(Lecture 16) Sampling Distributions: Understanding Sample Proportions and Their Variability

Sampling Distributions

Introduction to Sampling Distributions

Sampling distributions are a fundamental concept in statistics, describing how sample statistics (such as proportions or means) vary from sample to sample. They allow us to make probabilistic statements about how close a sample statistic is likely to be to the true population parameter.

• Sampling Distribution: The probability distribution of a statistic computed from a sample.

• Population Parameter: A numerical value that summarizes a characteristic for an entire population (e.g., population proportion).

• Sample Statistic: A numerical value that summarizes a characteristic for a sample (e.g., sample proportion).

Section 7.1: How Sample Proportions Vary Around the Population Proportion

This section explores how the proportion observed in a sample can be used to estimate the population proportion, and how much variability to expect in this estimate.

• Key Question: How close is the sample proportion to the population proportion?

• Influence of Sample Size: Larger samples tend to produce sample statistics closer to the population parameter.

• Sampling Distribution: Helps determine the likelihood that a sample statistic falls close to the population parameter.

Example: Predicting Election Results Using Exit Polls

Polling organizations use exit polls to predict election outcomes by sampling a small number of voters and estimating the proportion who voted for each candidate.

• Scenario: In the 2010 California gubernatorial election, 3889 voters were sampled out of over 9 million.

• Sample Results: 46.9% for Brown, 42.4% for Whitman, remainder for other/no answer.

• Population Proportion: The true proportion for Brown was 0.538 (53.8%).

• Unknowns: At the time of polling, the population proportion was unknown.

Data Distribution vs. Population Distribution:

• Data Distribution: The distribution of outcomes in the sample (e.g., proportion voting for Brown in the sample).

• Population Distribution: The distribution of outcomes in the entire population (e.g., proportion voting for Brown in all voters).

Table: Comparison of Population and Sample Distributions

Distribution Type	Proportion for Brown	Proportion for Not Brown	Sample Size
Population Distribution	0.538	0.462	9,500,000
Sample Distribution	0.469	0.531	3,889

Additional info: Table values inferred from context and slide images.

Definition and Properties of Sampling Distribution

The sampling distribution of a statistic describes the probabilities for the possible values the statistic can take, based on repeated sampling from the population.

• Random Variable: Each sample statistic (e.g., sample proportion) is a random variable with its own sampling distribution.

• Types of Sampling Distributions: Sample mean, sample median, sample standard deviation, sample proportion, etc.

Describing the Sampling Distribution of a Sample Proportion

To summarize the center and variability of the sampling distribution for a sample proportion, we use the mean and standard deviation.

• Mean of Sampling Distribution: The mean of the sampling distribution of the sample proportion is equal to the population proportion $p$.

• Standard Deviation of Sampling Distribution: The standard deviation is given by:

$\text{Standard deviation} = \sqrt{\frac{p(1-p)}{n}}$

• Where: $p$ is the population proportion, $n$ is the sample size.

Normal Approximation of the Sampling Distribution

When the sample size is sufficiently large, the sampling distribution of the sample proportion is approximately normal.

• Rule of Thumb: Both $np$ and $n(1-p)$ should be at least 10 for the normal approximation to be valid.

• Implication: This allows us to use normal probability calculations to estimate the likelihood that a sample proportion falls within a certain range of the population proportion.

Summary

• Sampling distributions describe the variability of sample statistics.

• The mean of the sampling distribution of the sample proportion equals the population proportion.

• The standard deviation depends on both the population proportion and the sample size.

• With large samples, the sampling distribution of the sample proportion is approximately normal.

Example Application

Suppose a poll samples 1,000 voters and finds that 52% support a candidate. If the true population proportion is 53%, the sampling distribution allows us to estimate how likely it is to observe a sample proportion this close to the population value, given the sample size.