Sampling Distributions – Statistics for Business

Sampling Distributions

Introduction to Sampling Distributions

Sampling distributions are a foundational concept in statistics, especially for business applications. They describe the probability distribution of a given sample statistic, such as the mean, for all possible samples of a specific size drawn from a population.

• Definition: A sampling distribution is the distribution of all possible values of a sample statistic for a given sample size selected from a population.

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

Developing a Sampling Distribution

Population and Sample Setup

To understand sampling distributions, consider a small population and all possible samples of a given size.

• Population size: N = 4

• Variable of interest: X (e.g., age of individuals)

• Values of X: 18, 20, 22, 24 (years)

Population Distribution and Summary Measures

First, calculate the mean and variance of the population.

• Population Mean: $\mu = \frac{18 + 20 + 22 + 24}{4} = 21$

• Population Variance: $\sigma^2 = \frac{(18-21)^2 + (20-21)^2 + (22-21)^2 + (24-21)^2}{4} = 2.236$

The population distribution is uniform, meaning each value is equally likely.

All Possible Samples of Size n=2

When sampling with replacement, list all possible pairs:

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

This results in 16 possible samples.

Sampling Distribution of the Sample Means

Calculate the mean for each sample and construct the distribution of these means.

Sample	Sample Mean
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

The resulting sampling distribution of sample means is no longer uniform.

Summary Measures of the Sampling Distribution

• Mean of Sampling Distribution: $\mu_{\bar{x}} = \frac{18 + 19 + 19 + \cdots + 24}{16} = 21$

• Standard Deviation of Sampling Distribution: $\sigma_{\bar{x}} = \sqrt{\frac{(18-21)^2 + (19-21)^2 + \cdots + (24-21)^2}{16}} = 1.5$

Note: The mean of the sampling distribution equals the population mean, making the sample mean an unbiased estimator of the population mean.

Properties of Sampling Distributions

Unbiased Estimator

• The sample mean $\bar{x}$ is an unbiased estimator of the population mean $\mu$.

Z-value for Sampling Distribution of the Mean

The Z-value standardizes the sample mean for probability calculations:

• Formula: $Z = \frac{\bar{X} - \mu}{\sigma / \sqrt{n}}$ where $\bar{X}$ = sample mean, $\mu$ = population mean, $\sigma$ = population standard deviation, $n$ = sample size.

Interval Estimation for Sample Means

To find an interval containing a fixed proportion (e.g., 95%) of sample means:

• Use the standard normal table to find Z-scores for the desired confidence level (e.g., ±1.96 for 95%).

• Lower limit: $\mu + (Z_{lower}) \frac{\sigma}{\sqrt{n}}$

• Upper limit: $\mu + (Z_{upper}) \frac{\sigma}{\sqrt{n}}$

Central Limit Theorem (CLT)

Statement and Importance

The Central Limit Theorem is a key result in statistics, stating that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the population's shape.

• For most distributions, $n > 30$ is considered large enough for the sampling distribution to be nearly normal.

• For symmetric distributions, $n > 15$ may suffice.

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

Example Application

• Population mean: $\mu = 8$ minutes

• Population standard deviation: $\sigma = 3$ minutes

• Sample size: $n = 36$

• Standard error: $\frac{3}{\sqrt{36}} = 0.5$ minutes

• Probability calculation: Use Z-scores to find the probability that the sample mean falls within a specified interval.

Example: Probability that the sample mean is between 7.8 and 8.2 minutes:

• Calculate Z-scores for 7.8 and 8.2:

• $Z_{7.8} = \frac{7.8 - 8}{0.5} = -0.4$

• $Z_{8.2} = \frac{8.2 - 8}{0.5} = 0.4$

• Find probabilities from the standard normal table and subtract to get the interval probability.

Summary Table: Key Formulas

Concept	Formula (LaTeX)
Population Mean	$\mu = \frac{\sum X_i}{N}$
Population Variance	$\sigma^2 = \frac{\sum (X_i - \mu)^2}{N}$
Sample Mean	$\bar{X} = \frac{\sum x_i}{n}$
Standard Error	$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$
Z-value	$Z = \frac{\bar{X} - \mu}{\sigma / \sqrt{n}}$

Additional info: These notes expand on the provided slides and text to ensure completeness and clarity for exam preparation in a Statistics for Business course.