Intro to Business Statistics: Sampling and Sampling Distributions Finite Population Correction FPC Factor

What This Is

The Finite Population Correction (FPC) Factor is a crucial concept in survey sampling, particularly when the sample size is a significant portion of the population. A retail chain wants to know if average daily sales exceed $10,000 in a small town with a population of 5,000. They collect a sample of 1,000 customers and find the sample mean daily sales to be $9,500. To make an accurate inference about the population, they need to apply the FPC Factor.

Key Formulas & Symbols

• FPC Factor = (N - n) / (N - 1) where N = population size, n = sample size.
• Standard Error (SE) = σ / √n where σ = population standard deviation, n = sample size.
• SE (FPC) = SE * √[(N - n) / (N - 1)] where SE = standard error, N = population size, n = sample size.
• Z = (x̄ - μ) / SE (FPC) where x̄ = sample mean, μ = population mean.
• t = (x̄ - μ) / SE (FPC) where x̄ = sample mean, μ = population mean, t = test statistic.
• p-value = P(t > |t|) where t = test statistic, p-value = probability of observing the data (or more extreme).
• Critical Value (t) = tα/2, df = N - n where t = critical value, α = significance level, df = degrees of freedom.

Step-by-Step Procedure

1. State hypotheses: H₀: μ = μ₀ (null hypothesis), H₁: μ ≠ μ₀ (alternative hypothesis).
2. Choose test: Select the appropriate test (t-test) and determine the significance level (α = 0.05).
3. Compute test statistic: Calculate the FPC Factor, standard error, and test statistic (t).
4. Find p-value or critical value: Determine the p-value or critical value (t) using a t-distribution table or calculator.
5. Compare to α: Compare the p-value or critical value to the significance level (α).
6. Conclude: Make a decision based on the comparison (reject H₀ or fail to reject H₀).

Common Mistakes

• Mistake: Failing to apply the FPC Factor when the sample size is a significant portion of the population.
• Correction: Always apply the FPC Factor when the sample size is greater than 5% of the population size to ensure accurate inferences.
• Mistake: Misinterpreting the p-value as the probability that 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: Using the wrong critical value (t) or degrees of freedom (df) for the t-test.
• Correction: Always use the correct critical value (t) and degrees of freedom (df) for the t-test, which depend on the sample size and population size.

Quick Practice Problems

1. A marketing firm wants to know if the average customer satisfaction rating exceeds 4.5 on a scale of 1 to 5. They collect a sample of 200 customers and find the sample mean rating to be 4.8. The population size is 1,000. Calculate the confidence interval.

Answer: (4.73, 4.87) Explanation: The FPC Factor is applied to calculate the standard error, and then the confidence interval is constructed using the sample mean and standard error.

1. A quality control team wants to know if the average defect rate exceeds 2% in a production line. They collect a sample of 500 units and find the sample mean defect rate to be 2.5%. The population size is 10,000. What is the p-value?

Answer: 0.001 Explanation: The FPC Factor is applied to calculate the standard error, and then the t-statistic is calculated and used to find the p-value.

1. A sales manager wants to know if the average sales revenue exceeds $50,000 in a region. They collect a sample of 300 customers and find the sample mean revenue to be $55,000. The population size is 5,000. Calculate the t-statistic.

Answer: 2.33 Explanation: The FPC Factor is applied to calculate the standard error, and then the t-statistic is calculated using the sample mean and standard error.

Last-Minute Cram Sheet

1. FPC Factor = (N - n) / (N - 1): Apply when sample size is greater than 5% of population size.
2. SE (FPC) = SE * √[(N - n) / (N - 1)]: Use to calculate standard error with FPC.
3. Z = (x̄ - μ) / SE (FPC): Use for large sample sizes (n > 30).
4. t = (x̄ - μ) / SE (FPC): Use for small sample sizes (n ≤ 30).
5. p-value = P(t > |t|): Probability of observing data (or more extreme) if H₀ is true.
6. Critical Value (t) = tα/2, df = N - n: Use to determine significance.
7. ⚠️ 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.
8. ⚠️ Always apply the FPC Factor when sample size is greater than 5% of population size.
9. ⚠️ Use correct critical value (t) and degrees of freedom (df) for t-test.
10. ⚠️ Misinterpretation of p-value can lead to incorrect conclusions.