Sampling Distributions: Concepts and Applications in Business Statistics

Chapter 7: Sampling Distributions

Objectives

This chapter introduces the concept of sampling distributions, their properties, and their importance in statistical inference for business applications. Students will learn to:

• Understand what a sampling distribution is.

• Calculate probabilities related to the sample mean and the sample proportion.

• Recognize the importance of the Central Limit Theorem (CLT).

Sampling Distributions

Definition and Importance

A sampling distribution is the probability distribution of a given statistic (such as the mean or proportion) based on a random sample of a specific size drawn from a population. It is fundamental in inferential statistics, as it allows us to estimate population parameters and assess the variability of sample statistics.

• Sample Statistic: A numerical value calculated from a sample (e.g., sample mean \( \bar{X} \)).

• Population Parameter: A numerical value that describes the entire population (e.g., population mean \( \mu \)).

• Example: If you repeatedly sample 50 students from a college and calculate the mean GPA for each sample, the distribution of all these sample means forms the sampling distribution of the mean.

Developing a Sampling Distribution

Illustrative Example

Suppose we have a small population with the following characteristics:

• Population size: N = 4

• Variable of interest: Age (X)

• Values: 18, 20, 22, 24 years

Population Summary Measures

• Population Mean: $\mu = \frac{\sum_{i=1}^{N} X_i}{N} = \frac{18 + 20 + 22 + 24}{4} = 21$

• Population Variance: $\sigma^2 = \frac{\sum_{i=1}^{N} (X_i - \mu)^2}{N} = 2.236$

Population Distribution Table

Value (X)	Probability P(X)
18	0.25
20	0.25
22	0.25
24	0.25

This is a uniform distribution since each value is equally likely.

All Possible Samples of Size n = 2 (with Replacement)

When sampling with replacement, each sample of size 2 can be any combination of the four values. There are 16 possible samples:

1st Observation	2nd Observation
18	18
18	20
18	22
18	24
20	18
20	20
20	22
20	24
22	18
22	20
22	22
22	24
24	18
24	20
24	22
24	24

Sample Means for Each Sample

Sample	Sample Mean (\( \bar{X} \))
18, 18	18
18, 20	19
18, 22	20
18, 24	21
20, 18	19
20, 20	20
20, 22	21
20, 24	22
22, 18	20
22, 20	21
22, 22	22
22, 24	23
24, 18	21
24, 20	22
24, 22	23
24, 24	24

Distribution of Sample Means

The sample means are not uniformly distributed, even though the original population is. This illustrates how the sampling distribution of the mean can differ in shape from the population distribution.

Summary Measures of the Sampling Distribution

• Mean of the Sampling Distribution: $\mu_{\bar{X}} = \mu = 21$

• Variance of the Sampling Distribution: $\sigma^2_{\bar{X}} = \frac{\sigma^2}{n}$ For this example, $n = 2$ and $\sigma^2 = 2.236$.

Properties of the Sample Mean Sampling Distribution

Standard Error of the Mean

The standard error of the mean measures the variability of sample means around the population mean. It is calculated as:

• $\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}}$

• As sample size n increases, the standard error decreases, making sample means more tightly clustered around the population mean.

If the Population is Normal

• If the population is normally distributed with mean $\mu$ and standard deviation $\sigma$, then the sampling distribution of the sample mean is also normal, regardless of sample size.

• Mean: $\mu_{\bar{X}} = \mu$

• Standard Error: $\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}}$

Z-Value for the Sampling Distribution of the Mean

To standardize a sample mean and find probabilities, use:

• $Z = \frac{\bar{X} - \mu}{\sigma / \sqrt{n}}$

• This converts the sample mean to a standard normal variable.

Effect of Sample Size

• Larger sample sizes result in smaller standard errors, making the sampling distribution narrower.

• Smaller sample sizes result in greater variability among sample means.

Determining an Interval for a Fixed Proportion of Sample Means

To find an interval that contains a certain percentage (e.g., 95%) of sample means:

• Use the formula: $\bar{X} = \mu \pm Z \frac{\sigma}{\sqrt{n}}$

• For 95%, $Z = 1.96$ (from the standard normal table).

• Example: If $\mu = 368$, $\sigma = 15$, $n = 25$:

  • Lower limit: $368 + (-1.96) \times \frac{15}{5} = 362.12$

  • Upper limit: $368 + (1.96) \times \frac{15}{5} = 373.88$

• 95% of sample means will fall between 362.12 and 373.88.

Central Limit Theorem (CLT)

Statement and Importance

The Central Limit Theorem states that, for a sufficiently large sample size, the sampling distribution of the sample mean will be approximately normal, regardless of the shape of the population distribution.

• For most distributions, n > 30 is considered large enough.

• For symmetric distributions, n > 15 may suffice.

• If the population is normal, the sampling distribution is always normal, regardless of sample size.

Properties of the Sampling Distribution (Non-Normal Population)

• Mean: $\mu_{\bar{X}} = \mu$

• Standard Error: $\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}}$

• As n increases, the distribution becomes more normal and the standard error decreases.

Sampling Distribution of the Sample Proportion

Definition and Properties

The sample proportion (\( p \)) estimates the proportion of the population (\( \pi \)) with a certain characteristic. For large samples, the sampling distribution of p is approximately normal.

• Mean: $\mu_p = \pi$

• Standard Error: $\sigma_p = \sqrt{\frac{\pi (1 - \pi)}{n}}$

• Normal approximation is valid if $n\pi \geq 5$ and $n(1-\pi) \geq 5$.

Z-Value for Proportions

• Standardize the sample proportion using:

• $Z = \frac{p - \pi}{\sqrt{\frac{\pi (1 - \pi)}{n}}}$

Example: Probability for a Sample Proportion

• If the true proportion of voters supporting a proposition is $\pi = 0.4$ and $n = 200$, what is $P(0.40 < p < 0.45)$?

• Calculate standard error: $\sigma_p = \sqrt{\frac{0.4 \times 0.6}{200}} = 0.0346$

• Standardize endpoints:

  • $Z_1 = \frac{0.40 - 0.4}{0.0346} = 0$

  • $Z_2 = \frac{0.45 - 0.4}{0.0346} = 1.44$

• From the standard normal table: $P(0 < Z < 1.44) = 0.4251$

Chapter Summary

• Sampling distributions describe the variability of sample statistics.

• The standard error quantifies the spread of the sampling distribution.

• The Central Limit Theorem allows normal approximation for sample means and proportions, facilitating probability calculations.

• Understanding these concepts is essential for making inferences about populations from samples in business statistics.