Sampling and Sampling Distributions Notes

1. Fundamental Definitions in Sampling

• Element: The specific entity on which data are collected during a study.

• Population: The complete collection of all the elements that are of interest to the researcher.

• Sample: A specific subset of the population chosen for data collection.

• Sampled Population: The specific population from which the sample is actually drawn.

• Frame: A comprehensive list of the elements that the sample will be selected from.

• Purpose of Sampling: The primary reason to select a sample is to collect data to answer research questions about a population. Sample results provide estimates of population characteristics because the sample only contains a portion of the population. With proper methods, samples provide "good" estimates.

Selecting a Sample from a Finite Population

• Finite Populations: These are populations often defined by existing lists, such as:

  • Organization membership rosters.

  • Credit card account numbers.

  • Inventory product numbers.

• Simple Random Sample (Finite): A sample of size $n$ from a finite population of size $N$ is selected such that every possible sample of size $n$ has the same probability of being selected.

• Automation: In large sampling projects, computer-generated random numbers are used to automate selection.

• Example: National Baseball League Teams:

  • Population ($N$): there were $15$ teams in the $2019$ National Baseball League.

  • Objective: Select a simple random sample of $n = 5$ teams for in-depth interviews regarding minor league franchise management.

  • Step 1: Assign a random number to each of the $15$ teams. Using Excel’s RAND function generates numbers following a uniform probability distribution between $0$ and $1$.

  • Step 2: Select the $5$ teams corresponding to the $5$ smallest random numbers as the sample.

• Excel Sorting Procedure:

  • Select a cell in range B2:B16.

  • Navigate to the Home tab on the Ribbon.

  • In the Editing group, click Sort & Filter.

  • Choose Sort smallest to largest to identify the sample elements.

Selecting a Sample from an Infinite Population

• Constraints: Sometimes it is impossible to obtain a list of all elements, meaning a frame cannot be constructed. This typically occurring in infinite population cases.

• Ongoing Processes: Infinite populations are often generated by processes with no upper limit on units produced. Examples include:

  • Parts being manufactured on a production line.

  • Transactions occurring at a bank.

  • Telephone calls arriving at a technical help desk.

  • Customers entering a retail store.

• Random Sample (Infinite): To make valid inferences, a random sample must satisfy two conditions:

  • Each element selected comes from the population of interest.

  • Each element is selected independently.

Point Estimation

• Definition: Point estimation is a form of statistical inference where sample data is used to compute a value of a sample statistic that serves as an estimate of a population parameter.

• Point Estimators:

  • $\bar{x}$ is the point estimator of the population mean $\mu$.

  • $s$ is the point estimator of the population standard deviation $\sigma$.

  • $\bar{p}$ is the point estimator of the population proportion $p$.

• Example: EAI Employee Data:

  • Total employees ($N$): $2,500$.

  • Simple random sample size ($n$): $30$.

  • Data collected: Annual salary ($x_1, x_2, \dots, x_n$) and participation in a Management Training Program (MTP) [Yes/No].

  • Unbiasedness: When the expected value of the point estimator equals the population parameter, the estimator is considered unbiased.

• Summary of EAI Point Estimates:

  • Population Mean ($\mu$): $\$71,800$ | Point Estimate ($\bar{x}$): $\$71,814$

  • Population Standard Deviation ($\sigma$): $\$4,000$ | Point Estimate ($s$): $\$3,348$

  • Population Proportion ($p$): $0.60$ | Point Estimate ($\bar{p}$): $0.63$

• Important Note: Using different random numbers would identify a different sample, resulting in different point estimates.

Sampling Distribution of $\bar{x}$

• Definition: The probability distribution of all possible values of the sample mean $\bar{x}$.

• Expected Value: $E(\bar{x}) = \mu$, where $\mu$ is the population mean.

• Standard Deviation (Standard Error of the Mean):

  • For a Finite Population: $\sigma_{\bar{x}} = \sqrt{\frac{N-n}{N-1}} \left( \frac{\sigma}{\sqrt{n}} \right)$.

  • For an Infinite Population: $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$.

  • Finite Population Correction Factor: $\sqrt{\frac{N-n}{N-1}}$. This is used unless the sample size is small relative to the population ($n/N \le 0.05$).

• Shape of the Distribution:

  • If the population has a normal distribution, the sampling distribution of $\bar{x}$ is normal for any sample size.

  • If the population is not normal, the distribution of $\bar{x}$ can be approximated by a normal distribution for $n \ge 30$.

  • If the population is highly skewed or contains outliers, a sample size of $n=50$ may be required.

Central Limit Theorem (CLT)

• Definition: In selecting random samples of size $n$ from a population, the sampling distribution of the sample mean $\bar{x}$ can be approximated by a normal distribution as the sample size becomes large, regardless of the shape of the original population distribution.

Probability Calculations for $\bar{x}$ (EAI Example)

• Problem: What is the probability that a sample mean of $n=30$ is within $\$500$ of the population mean (Note: $\sigma = \$4,000$, $\mu = \$71,800$)?

• Standard Error Calculation: $\sigma_{\bar{x}} = \frac{4,000}{\sqrt{30}} \approx 730.30$.

• Step 1 (Upper Endpoint): $z = \frac{72,300 - 71,800}{730.30} = 0.68$. Cumulative probability for $z=0.68$ is $0.7517$.

• Step 2 (Lower Endpoint): $z = \frac{71,300 - 71,800}{730.30} = -0.68$. Cumulative probability for $z=-0.68$ is $0.2483$.

• Step 3 (Interval Probability): $P(71,300 \le \bar{x} \le 72,300) = 0.7517 - 0.2483 = 0.5034$.

• Excel Implementation: Using =NORM.DIST(value, mean, standard_error, TRUE) provides more accurate results than rounded tables.

  • Upper: =NORM.DIST(72300, 71800, 730.30, TRUE) yields $0.7532$.

  • Lower: =NORM.DIST(71300, 71800, 730.30, TRUE) yields $0.2468$.

  • Total Probability: $0.7532 - 0.2468 = 0.5064$.

• Effect of Sample Size: If sample size increases to $n=100$, the standard error decreases from $730.30$ to $400.00$ ($\frac{4,000}{\sqrt{100}}$). Smaller standard error means values of $\bar{x}$ have less variability and stay closer to $\mu$.

Sampling Distribution of $\bar{p}$

• Definition: The probability distribution of all possible values of the sample proportion $\bar{p}$.

• Expected Value: $E(\bar{p}) = p$, where $p$ is the population proportion.

• Standard Deviation (Standard Error of the Proportion):

  • For a Finite Population: $\sigma_{\bar{p}} = \sqrt{\frac{N-n}{N-1}} \sqrt{\frac{p(1-p)}{n}}$.

  • For an Infinite Population: $\sigma_{\bar{p}} = \sqrt{\frac{p(1-p)}{n}}$.

• Normal Approximation Condition: The sampling distribution of $\bar{p}$ can be approximated by a normal distribution if:

  • $np \ge 5$

  • $n(1-p) \ge 5$

• EAI Example (Proportions):

  • $p = 0.60$, $n = 30$.

  • Check conditions: $30 \times 0.60 = 18 \ge 5$ and $30 \times 0.40 = 12 \ge 5$. Normal approximation is valid.

  • Standard error: $\sigma_{\bar{p}} = \sqrt{\frac{0.60 \times 0.40}{30}} = 0.0894$.

  • Probability within $\pm 0.05$ of $p$: Calculated using NORM.DIST with endpoints $0.55$ and $0.65$. Cumulative probability at $0.55$ is $0.2880$. Interval probability is $2 \times (0.5000 - 0.2880) = 0.424$.

Other Sampling Methods

• Stratified Random Sampling:

  • The population is divided into groups called strata.

  • Elements within strata should be as homogeneous (alike) as possible.

  • A simple random sample is taken from each stratum.

  • Advantage: Can be more precise than simple random sampling with a smaller total sample size.

  • Examples: Age, department, industry type.

• Cluster Sampling:

  • The population is divided into separate groups called clusters.

  • Ideally, each cluster is a miniature, heterogeneous (diverse) version of the population.

  • A simple random sample of clusters is taken, and every element within the chosen clusters is sampled.

  • Advantage: Cost-effective for area sampling (e.g., city blocks).

  • Disadvantage: Usually requires a larger total sample size than simple or stratified methods.

• Systematic Sampling:

  • For sample size $n$ and population $N$, select one element for every $k = N/n$ elements.

  • Randomly select one of the first $k$ elements, then pick every $k$-th element thereafter.

  • Advantage: Easier to identify than a simple random sample.

• Convenience Sampling:

  • Non-probability technique where items are included based on ease of access.

  • Advantage: Easy data collection.

  • Disadvantage: Impossible to determine representativeness.

• Judgment Sampling:

  • Non-probability technique where a knowledgeable person selects elements they feel represent the population.

  • Advantage: Easy to select.

  • Disadvantage: Results depend entirely on the quality of the selector's judgment.

Errors in Sampling

• Sampling Error: The inherent difference between the value of a sample statistic (e.g., $\bar{x}$) and the corresponding population parameter (e.g., $\mu$).

• Nonsampling Errors: Deviations from the population occurring for reasons other than random sampling. These can occur in both samples and censuses.

  • Coverage Error: Incorrectly defining the population or frame.

  • Non-response Error: Failure to obtain data from selected elements.

  • Interviewer Error: Errors introduced by the person conducting the interview.

  • Processing Error: Data entry or coding mistakes.

  • Measurement Error: Inaccuracy in the measurement tool or response.

• Minimizing Nonsampling Errors:

  • Carefully define the target population.

  • Train data collectors properly.

  • Pretest data collection procedures.

  • Use stratified sampling for qualitative characteristics or systematic sampling for quantitative characteristics.