Section 8.1: Distribution of the Sample Mean

Introduction to Sampling Distributions

Sampling distributions are fundamental in inferential statistics, allowing us to understand how sample statistics (such as the sample mean) behave when repeatedly sampling from a population. This section focuses on the distribution of the sample mean, both for normal and non-normal populations, and introduces the Central Limit Theorem.

• Sample Mean (𝑛): The average value calculated from a sample drawn from a population.

• Sampling Distribution: The probability distribution of a given statistic based on a random sample.

• Central Limit Theorem (CLT): A key result describing the shape of the sampling distribution of the sample mean as the sample size increases.

Key Questions

• What is the sampling distribution of a statistic?

• What is the sampling distribution of the sample mean 𝑛?

• What are the three ways for determining the sampling distribution of the sample mean?

Objective 1: Distribution of the Sample Mean - Normal Population

Shape and Properties of the Sampling Distribution

When the population is normally distributed, the sampling distribution of the sample mean is also normal, regardless of sample size.

• Shape: The distribution of the sample mean is normal if the population is normal.

• Mean of the Sampling Distribution: The mean of the sampling distribution of the sample mean equals the population mean ().

• Standard Deviation (Standard Error): The standard deviation of the sampling distribution of the sample mean is called the standard error and is given by , where is the population standard deviation and is the sample size.

Formulas

• Mean of the sampling distribution:

• Standard deviation (standard error):

Effects of Sample Size

• As sample size increases, the standard error decreases, making the distribution of the sample mean narrower and more concentrated around the population mean.

Example: Finding Probabilities of a Sample Mean

Suppose IQ scores are normally distributed with mean and standard deviation . For a sample of , the probability that the sample mean exceeds 110 is:

• Calculate the standard error:

• Find using the normal distribution.

Objective 2: Distribution of the Sample Mean - Non-normal Population

Central Limit Theorem and Non-normal Populations

When the population is not normal, the sampling distribution of the sample mean approaches normality as the sample size increases, due to the Central Limit Theorem.

• Central Limit Theorem (CLT): For a sufficiently large sample size ( is a common rule of thumb), the sampling distribution of the sample mean will be approximately normal, regardless of the population's distribution.

• Mean and Standard Error: The mean and standard error formulas remain the same: , .

• Shape: As increases, the distribution becomes more symmetric and bell-shaped.

Rule of Thumb for Normality

• If the population is highly skewed, a larger sample size may be needed for the sampling distribution to be approximately normal.

• For most distributions, is sufficient.

Example: Weight Gain during Pregnancy

The mean weight gain during pregnancy is 30 pounds, with a standard deviation of 12.5 pounds. For a sample of 15 underweight women, the sample mean is 36.2 pounds. The sampling distribution of the sample mean can be used to assess whether this observed mean is unusual.

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

Central Limit Theorem (CLT)

• Statement: If random samples of size are drawn from any population with mean and standard deviation , then as increases, the sampling distribution of the sample mean approaches a normal distribution with mean and standard deviation .

• Implication: Allows for the use of normal probability methods even when the population is not normal, provided the sample size is large.

Key Terms

• Sample Mean (): The average value in a sample.

• Population Mean (): The average value in the entire population.

• Standard Error (): The standard deviation of the sampling distribution of the sample mean.

• Central Limit Theorem: The theorem describing the normality of the sampling distribution as sample size increases.

Applications

• Estimating population parameters using sample statistics.

• Calculating probabilities for sample means.

• Assessing whether observed sample means are unusual.

Summary

• The sampling distribution of the sample mean is a cornerstone of inferential statistics.

• For normal populations, the sampling distribution is always normal.

• For non-normal populations, the Central Limit Theorem ensures approximate normality for large samples.

• Mean and standard error formulas are consistent across cases.