Intro to Business Statistics: Hypothesis Testing TwoSample Proportion Test ZTest

What This Is

The Two-Sample Proportion Test (Z-Test) is used to compare the proportions of two independent groups. A retail chain wants to know if the proportion of customers who prefer online shopping is higher than 0.6. They collect data from two random samples of customers: one from their online store and one from their physical store. The Z-Test will help them determine if the difference in proportions is statistically significant.

Key Formulas & Symbols

• p̂1 = sample proportion 1, where p̂1 = (number of successes in sample 1) / (sample size 1)
• p̂2 = sample proportion 2, where p̂2 = (number of successes in sample 2) / (sample size 2)
• p̂ = pooled proportion, where p̂ = ((number of successes in sample 1) + (number of successes in sample 2)) / (total sample size)
• Z = (p̂1 - p̂2) / √(p̂(1-p̂)(1/n1 + 1/n2)) where p̂ = pooled proportion, n1 = sample size 1, n2 = sample size 2
• p-value = probability of observing the data (or more extreme) if H₀ is true
• α = significance level (default = 0.05)
• H₀ = null hypothesis (e.g., p̂1 = p̂2)
• H₁ = alternative hypothesis (e.g., p̂1 ≠ p̂2)

Step-by-Step Procedure

1. State hypotheses: Define the null and alternative hypotheses. For example, H₀: p̂1 = p̂2 and H₁: p̂1 ≠ p̂2.
2. Choose test: Select the Two-Sample Proportion Test (Z-Test) since the sample sizes are large and the population proportions are known.
3. Compute test statistic: Calculate the Z-statistic using the formula Z = (p̂1 - p̂2) / √(p̂(1-p̂)(1/n1 + 1/n2)).
4. Find p-value or critical value: Determine the p-value using a standard normal distribution table or calculator. Alternatively, find the critical value from a standard normal distribution table.
5. Compare to α: Compare the p-value to the significance level (α = 0.05). If p-value < α, reject H₀.
6. Conclude: If H₀ is rejected, conclude that the difference in proportions is statistically significant.

Common Mistakes

• 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. It does not represent the probability that H₀ is true.
• Mistake: Failing to check the assumptions of the Z-Test (large sample sizes and known population proportions).
• Correction: The Z-Test assumes large sample sizes (n1 ≥ 30 and n2 ≥ 30) and known population proportions. If these assumptions are not met, use alternative tests like the Chi-Square Test.
• Mistake: Using the Z-Test with small sample sizes or unknown population proportions.
• Correction: Use alternative tests like the Chi-Square Test or Fisher's Exact Test for small sample sizes or unknown population proportions.

Quick Practice Problems

1. A marketing firm wants to compare the proportions of customers who prefer product A and product B. They collect data from two random samples of customers: one for product A and one for product B. The sample proportions are p̂1 = 0.7 and p̂2 = 0.5. The sample sizes are n1 = 100 and n2 = 150. What is the p-value?

p-value = 0.012 (using a standard normal distribution table or calculator)

1. A quality control team wants to compare the proportions of defective products in two manufacturing lines. They collect data from two random samples of products: one from line A and one from line B. The sample proportions are p̂1 = 0.02 and p̂2 = 0.05. The sample sizes are n1 = 50 and n2 = 75. What is the Z-statistic?

Z = (0.02 - 0.05) / √(0.03(1-0.03)(1/50 + 1/75)) = -1.43

1. A sales team wants to compare the proportions of customers who purchase product X in two different regions. They collect data from two random samples of customers: one from region A and one from region B. The sample proportions are p̂1 = 0.8 and p̂2 = 0.6. The sample sizes are n1 = 200 and n2 = 300. What is the critical value for α = 0.05?

Critical value = 1.96 (using a standard normal distribution table)

Last-Minute Cram Sheet

1. Z-Test: used for comparing proportions of two independent groups.
2. p̂1 = sample proportion 1, p̂2 = sample proportion 2, p̂ = pooled proportion.
3. Z = (p̂1 - p̂2) / √(p̂(1-p̂)(1/n1 + 1/n2)).
4. p-value = probability of observing the data (or more extreme) if H₀ is true.
5. α = significance level (default = 0.05).
6. H₀ = null hypothesis (e.g., p̂1 = p̂2).
7. H₁ = alternative hypothesis (e.g., p̂1 ≠ p̂2).
8. Assumptions: large sample sizes (n1 ≥ 30 and n2 ≥ 30) and known population proportions.
9. Trap: 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.
10. Critical value: 1.96 for α = 0.05.