Before we begin, let’s make sure we review the terms and notation associated with proportions:
- \(p\) is the population proportion. It is a fixed value.
- \(n\) is the size of the random sample.
- \(\hat{p}\) is the sample proportion. It varies based on the sample.
The following example will illustrate how to find the sampling distribution for an example where the population is small.
Sample Proportions with a Small Population: Favorite Color Section
In a particular family, there are five children. Their names are Alex (A), Betina (B), Carly (C), Debbie (D), and Edward (E). The table below shows the child’s name and their favorite color.
|
Name |
Alex (A) |
Betina (B) |
Carly (C) |
Debbie (D) |
Edward (E) |
|---|---|---|---|---|---|
|
Color |
Green |
Blue |
Yellow |
Purple |
Blue |
We are interested in the proportion of children in the family who prefer the color blue, and from the table, we can see that \(p = .40\) of the children prefer blue.
Similar to the pumpkin example earlier in the lesson, let's say we didn't know the proportion of children who like blue as their favorite color. We'll use resampling methods to estimate the proportion. Let’s take \(n=2\) repeated samples, taken without replacement. Here are all the possible samples of size \(n=2\) and their respective probabilities of the proportion of children who like blue.
|
Sample |
P(Blue) |
Probability |
|---|---|---|
|
AB |
1/2 |
1/10 |
|
AC |
0 |
1/10 |
|
AD |
0 |
1/10 |
|
AE |
1/2 |
1/10 |
|
BC |
1/2 |
1/10 |
|
BD |
1/2 |
1/10 |
|
BE |
1 |
1/10 |
|
CD |
0 |
1/10 |
|
CE |
1/2 |
1/10 |
|
DE |
1/2 |
1/10 |
The probability mass function (PMF) is:
|
P(Blue) |
0 |
1/2 |
1 |
|---|---|---|---|
|
Probability |
3/10 |
6/10 |
1/10 |
The graph of the PMF:
Sampling Distribution of P(Blue)
Bar graph showing three bars (0 with a length of 0.3, 0.5 with length of 0.5 and 1 with a lenght of 0.1).
The true proportion is \(p=P(Blue)=\frac{2}{5}\). When the sample size is \(n=2\), you can see from the PMF, it is not possible to get a sampling proportion that is equal to the true proportion.
Although not presented in detail here, we could find the sampling distribution for a larger sample size, say \(n=4\). The PMF for n=4 is...
|
P(Blue) |
1/4 |
1/2 |
|---|---|---|
|
Probability |
2/5 |
3/5 |
As with the sampling distribution of the sample mean, the sampling distribution of the sample proportion will have sampling error. It is also the case that the larger the sample size, the smaller the spread of the distribution.