# 6.3: The Sample Proportion

Skills to Develop

• To recognize that the sample proportion $$\hat{p}$$ is a random variable.
• To understand the meaning of the formulas for the mean and standard deviation of the sample proportion.
• To learn what the sampling distribution of $$\hat{p}$$ is when the sample size is large.

Often sampling is done in order to estimate the proportion of a population that has a specific characteristic, such as the proportion of all items coming off an assembly line that are defective or the proportion of all people entering a retail store who make a purchase before leaving. The population proportion is denoted $$p$$ and the sample proportion is denoted $$\hat{p}$$. Thus if in reality $$43\%$$ of people entering a store make a purchase before leaving,

$p = 0.43 \nonumber$

if in a sample of $$200$$ people entering the store, $$78$$ make a purchase,

$\hat{p}=\dfrac{78}{200}=0.39. \nonumber$

The sample proportion is a random variable: it varies from sample to sample in a way that cannot be predicted with certainty. Viewed as a random variable it will be written $$\hat{P}$$. It has a mean $$μ_{\hat{P}}$$ and a standard deviation $$σ_{\hat{P}}$$. Here are formulas for their values.

mean and standard deviation of the sample proportion

Suppose random samples of size $$n$$ are drawn from a population in which the proportion with a characteristic of interest is $$p$$. The mean $$μ_{\hat{P}}$$ and standard deviation $$σ_{\hat{P}}$$ of the sample proportion $$\hat{P}$$ satisfy

$μ_{\hat{P}}=p$

and

$σ_{\hat{P}}= \sqrt{\dfrac{pq}{n}}$

where $$q=1−p$$.

The Central Limit Theorem has an analogue for the population proportion $$\hat{p}$$. To see how, imagine that every element of the population that has the characteristic of interest is labeled with a $$1$$, and that every element that does not is labeled with a $$0$$. This gives a numerical population consisting entirely of zeros and ones. Clearly the proportion of the population with the special characteristic is the proportion of the numerical population that are ones; in symbols,

$p=\dfrac{\text{number of 1s}}{N}$

But of course the sum of all the zeros and ones is simply the number of ones, so the mean $$μ$$ of the numerical population is

$μ=\dfrac{ \sum x}{N}= \dfrac{\text{number of 1s}}{N}$

Thus the population proportion $$p$$ is the same as the mean $$μ$$ of the corresponding population of zeros and ones. In the same way the sample proportion $$\hat{p}$$ is the same as the sample mean $$\bar{x}$$. Thus the Central Limit Theorem applies to $$\hat{p}$$ . However, the condition that the sample be large is a little more complicated than just being of size at least $$30$$.

## The Sampling Distribution of the Sample Proportion

For large samples, the sample proportion is approximately normally distributed, with mean $$μ_{\hat{P}}=p$$ and standard deviation $$\sigma _{\hat{P}}=\sqrt{\frac{pq}{n}}$$.

A sample is large if the interval $$\left [ p-3\sigma _{\hat{p}},\, p+3\sigma _{\hat{p}} \right ]$$ lies wholly within the interval $$[0,1]$$.

In actual practice $$p$$ is not known, hence neither is $$σ_{\hat{P}}$$. In that case in order to check that the sample is sufficiently large we substitute the known quantity $$\hat{p}$$ for $$p$$. This means checking that the interval

$\left [ \hat{p}-3\sqrt{\frac{\hat{p}(1-\hat{p})}{n}},\, \hat{p}+3\sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \right ]$

lies wholly within the interval $$[0,1]$$. This is illustrated in the examples.

Figure $$\PageIndex{1}$$ shows that when $$p = 0.1$$, a sample of size $$15$$ is too small but a sample of size $$100$$ is acceptable.

Figure $$\PageIndex{1}$$: Distribution of Sample Proportions

Figure $$\PageIndex{2}$$ shows that when $$p=0.5$$ a sample of size $$15$$ is acceptable.

Figure $$\PageIndex{2}$$: Distribution of Sample Proportions for $$p=0.5$$ and $$n=15$$

Example $$\PageIndex{1}$$

Suppose that in a population of voters in a certain region $$38\%$$ are in favor of particular bond issue. Nine hundred randomly selected voters are asked if they favor the bond issue.

1. Verify that the sample proportion $$\hat{p}$$ computed from samples of size $$900$$ meets the condition that its sampling distribution be approximately normal.
2. Find the probability that the sample proportion computed from a sample of size $$900$$ will be within $$5$$ percentage points of the true population proportion.

Solution:

1. The information given is that $$p=0.38$$, hence $$q=1-p=0.62$$. First we use the formulas to compute the mean and standard deviation of $$\hat{p}$$:

$\mu _{\hat{p}}=p=0.38\; \text{and}\; \sigma _{\hat{P}}=\sqrt{\frac{pq}{n}}=\sqrt{\frac{(0.38)(0.62)}{900}}=0.01618 \nonumber$

Then $$3\sigma _{\hat{P}}=3(0.01618)=0.04854\approx 0.05$$ so

$\left [ \hat{p}-3\sqrt{\frac{\hat{p}(1-\hat{p})}{n}},\, \hat{p}+3\sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \right ]=[0.38-0.05,0.38+0.05]=[0.33,0.43] \nonumber$

which lies wholly within the interval $$[0,1]$$, so it is safe to assume that $$\hat{p}$$ is approximately normally distributed.
1. To be within $$5$$ percentage points of the true population proportion $$0.38$$ means to be between $$0.38-0.05=0.33$$ and $$0.38+0.05=0.43$$. Thus

\begin{align*} P(0.33<\hat{P}<0.43) &= P\left ( \frac{0.33-\mu _{\hat{P}}}{\sigma _{\hat{P}}} <Z< \frac{0.43-\mu _{\hat{P}}}{\sigma _{\hat{P}}} \right )\\[4pt] &= P\left ( \frac{0.33-0.38}{0.01618} <Z< \frac{0.43-0.38}{0.01618}\right )\\[4pt] &= P(-3.09<Z<3.09)\\[4pt] &= P(3.09)-P(-3.09)\\[4pt] &= 0.9990-0.0010\\[4pt] &= 0.9980 \end{align*}

Example $$\PageIndex{2}$$

An online retailer claims that $$90\%$$ of all orders are shipped within $$12$$ hours of being received. A consumer group placed $$121$$ orders of different sizes and at different times of day; $$102$$ orders were shipped within $$12$$ hours.

1. Compute the sample proportion of items shipped within $$12$$ hours.
2. Confirm that the sample is large enough to assume that the sample proportion is normally distributed. Use $$p=0.90$$, corresponding to the assumption that the retailer’s claim is valid.
3. Assuming the retailer’s claim is true, find the probability that a sample of size $$121$$ would produce a sample proportion so low as was observed in this sample.
4. Based on the answer to part (c), draw a conclusion about the retailer’s claim.

Solution:

1. The sample proportion is the number $$x$$ of orders that are shipped within $$12$$ hours divided by the number $$n$$ of orders in the sample:

$\hat{p} =\frac{x}{n}=\frac{102}{121}=0.84\nonumber$

1. Since $$p=0.90$$, $$q=1-p=0.10$$, and $$n=121$$,

$\sigma _{\hat{P}}=\sqrt{\frac{(0.90)(0.10)}{121}}=0.0\overline{27}\nonumber$

hence

$\left [ p-3\sigma _{\hat{P}},\, p+3\sigma _{\hat{P}} \right ]=[0.90-0.08,0.90+0.08]=[0.82,0.98]\nonumber$

Because

$[0.82,0.98]⊂[0,1]\nonumber$

it is appropriate to use the normal distribution to compute probabilities related to the sample proportion $$\hat{P}$$.

1. Using the value of $$\hat{P}$$ from part (a) and the computation in part (b),

\begin{align*} P(\hat{P}\leq 0.84) &= P\left ( Z\leq \frac{0.84-\mu _{\hat{P}}}{\sigma _{\hat{P}}} \right )\\[4pt] &= P\left ( Z\leq \frac{0.84-0.90}{0.0\overline{27}} \right )\\[4pt] &= P(Z\leq -2.20)\\[4pt] &= 0.0139 \end{align*}

1. The computation shows that a random sample of size $$121$$ has only about a $$1.4\%$$ chance of producing a sample proportion as the one that was observed, $$\hat{p} =0.84$$, when taken from a population in which the actual proportion is $$0.90$$. This is so unlikely that it is reasonable to conclude that the actual value of $$p$$ is less than the $$90\%$$ claimed.

## Key Takeaway

• The sample proportion is a random variable $$\hat{P}$$.
• There are formulas for the mean $$μ_{\hat{P}}$$, and standard deviation $$σ_{\hat{P}}$$ of the sample proportion.
• When the sample size is large the sample proportion is normally distributed.