Chapter 8: Sampling Distributions

8.1 Distribution of the Sample Mean

Sampling distributions describe the probability distributions of statistics calculated from random samples. This section focuses on the distribution of the sample mean, both for normal and non-normal populations.

Definition: Sampling Distribution of the Sample Mean

• Sampling Distribution of the Sample Mean: The probability distribution of all possible values of the sample mean, \( \overline{x} \), computed from samples of size n drawn from a population.

• The mean of the sampling distribution of the sample mean is equal to the population mean \( \mu \).

• The standard deviation of the sampling distribution of the sample mean is called the standard error:

• Where \( \sigma \) is the population standard deviation and n is the sample size.

Describing the Distribution of the Sample Mean: Normal Population

• If the population is normally distributed, the sampling distribution of the sample mean is also normal for any sample size n.

• For a normal population with mean \( \mu \) and standard deviation \( \sigma \):

• Mean of \( \overline{x} \): \( \mu_{\overline{x}} = \mu \)

• Standard deviation of \( \overline{x} \): \( \sigma_{\overline{x}} = \frac{\sigma}{\sqrt{n}} \)

Describing the Distribution of the Sample Mean: Non-Normal Population

• If the population is not normal, the sampling distribution of the sample mean becomes approximately normal as the sample size increases, due to the Central Limit Theorem.

• For large enough n (commonly n \geq 30), the distribution of \( \overline{x} \) is approximately normal, regardless of the population's shape.

The Central Limit Theorem (CLT)

• The CLT states that for a random variable X with mean \( \mu \) and standard deviation \( \sigma \), the sampling distribution of \( \overline{x} \) approaches a normal distribution as n increases.

Central Limit Theorem: If samples of size n are drawn from any population with mean \( \mu \) and standard deviation \( \sigma \), then as n increases, the distribution of the sample mean \( \overline{x} \) approaches a normal distribution with mean \( \mu \) and standard deviation \( \frac{\sigma}{\sqrt{n}} \).

Example: Using the Central Limit Theorem

• Suppose the distance traveled (in miles) before a tire wears out is normally distributed with mean 36,500 and standard deviation 5,000. For a sample of size 50, the sampling distribution of the sample mean is normal with:

• Mean:

• Standard deviation:

Probability Calculations with the Sample Mean

• To find probabilities involving the sample mean, standardize using the z-score:

• Use the standard normal table to find probabilities.

Summary Table: Shape, Center, and Spread of the Sampling Distribution of \( \overline{x} \)

8.2 Distribution of the Sample Proportion

The sample proportion is another important statistic. This section describes its sampling distribution and how to compute probabilities involving sample proportions.

Definition: Sampling Distribution of the Sample Proportion

• For a population proportion p, the sample proportion \( \hat{p} \) is the proportion of successes in a random sample of size n.

• The sampling distribution of \( \hat{p} \) is approximately normal if:

• \( np \geq 10 \) and \( n(1-p) \geq 10 \)

• Mean of \( \hat{p} \): \( \mu_{\hat{p}} = p \)

• Standard deviation of \( \hat{p} \) (standard error):

Probability Calculations with the Sample Proportion

• To find probabilities involving the sample proportion, standardize using the z-score:

• Use the standard normal table to find probabilities.

Example: Describing the Sampling Distribution of a Sample Proportion

• Suppose 24% of young adults in the U.S. are financially independent. For a sample of 200, the sampling distribution of \( \hat{p} \) is approximately normal because:

• \( np = 200 \times 0.24 = 48 \geq 10 \) and \( n(1-p) = 200 \times 0.76 = 152 \geq 10 \)

• Mean:

• Standard deviation:

Summary Table: Sampling Distribution of \( \hat{p} \)

Key Points

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

• The standard deviation (standard error) decreases as sample size increases.

• The Central Limit Theorem allows us to use normal probability methods for sample means and proportions when sample sizes are large.

Additional info:

• These notes include expanded definitions, formulas, and examples for clarity and completeness.