Question

Male workers A factory employs $3000$unionized workers, $90\%$of whom are male. A random sample of 15 workers is selected for a survey about worker satisfaction. Let $\hat{p}$= the proportion of males in the sample.

a. Identify the mean of the sampling distribution of $\hat{p}$

b. Calculate and interpret the standard deviation of the sampling distribution of $\hat{p}$.

Verify that the $10\%$condition is met.

c. Describe the shape of the sampling distribution of $\hat{p}$. Justify your answer.

Short answer

a. Mean of sampling distribution is $0.90$

b. Standard deviation is $0.0775$.

c. Shape is skewed to left.

Step-by-step solution

Given Information

It is given that $p=90\%=0.90$

$n=15$

Mean of sampling distribution

The mean is${\mu }_{\hat{p}}=p=90\%=0.90$

Standard Deviation

We know that standard deviation is:

${\sigma }_{\hat{p}}=\sqrt{\frac{p(1-p)}{n}}$

$=\sqrt{\frac{0.90(1-0.90)}{15}}$

$=0.0775$

The standard deviation is role="math" localid="1654889468124" $0.0775$

Shape of sampling distribution

Condition of normality is $np\ge 10\text{and}n(1-p)\ge 10$

Here, $np=15(0.90)=13.5\ge 10$

$n(1-n)=15(1-0.90)=1.5<10$

Population proportion is $0.90$which is closer to $1$. Hence, sampling distribution is skewed to left.