Chapter 10.1, Problem 1.1CYU

To determine

To find: The shape of the sampling distribution of $\left({\widehat{p}}_1-{\widehat{p}}_2\right)$ .

Answer to Problem 1.1CYU

It is approximately normally distributed.

Explanation of Solution

Given:

Proportion of red crackers in Bag 1 $\left(p_1\right)=25\%=0.25$

Proportion of red crackers in Bag 2 $\left(p_2\right)=35\%=0.35$

Number of crackers in Bag 1 $\left(n_1\right)=50$

Number of crackers in Bag 2 $\left(n_2\right)=40$

Calculation:

The required conditions could be checked as:

  $\begin{array}{l}{n}_1{p}_1=50\left(0.25\right)=12.5>10\\ n_1\left(1-p_1\right)=50\left(1-0.25\right)>10\\ n_2{p}_2=40\left(0.35\right)=14>10\\ n_2\left(1-p_2\right)=40\left(1-0.65\right)=26>10\end{array}$

Since, all the values are greater than 10. Thus, it could be said that distribution is approximately normally distributed.