(Lecture 17) Sampling Distributions and the Central Limit Theorem

Sampling Distributions of the Sample Mean

Introduction to Sampling Distributions

Sampling distributions describe how a statistic, such as the sample mean, varies from sample to sample when drawn from a population. Understanding these distributions is essential for making statistical inferences about population parameters.

• Sampling Distribution: The probability distribution of a statistic (e.g., sample mean) based on all possible samples of a given size from a population.

• Sample Mean (̅X): The average value calculated from a sample, used to estimate the population mean (μ).

Key Results for the Sampling Distribution of the Sample Mean

There are two main results regarding the sampling distribution of the sample mean:

1. Formulas for Mean and Standard Deviation: The mean and standard deviation of the sampling distribution can be calculated using population parameters.

2. Shape of the Distribution: The sampling distribution of the sample mean is often approximately normal, especially for large sample sizes.

Sampling Distribution When the Population is Normally Distributed

Properties of the Sampling Distribution

If a random sample of size n is drawn from a normally distributed population with mean μ and standard deviation σ, then:

• The sampling distribution of the sample mean ̅X is also normally distributed.

• Mean of the sampling distribution: $ext{E}(ar{X}) = ext{Population Mean} = bc$

• Standard deviation of the sampling distribution (Standard Error): $ext{SE}(ar{X}) = rac{c}{sqrt{n}}$

Sampling Distribution for Any Population

Central Limit Theorem (CLT)

When the population distribution is not normal (not bell-shaped), the sampling distribution of the sample mean can still become approximately normal as the sample size increases. This phenomenon is explained by the Central Limit Theorem (CLT).

• Central Limit Theorem (CLT): For a random sample of size n from any population with mean μ and standard deviation σ, as n increases, the sampling distribution of the sample mean approaches a normal distribution.

• Implication: The mean of the sampling distribution remains μ, and the standard deviation is $rac{c}{sqrt{n}}$.

Visualizing the CLT

Regardless of the shape of the population distribution, the sampling distribution of the sample mean becomes more bell-shaped (normal) as the sample size n increases.

Population Distribution	Sampling Distribution of ̅X (n=2)	Sampling Distribution of ̅X (n=30)
Skewed	Skewed	Approximately Normal
Uniform	Uniform	Approximately Normal
Normal	Normal	Normal
Irregular	Irregular	Approximately Normal

Additional info: The table above summarizes how the sampling distribution of the mean becomes normal as sample size increases, regardless of the original population shape.

Example: Average Salary

Scenario Description

Peter works as a seasonal waiter. The salary for each shift follows a probability distribution with mean μ = €150 and standard deviation σ = €60. At the end of the season, Peter randomly selects pay stubs from 32 shifts and calculates the mean salary per shift.

• Population Mean: μ = €150

• Population Standard Deviation: σ = €60

• Sample Size: n = 32

Expected Value and Variability

• Expected Mean Salary per Shift: The sample mean is expected to fluctuate around the population mean, μ = €150.

• Standard Deviation of the Sampling Distribution: $ext{SE}(ar{X}) = rac{60}{sqrt{32}} = 10.6$

• Interpretation: From one season to the next, Peter's mean salary per shift will vary around €150, with variability described by a standard deviation of €10.6.

Effect of Sample Size on Standard Deviation

Relationship Between Sample Size and Variability

The standard deviation of the sample mean decreases as the sample size increases, making the sample mean a more precise estimate of the population mean.

• Formula: $ext{SE}(ar{X}) = rac{c}{sqrt{n}}$

• Implication: Larger samples yield smaller standard errors, so the sample mean is more likely to be close to the population mean.

Practical Implications of the Central Limit Theorem

Making Inferences

The CLT and the formula for the standard error of the sample mean have important implications for statistical inference:

• If the sampling distribution of the sample mean is approximately normal, then the sample mean falls within 2 standard deviations of the population mean with probability close to 0.95, and almost certainly within 3 standard deviations.

• For large n, the sampling distribution is approximately normal even if the population distribution is not. This allows us to make inferences about population means regardless of the population's shape.

• This is especially useful in practice, as the shape of the population distribution is often unknown or irregular.

Summary Table: Key Properties of the Sampling Distribution of the Sample Mean

Property	Formula	Description
Mean	$bc$	Equals the population mean
Standard Deviation (Standard Error)	$rac{c}{sqrt{n}}$	Population standard deviation divided by square root of sample size
Shape	Normal (for large n)	Approximately normal for large samples, regardless of population shape

Additional info: These properties are foundational for constructing confidence intervals and conducting hypothesis tests about population means.