Intro to Business Statistics: Estimation Confidence Intervals for Population Proportion p

What This Is

Confidence intervals for population proportion (p̂) are used in business to estimate the true proportion of a population that possesses a certain characteristic. For example, a marketing firm wants to know the true proportion of customers who prefer a new product feature. By constructing a confidence interval, the firm can estimate this proportion with a certain level of confidence, say 95%.

Key Formulas & Symbols

• p̂ = x̄ / n where x̄ = sample proportion, n = sample size.
• p̂ = (number of successes) / (sample size).
• SE = √(p̂(1-p̂)/n) where SE = standard error.
• CI = p̂ ± (Z * SE) where CI = confidence interval, Z = Z-score.
• Z = (p̂ - p) / SE where p = population proportion.
• p̂ = (number of successes) / (sample size).
• SE = √(p̂(1-p̂)/n) where SE = standard error.
• df = n - 1 where df = degrees of freedom.

Step-by-Step Procedure

1. State hypotheses: H₀: p = p₀ (null hypothesis) and H₁: p ≠ p₀ (alternative hypothesis).
2. Choose test: Select the appropriate test (e.g., Z-test for large samples, t-test for small samples).
3. Compute test statistic: Calculate the Z-score or t-statistic using the formula Z = (p̂ - p) / SE or t = (p̂ - p) / SE.
4. Find p-value or critical value: Determine the p-value or critical value using a standard normal distribution (Z-table) or t-distribution table.
5. Compare to α: Compare the p-value or critical value to the significance level α (default = 0.05).
6. Conclude: If p-value < α or critical value > Z-score, reject H₀; otherwise, fail to reject H₀.

Common Mistakes

• Mistake: Using Z when σ is unknown.
• Correction: Use t-test instead, which is more robust for small samples.
• 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: Failing to check assumptions (e.g., large sample size, normal distribution).
• Correction: Verify that the sample size is sufficiently large and the data follow a normal distribution.

Quick Practice Problems

1. A marketing firm conducts a survey of 100 customers and finds that 60 prefer a new product feature. Calculate the 95% confidence interval for the true proportion of customers who prefer the feature.

Answer: 0.55 ≤ p ≤ 0.65 (Explanation: p̂ = 0.6, SE = √(0.6(1-0.6)/100) = 0.05, Z = 1.96, CI = 0.6 ± (1.96 * 0.05) = 0.55 ≤ p ≤ 0.65)

1. A retail chain wants to know if the true proportion of customers who prefer online shopping is greater than 0.7. Conduct a hypothesis test using a sample of 50 customers, where 35 prefer online shopping.

Answer: p-value = 0.02 (Explanation: p̂ = 0.7, SE = √(0.7(1-0.7)/50) = 0.07, Z = (0.7 - 0.7) / 0.07 = 0, p-value = 0.5)

1. A quality control team wants to estimate the true proportion of defective products in a manufacturing process. Conduct a confidence interval using a sample of 200 products, where 15 are defective.

Answer: 0.06 ≤ p ≤ 0.14 (Explanation: p̂ = 0.075, SE = √(0.075(1-0.075)/200) = 0.02, Z = 1.96, CI = 0.075 ± (1.96 * 0.02) = 0.06 ≤ p ≤ 0.14)

Last-Minute Cram Sheet

1. Z-test: Use for large samples (n ≥ 30) and known population proportion.
2. t-test: Use for small samples (n < 30) and unknown population proportion.
3. p̂ = (number of successes) / (sample size).
4. SE = √(p̂(1-p̂)/n).
5. CI = p̂ ± (Z * SE).
6. Z = (p̂ - p) / SE.
7. df = n - 1.
8. 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 ⚠️
9. Use t-test instead of Z-test when σ is unknown ⚠️
10. Verify assumptions (e.g., large sample size, normal distribution) ⚠️