Question

Your teacher brings two bags of colored goldfish crackers to class. She tells you that Bag $1$has $25\%$red crackers and Bag $2$has $35\%$red crackers. Each bag contains more than $500$crackers. Using a paper cup, your teacher takes an SRS of $50$crackers from Bag 1 and a separate SRS of $40$crackers from Bag $2$. Let $\widehat{p_1}-\widehat{p_2}$be the difference in the sample proportions of red crackers.

What is the shape of the sampling distribution of$\widehat{p_1}-\widehat{p_2}$ ? Why?

Short answer

It is approximately normally distributed.

Step-by-step solution

Given Information

The proportion of red crackers in $Bag1\left(p_1\right)=25\%$

$=0.25$.

The 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 $2\left(n_2\right)=40$.

Explanation

The required conditions could be checked as:

$n_1{p}_1=50(0.25)$

$=12.5>10$

$n_1\left(1-p_1\right)=50(1-0.25)>10$

localid="1650365073121" $n_2{p}_2=40(0.35)$

$=14>10$

localid="1650365094942" $n_2\left(1-p_2\right)=40(1-0.65)$

$=26>10$

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

Final Answer

It is approximately normally distributed.