Intro to Business Statistics: Sampling and Sampling Distributions Sampling Methods Simple Random Systematic Stratified Cluster Convenience Quota

What This Is

Sampling methods are used to select a subset of data from a larger population to make inferences about the population. A retail chain wants to know if average daily sales exceed $10,000 at its new store location. To answer this question, the chain can use sampling methods to collect data from a representative sample of customers.

Key Formulas & Symbols

• Simple Random Sampling (SRS): n = N / f where n = sample size, N = population size, f = sampling fraction.
• Systematic Sampling: n = N / k where n = sample size, N = population size, k = sampling interval.
• Stratified Sampling: n = N / h where n = sample size, N = population size, h = number of strata.
• Cluster Sampling: n = N / c where n = sample size, N = population size, c = number of clusters.
• Convenience Sampling: No formula, as it's a non-probability sampling method.
• Quota Sampling: n = N / q where n = sample size, N = population size, q = number of quotas.
• Sample Size Formula: n = (Z^2 * σ^2) / E^2 where n = sample size, Z = Z-score, σ = population standard deviation, E = margin of error.

Step-by-Step Procedure

1. Define the problem: Identify the research question and the population of interest.
2. Choose the sampling method: Select the appropriate sampling method based on the research question and population characteristics.
3. Determine the sample size: Calculate the required sample size using the sample size formula or a predetermined sample size.
4. Select the sample: Use the chosen sampling method to select the sample from the population.
5. Collect data: Collect data from the selected sample.
6. Analyze data: Analyze the collected data to answer the research question.

Common Mistakes

• Mistake: Using Z when σ is unknown.
• Correction: Use t-distribution instead of Z when σ is unknown, and specify the degrees of freedom (df = n - 1).
• Mistake: Misinterpreting p-value as probability H₀ is true.
• Correction: The p-value is the probability of observing the data (or more extreme) if H₀ is true, not the probability that H₀ is true.
• Mistake: Not accounting for clustering in cluster sampling.
• Correction: Use the cluster sampling formula to calculate the sample size, and account for clustering in the analysis.

Quick Practice Problems

1. A marketing firm wants to estimate the average monthly expenditure on advertising for a new product. They use a simple random sample of 100 customers. If the population standard deviation is $500, and the desired margin of error is $100, what is the estimated average monthly expenditure?

Answer: $4,900. The sample size was calculated using the sample size formula, and the estimated average monthly expenditure was calculated using the sample mean.

1. A quality control team wants to monitor the average weight of a new product. They use a systematic sample of 50 products, with a sampling interval of 5. If the population standard deviation is 0.5 kg, and the desired margin of error is 0.1 kg, what is the estimated average weight?

Answer: 2.5 kg. The sample size was calculated using the sample size formula, and the estimated average weight was calculated using the sample mean.

1. A retail chain wants to estimate the average daily sales at its new store location. They use a stratified sample of 100 customers, with 20 customers in each stratum. If the population standard deviation is $1,000, and the desired margin of error is $200, what is the estimated average daily sales?

Answer: $10,000. The sample size was calculated using the sample size formula, and the estimated average daily sales was calculated using the sample mean.

Last-Minute Cram Sheet

• ⚠️ p-value is NOT the probability that H₀ is true – it’s the probability of observing the data (or more extreme) if H₀ is true.
• Use Z when σ is known, and t-distribution when σ is unknown.
• Specify the degrees of freedom (df = n - 1) when using t-distribution.
• Use cluster sampling formula to calculate sample size.
• Account for clustering in analysis.
• Use stratified sampling when population is heterogeneous.
• Use systematic sampling when population is homogeneous.
• Use convenience sampling when time or resources are limited.
• Use quota sampling when population is difficult to access.
• ⚠️ Convenience sampling is a non-probability sampling method.
• ⚠️ Quota sampling is a non-probability sampling method.