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 the definition and role of sampling distributions.

• Calculate probabilities related to sample means and sample proportions.

• Appreciate the significance of the Central Limit Theorem.

Sampling Distributions

Definition and Importance

A sampling distribution is the probability distribution of a given sample statistic (such as the mean or proportion) computed from all possible samples of a specific size drawn from a population. This concept is fundamental in inferential statistics, as it allows us to estimate population parameters and assess the variability of sample statistics.

• Sample Statistic: A numerical summary (e.g., mean, proportion) calculated from a sample.

• Population Parameter: A numerical summary (e.g., mean, proportion) describing the entire population.

• Example: If we sample 50 students from a college and calculate their mean GPA, repeating this process many times yields a distribution of sample means.

Developing a Sampling Distribution

Population and Sample Construction

To illustrate sampling distributions, consider a small population:

• Population size: $N = 4$

• Variable of interest: $X$ (age of individuals)

• Values of $X$: 18, 20, 22, 24 (years)

Population Distribution Measures

• Population Mean ($ar{X}$): $\bar{X} = \frac{\sum_{i=1}^{N} X_i}{N} = \frac{18 + 20 + 22 + 24}{4} = 21$

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

• Distribution Shape: Uniform (each value equally likely)

Sampling All Possible Samples

Consider all possible samples of size $n = 2$ (with replacement):

1st Observation	2nd Observation	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

Additional info: The table above shows all 16 possible samples and their means.

Sampling Distribution of the Sample Mean

• The distribution of sample means is not uniform, even if the population distribution is uniform.

• Mean of the Sampling Distribution: $\mu_{\bar{X}} = \mu$ (the sample mean is an unbiased estimator of the population mean).

• Standard Error of the Mean: Measures the variability of sample means. $\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}}$

• As sample size $n$ increases, the standard error decreases.

Properties of Sampling Distributions

Unbiasedness and Variability

• Unbiased Estimator: The expected value of the sample mean equals the population mean.

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

• Smaller sample sizes yield greater variability in sample means.

• Larger sample sizes yield less variability.

Sampling Distribution When Population is Normal

Normal Population Case

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

• Mean: $\mu_{\bar{X}} = \mu$

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

Z-Value for the Sample Mean

• To standardize a sample mean, use: $Z = \frac{\bar{X} - \mu}{\sigma_{\bar{X}}} = \frac{\bar{X} - \mu}{\sigma / \sqrt{n}}$

• This allows calculation of probabilities using the standard normal distribution.

Central Limit Theorem (CLT)

Importance and Application

The Central Limit Theorem states that, regardless of the population's distribution, the sampling distribution of the sample mean approaches a normal distribution as the sample size increases (typically $n > 30$ is considered sufficient).

• For symmetric distributions, $n > 15$ may be sufficient.

• For normal populations, the sampling distribution is always normal, regardless of sample size.

• Mean: $\mu_{\bar{X}} = \mu$

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

Interval Estimation for Sample Means

Calculating Confidence Intervals

• To find an interval containing a fixed proportion (e.g., 95%) of sample means, use the standard normal table.

• For 95% confidence, the critical Z-values are $-1.96$ and $1.96$.

• Lower Limit: $X_L = \mu + Z_{lower} \cdot \sigma_{\bar{X}}$

• Upper Limit: $X_U = \mu + Z_{upper} \cdot \sigma_{\bar{X}}$

• Example: If $\mu = 368$, $\sigma = 15$, $n = 25$: $\sigma_{\bar{X}} = \frac{15}{\sqrt{25}} = 3$ $X_L = 368 + (-1.96) \times 3 = 362.12$ $X_U = 368 + (1.96) \times 3 = 373.88$ Thus, 95% of sample means fall between 362.12 and 373.88.

Sampling Distribution of Proportions

Population Proportion and Sample Proportion

• Population Proportion ($\pi$): The proportion of the population with a certain characteristic.

• Sample Proportion ($p$): The proportion in a sample with that characteristic; used to estimate $\pi$.

• For large $n$, the sampling distribution of $p$ is approximately normal.

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

Z-Value for Proportions

• To standardize a sample proportion: $Z = \frac{p - \pi}{\sigma_p} = \frac{p - \pi}{\sqrt{\frac{\pi (1 - \pi)}{n}}}$

Example: Probability for Sample Proportion

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

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

• Standardize: $Z_{0.40} = \frac{0.40 - 0.40}{0.0346} = 0$ $Z_{0.45} = \frac{0.45 - 0.40}{0.0346} = 1.44$

• From the standard normal table: $P(0 < Z < 1.44) = 0.9251 - 0.5000 = 0.4251$

Chapter Summary

• Sampling distributions are essential for making inferences about population parameters.

• The Central Limit Theorem allows the use of normal probability models for sample means and proportions, even when the population is not normal, provided the sample size is sufficiently large.

• Understanding standard error and unbiasedness is crucial for interpreting sample statistics.