Chapter 7, Problem 3PT

To determine

The correct answer from the options that for which value of $\widehat{p}$ would it be safe to use the Normal approximation to the sampling distribution of $\widehat{p}$ is (a) 0.01, (b) $\frac{1}{11}$ , (c) 0.85, (d) 0.975, (e) 0.999.

Answer to Problem 3PT

The correct answer is (c).

Explanation of Solution

Given information:

Sample size for SRS, $n=100$

Proportion of successes in the sample $=\widehat{p}$

Explanation:

Know that the sampling distribution of $\widehat{p}$ is approximately normal if both $np$ and $n\left(1-p\right)$ are at least 10.

For option (a):

  $np=100\times 0.01=1$

  $\begin{array}{c}n\left(1-p\right)=100\times \left(1-0.01\right)\\ =99\end{array}$

For option (b):

  $np=100\times \frac{1}{11}\approx 9.0909$

  $\begin{array}{c}n\left(1-p\right)=100\times \left(1-\frac{1}{11}\right)\\ \approx 90.9091\end{array}$

For option (c):

  $np=100\times 0.85=85$

  $\begin{array}{c}n\left(1-p\right)=100\times \left(1-0.85\right)\\ =15\end{array}$

For option (d):

  $np=100\times 0.975=97.5$

  $\begin{array}{c}n\left(1-p\right)=100\times \left(1-0.975\right)\\ =2.5\end{array}$

For option (e):

  $np=100\times 0.999=99.9$

  $\begin{array}{c}n\left(1-p\right)=100\times \left(1-0.999\right)\\ =0.1\end{array}$

It is seen that in option (c) both $np$ and $n\left(1-p\right)$ are greater than 10.

Hence, the correct answer is (c).