Question

This exercise involves the use of an unrealistically small population to provide a concrete illustration of the exact distribution of a sample proportion. The population consists of three men and two women. The first names of the men are Jose, Pete, and Carlo; the first names of the women are Gail and Frances, Suppose that the specified attribute is "female."

a. Determine the population proportion, $p$.

b. The first column of the following table provides the possible samples of size 2, where each person is represented by the first letter of his or her first name; the second column gives the number of successes - the number of females obtained-for each sample, and the third column shows the sample proportion. Complete the table.

c. Construct a dot-plot for the sampling distribution of the proportion for samples of size 2. Mark the position of the population proportion on the dot-plot.

d. Use the third column of the table to obtain the mean of the variable $\hat{p}$.

e. Compare your answers from parts (a) and (d). Why are they the Same?

Short answer

(a) The population proportion is $0.4$

(b)

(c)

(d) The mean of the variable is $0.4$

(e) Both values are the same.

Step-by-step solution

Part (a) Step 1: Given information

The given data is

Part (a) Step 2: Explanation

According to the information provided, the group consists of three men and two women.

Being a woman is the specified characteristic.

As a result, the success rate is $x=2$

The population size is$n=5$

As a result, the population proportion is

$p=\frac{x}{n}$

$$\begin{gathered} =\frac{2}{5} \\ =0.4. \end{gathered}$$

Part (b) Step 1: Given information

The given data is

Part (b) Step 2: Explanation

According to the information provided, the group consists of three men and two women.

Being a woman is the specified characteristic.

Population size is $n=5$

Part (c) Step 1: Given information

The given data is

Part (c) Step 2: Explanation

Population proportion is $p=0.4$

The sample proportion is calculated for each sample of size $2$in section b of this activity, and a dot plot is constructed below.

Part (d) Step 1: Given information

The given data is

Part (d) Step 2: Explanation

From the above values

Mean of sample proportions is

${\mu }_{\hat{p}}=\frac{\sum \hat{p}}{10}$

${\mu }_{\hat{p}}=\frac{0.5+0+0+0.5+0.5+0.5+1+0+0.5+0.5}{10}$

$=0.4$

Part (e) Step 1: Given information

The given data is

Part (e) Step 2: Explanation

Population proportion is $p=0.4$in part a of this exercise.

Mean of sample proportions, ${\mu }_{\hat{p}}=0.4$, from section d of this exercise.

Because the mean of all sample distributions, $\hat{p}$, is the same as the population percentage,$p$, both values are the same.