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3.6 Sampling Distributions of Sample Proportions

Section Overview

  • Gallery 3.6.1 – Samp. Dist. of Samp. Prop.
  • Example 3.6.1
  • Example 3.6.2
  • Pop Quiz 3.6.1

Concept of Sampling Distributions

So far, we’ve gone into great detail about how important random sampling is. If asked why it’s important and you had been paying attention up until this point, you might say something like “Random sampling allows us to make inferences about the population as a whole.” Your answer would be correct, but incomplete. What sort of things specifically would we be able to infer about the population? If these “things” are values, then they are called parameters. One example of a parameter that we’ll discuss further in this section is called the population proportion, denoted as pp. If we wanted to estimate the population proportion, we would find its corresponding statistic, the sample proportion p^\hat p, by taking a simple random sample.

Every time an SRS is taken, we get a different value for the statistic, in this case, a different value of the sample proportion. If we took many random samples of a fixed sample size and recorded the frequency of all the values of the sample proportion values in a graph, the graph would start to resemble the density curve of a Normal distribution.

When an infinite number of random samples are taken, the distribution is called the sampling distribution of the sample proportion. Gallery 3.5.1 shows how changing the population parameter pp, the sample size, and the number of samples can change the center and spread of the sampling distributions.

Source gap (S0302): Source reference: This paragraph refers to Gallery 3.5.1; the gallery shown on the same page is labelled 3.6.1. Both captured labels are retained.

The Sampling Distribution of the Sample Proportion

Much like how we described distributions back in Chapter 1.2 in terms of their Shape, Center, and Spread, we can also use these terms to describe the sampling distribution of the sample proportion:

Shape – As we mentioned earlier, the sampling distribution of the sample proportion takes a shape similar to a Normal distribution. This only applies when the sample has at least 10 “successes” (np≥10np\ge10) and 10 “failures”, (n(1−p)≥10n(1-p)\ge10).

Center – The mean of the sampling distribution of \hat p is equal to the population proportion pp.

Source gap (S0308): Source notation: The Center sentence prints the literal text “\hat p”; other occurrences on the page display p̂. The literal text is preserved here.

Spread – The standard deviation of the sampling distribution of p^\hat p is p(1−p)n\sqrt{\frac{p(1-p)}{n}} for an SRS of size nn. This only applies if the population is at least 10 times as large as the sample, or 10n≤N10n\le N.

The Sampling Distribution of the Difference Between Two Sample Proportions

If you haven’t already been scared off by the long title, the sampling distribution of the difference between two sample proportions comes from the statistic called the difference between two sample proportions p^1−p^2\hat p_1-\hat p_2, which is commonly used when we want to directly compare proportions from two different populations. It is very similar to its single proportion counterpart, as shown by its Shape, Center, and Spread:

Shape – The sampling distribution of the difference between two sample proportions also takes a shape similar to a Normal distribution. This only applies when both samples have at least 10 “successes” (np≥10np\ge10) and 10 “failures”, (n(1−p)≥10n(1-p)\ge10).

Center – The mean of the sampling distribution of the difference between two sample proportions is equal to p1−p2p_1-p_2.

Spread – The standard deviation of the sampling distribution of p^1−p^2\hat p_1-\hat p_2 is

p1(1−p1)n1+p2(1−p2)n2\sqrt{\frac{p_1(1-p_1)}{n_1}+\frac{p_2(1-p_2)}{n_2}}

Again this only applies if both populations are at least 10 times as large as their respective samples, or 10n≤N10n\le N.

Example 3.6.1

Briny Beds, a large company that sells water beds that also function as aquariums, takes a (small) SRS of 400 water beds manufactured in the last month and finds that 3% of them puncture before making it to the consumer. What is the probability that the random sample will result in a value that is within 1% of the population proportion?

Example 3.6.1 — recorded answer

All three descriptions of the sampling distribution: Shape, Center, and Spread, will be used to find the probability.

Shape: We know that the sampling distribution of the sample proportion closely follows a Normal distribution. This means we can use calculations similar to what we learned in Chapter 3.4, but only if the sample has at least 10 “successes” and 10 “failures,” where a “success” in this context is defined as a non-punctured bed. 400(0.97)=388≥10400(0.97)=388\ge10 and 400(0.3)=12≥10400(0.3)=12\ge10, so we are clear to use Normal calculations.

Source gap (S0309): Source inconsistency: The failure calculation is printed as 400(0.3) = 12. The other calculations use a puncture rate of 0.03. The captured decimal and result are retained.

Center: We know that the mean of the sampling distribution of the sample proportion is p=0.97p=0.97.

Spread: We know that the standard deviation of the sampling distribution of the sample proportion is

p(1−p)n\sqrt{\frac{p(1-p)}{n}}

However, we can only use this calculation if the population is at least 10 times as large as the sample, or 10(400)=400010(400)=4000. Briny Beds is described as a large company and the SRS taken is also described as small, so we can assume that there have been at least 4000 waterbeds made in the last month. Therefore we have:

0.97(0.03)400=0.0085\sqrt{\frac{0.97(0.03)}{400}}=0.0085

Using these numbers, we can find the corresponding z-scores for 0.97−0.01=0.960.97-0.01=0.96 and 0.97+0.01=0.980.97+0.01=0.98:

z=xˉ−μσ=0.96−0.970.0085=−1.17z=\frac{\bar x-\mu}{\sigma}=\frac{0.96-0.97}{0.0085}=-1.17
z=xˉ−μσ=0.98−0.970.0085=1.17z=\frac{\bar x-\mu}{\sigma}=\frac{0.98-0.97}{0.0085}=1.17

We are looking for the area between z=−1.17z=-1.17 and z=1.17z=1.17 on a standard Normal distribution graph, which we can find by calculating the difference between their corresponding probabilities. After looking up the probabilities in the z-table, we get:

0.879−0.121≈0.760.879-0.121\approx0.76

This means that about 76% of all SRS’s of size 400 will give a result within 1% of the true population proportion.

Example 3.6.2

Briny Beds has decided to try another manufacturer whose puncture rate is 2% from a SRS of 525 water beds. What is the probability of getting a difference in puncture rate of 0.5% or less?

Example 3.6.2 — recorded answer

Like in the previous example, all three descriptions of the sampling distribution: Shape, Center, and Spread, will be used to find the probability.

Shape: We know that the sampling distribution of the difference between two sample proportions also closely follows a Normal distribution if both samples have at least 10 “successes” and 10 “failures.” We already know the first manufacturer meets the condition, so we just have to check for the second manufacturer. 550(0.98)=539≥10550(0.98)=539\ge10 and 550(0.02)=11≥10550(0.02)=11\ge10, so we are clear to use Normal calculations.

Center: We know that the mean of the sampling distribution of the difference between two sample proportions is p1−p2=0.005p_1-p_2=0.005.

Spread: We know that the standard deviation of the sampling distribution of the difference between two sample proportions is

p1(1−p1)n1+p2(1−p2)n2\sqrt{\frac{p_1(1-p_1)}{n_1}+\frac{p_2(1-p_2)}{n_2}}

However, we can only use this calculation if the populations are at least 10 times as large as their respective samples. We already know this holds true for the first manufacturer. For the second manufacturer this would be at least 10(550)=550010(550)=5500, which, based on everything we know about Briny Beds, we can assume to be true. Therefore we have:

0.97(0.03)400+0.98(0.02)550=0.0104\sqrt{\frac{0.97(0.03)}{400}+\frac{0.98(0.02)}{550}}=0.0104

Using these numbers, we can find the corresponding z-score for p^1−p^2=0.005\hat p_1-\hat p_2=0.005.

z=xˉ−μσ=0.005−0.0100.0104=−0.4803z=\frac{\bar x-\mu}{\sigma}=\frac{0.005-0.010}{0.0104}=-0.4803

We are looking for the area to the left of z=−0.4803z=-0.4803 on a standard Normal distribution graph. According to the z-table, we get the corresponding probability of 0.32.

This means that about there is a 32% chance that the difference in puncture rates will be as large as 0.5%.

Source gap (S0311): Source inconsistencies: The prompt uses a sample of 525; the recorded work uses 550. The Center line gives p₁ − p₂ = 0.005, while the z-score substitutes 0.010 as the mean. The formula combines non-puncture proportions 0.97 and 0.98, while the prompt asks about puncture rates. The conclusion says “as large as 0.5%” after the prompt asks “0.5% or less” and the work uses a left-tail area. All captured statements and calculations are retained.

Pop Quiz 3.6.1

Question 1 of 5

Suppose the proportion of all college students who have drank alcohol in the last month is pN=200pN=200, representative of all college students, what is the mean and standard error of the sample proportion?

Source gap (S0313): Source stem gap: The screenshot literally displays “pN = 200” at this point in the question. It does not supply a readable population proportion or the missing wording that would distinguish p from a sample-size statement. No value or wording has been inferred from the choices.
Question 2 of 5

Considering the scenario above, what is the chance that the proportion of students who have drank alcohol in the last month is less than 32%?

Question 3 of 5

Which of the following are correct? (1) The sampling distribution of a statistic p^\hat p can be approximated by a normal curve in some cases. (2) The sample proportion is an unbiased estimator of the population proportion of successes. (3) The standard deviation of the sample proportion gets larger as the sample size gets larger.

Question 4 of 5

For non-normal populations, as the sample size gets __________, the distribution of sample means approaches a __________ distribution.

Question 5 of 5

If the sample size were decreased from 1000 to 250, the standard deviation would be multiplied by a factor of:

Source gap (S0317): Source answer gap: All five quiz questions and choices are captured, but no selected, checked, or feedback state is shown. No quiz answer key has been supplied.
Glossary and deeper reading · 11 refreshers

These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.

  • Random sampling

    Random sampling uses a specified chance process to select individuals from a population. It supports population inference when the sampling frame and data collection are appropriate, but does not guarantee perfect balance or eliminate every source of bias. For example, a random-number generator can select student identifiers from an enrollment list.

    Read more: Random sampling in the captured Statistics reading →

  • Statistical inference

    Statistical inference uses data and a study design to draw conclusions beyond the observed values. Random sampling supports population generalization, while well-conducted random assignment supports treatment-effect conclusions. For example, a random sample can estimate a school-wide preference without surveying every student.

    Read more: Statistical inference in the captured Statistics reading →

  • Simple random sample

    A simple random sample of a fixed size gives every possible subset of that size the same chance of selection. Giving individuals equal selection chances alone does not establish that the design is an SRS. For example, randomly choosing three identifiers without replacement from ten gives an SRS when every three-identifier set is equally likely.

    Read more: Simple random sample in the captured Statistics reading →

  • Sample statistic

    A statistic is a numerical summary calculated from sample data. Its value can change when a different sample is selected from the same population. For example, the proportion of walkers among 60 sampled students is a statistic.

  • Population parameter

    A population parameter is a numerical feature of the population or probability model being studied. It is fixed for that defined population, although its value may be unknown. For example, the true proportion of all enrolled students who walk to school is a parameter.

  • Sample

    A sample is the subset of a population actually studied. How it is selected matters for the conclusions that can be drawn about the population. For example, 60 selected students may be a sample from a school’s full enrollment.

    Read more: Sample in the captured Statistics reading →

  • Normal distribution

    A Normal distribution is a continuous, symmetric bell-shaped distribution determined by its mean and standard deviation. Many symmetric distributions are not Normal, so symmetry by itself is insufficient. For example, a Normal model centered at 50 with standard deviation 5 places most values relatively close to 50.

    Read more: Normal distribution in the captured Statistics reading →

  • Mean

    The mean is a measure of center often called an average. For a finite data set, the arithmetic mean is the sum of the values divided by their number; for a random variable, the mean is its probability-weighted expected value when that value exists. Means can be pulled toward extreme values and are not generally resistant. For example, the arithmetic mean of 2, 4, and 9 is 5.

    Read more: Mean in the captured Statistics reading →

  • Population proportion

    A population proportion is the fraction of the entire population with a specified characteristic. It is a parameter, commonly written p, that a sample proportion can estimate. For example, the fraction of all enrolled students who walk to school is the population proportion of walkers.

  • Standard deviation

    Standard deviation measures spread around the mean using squared deviations and a square root. A sample standard deviation uses the sum of squared deviations divided by one less than the sample size before taking the square root. For example, a larger standard deviation of travel times indicates greater variability, in the original time units.

    Read more: Standard deviation in the captured Statistics reading →

  • Population

    A population is the full set of individuals or measurements a study aims to describe. It must be defined clearly before deciding whether a sample supports generalization. For example, all students enrolled in a particular school this year form a population.

    Read more: Population in the captured Statistics reading →

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