Question

The Large Counts condition When constructing a confidence interval for a population proportion, we check that both $n{p}^{\wedge }\text{and}n\left(1-p^{\wedge }\right)$are at least $10$

a. Why is it necessary to check this condition?

b. What happens to the capture rate if this condition is violated?

Short answer

a. The sample distribution is approximately normal.

b. The chances are less to catch correct population parameter.

Step-by-step solution

Given Information

We check $n\hat{p}\text{and}n(1-\hat{p})\ge 10$during construction of confidence interval for population proportion.

Necessity of Conditions

Large Count condition, $n\hat{p}\text{and}n(1-\hat{p})$should be at least $10$. We require large count condition to be satisfied, so that sampling distribution is normal approximately.

If it is not met, sampling distribution of proportion is skewed. Hence, we can't use normal distribution for estimation of confidence interval.

If the above Condition is Violated

If the condition is not satisfied, sampling distribution of sample proportion is skewed, hence normal distribution is not used to estimate confidence interval.

Correct population parameter won't be captured.