be used to calculate the confidence interval for the population proportion.

Answer:

D. It is necessary to check that \(n\hat{p}\geq5\) and \(n\hat{q}\geq5\) because, if either of the values are less than 5, the distribution may not be normally distributed, thus \(z_{c}\) cannot be used to calculate the confidence interval.