Question

Repeat part (b)-(e) of Exercise \(11.9\) for samples of size \(5\).

Short answer

Part a. Population proportion \(p=0.4\)

Part b. A table is created with all the possible samples of size \(5\).

Part c.

Part d. Mean of sample proportions is \(\mu_{p}=0.4\)

part e. The answer to part (a) and (d) are the same.

Step-by-step solution

Part a. Step 1. Given information 

A population consists of three men Jose, Peter, Carlo and two women Gail and Frances. Specified attribute is "Female".

Part a. Step 2. Calculation

As per the given information, group has three men and two women. Specified attribute is being a female.

Therefore, number of success is \(x=2\) and population size is \(n=5\).

So, population proportion would be \(p=\frac{x}{n}=\frac{2}{5}=0.4\).

Part b. Step 1. Calculation

As per the given information, group has three men and two women. Specified attribute is being a female. Sample proportion size is \(n=5\).

A table is created with all the possible samples of size \(5\).

Sample	Number of females \((x)\)	Sample proportion \(\hat{p}=\frac{x}{n}\)
J, P, C, G, F	\(2\)	\(\frac{2}{5}=0.4\)

Part c. Step 1. Calculation

From part (a) of this exercise, Population proportion is \(p=0.4\).

From part (b) of this exercise, sample proportion is obtained for each sample of size \(5\) and below dot plot is created.

Part d. Step 1. Calculation

From part (a) of this exercise, Population proportion is \(p=0.4\).

From part (b) of this exercise, sample proportion is obtained for each sample of size \(5\).

Mean of sample proportions, \(\mu _{\hat{p}}=\frac{\sum \hat{p}}{10}\)

\(\Rightarrow \mu _{\hat{p}}= \frac{0.4}{1}\)

\(\Rightarrow \mu _{\hat{p}}= 0.4\)

Part e. Step 1. Calculation

From part (a) of this exercise, Population proportion is \(p=0.4\) .

From part (b) of this exercise, mean of sample proportions, \(\mu _{\hat{p}}= 0.4\)

Both value are same because, the mean of all sampling distribution, \(\hat{p} \) , is same as the population proportion, \(p\).