Intro to Business Statistics: Hypothesis Testing OneSample ZTest for Proportion

What This Is

The one-sample Z-test for proportion is a statistical method used to determine if a sample proportion is significantly different from a known population proportion. A retail chain wants to know if the proportion of customers who prefer their new online shopping platform exceeds 0.7. They collect a random sample of 500 customers and find that 380 prefer the new platform. The retail chain wants to use a one-sample Z-test for proportion to determine if the sample proportion is significantly greater than 0.7.

Key Formulas & Symbols

• p̂ = (x / n) where x = number of successes, n = sample size.
• p = population proportion (known value).
• Z = (p̂ - p) / √(p(1-p)/n) where p̂ = sample proportion, p = population proportion, n = sample size.
• p-value = P(Z > |Z|) where Z = test statistic, |Z| = absolute value of Z.
• Decision Rule: Reject H₀ if p-value < α where α = 0.05 (default).
• Assumption: Sample size must be sufficiently large (np ≥ 10, n(1-p) ≥ 10).
• Critical Value: Zα/2 = ±1.96 for α = 0.05 (two-tailed test).

Step-by-Step Procedure

1. State hypotheses: H₀: p = 0.7 (null hypothesis), H₁: p > 0.7 (alternative hypothesis).
2. Choose test: One-sample Z-test for proportion.
3. Compute test statistic: Z = (p̂ - p) / √(p(1-p)/n) = (380/500 - 0.7) / √(0.7(1-0.7)/500) = 1.44.
4. Find p-value or critical value: p-value = P(Z > |1.44|) = 0.0749 (using a standard normal distribution table or calculator).
5. Compare to α: Since p-value (0.0749) > α (0.05), do not reject H₀.
6. Conclude: There is insufficient evidence to conclude that the proportion of customers who prefer the new online shopping platform exceeds 0.7.

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 provide information about the probability of H₀ being true.
• Mistake: Failing to check the assumption of sufficiently large sample size.
• Correction: Check that np ≥ 10 and n(1-p) ≥ 10 before conducting the test.
• Mistake: Using a Z-test when the population proportion is unknown.
• Correction: Use a one-sample t-test for proportion when the population proportion is unknown.

Quick Practice Problems

1. A company wants to know if the proportion of customers who prefer their new product is greater than 0.6. They collect a random sample of 800 customers and find that 520 prefer the new product. Calculate the p-value.

p-value = P(Z > |(520/800 - 0.6) / √(0.6(1-0.6)/800)|) = 0.0173

1. A marketing firm wants to know if the proportion of customers who prefer their new advertising campaign is less than 0.4. They collect a random sample of 600 customers and find that 240 prefer the new campaign. What is the test statistic?

Z = (240/600 - 0.4) / √(0.4(1-0.4)/600) = -2.24

1. A quality control team wants to know if the proportion of defective products is greater than 0.05. They collect a random sample of 1000 products and find that 60 are defective. Calculate the confidence interval.

p̂ = 60/1000 = 0.06 SE = √(p(1-p)/n) = √(0.05(1-0.05)/1000) = 0.008 CI = (p̂ - 1.96SE, p̂ + 1.96SE) = (0.06 - 0.0156, 0.06 + 0.0156) = (0.0444, 0.0756)

Last-Minute Cram Sheet

1. One-sample Z-test for proportion: used to compare a sample proportion to a known population proportion.
2. p̂ = (x / n): sample proportion.
3. p = population proportion: known value.
4. Z = (p̂ - p) / √(p(1-p)/n): test statistic.
5. p-value = P(Z > |Z|): probability of observing the data (or more extreme) if H₀ is true.
6. Decision Rule: Reject H₀ if p-value < α: reject H₀ if the p-value is less than the significance level (α).
7. Assumption: Sample size must be sufficiently large (np ≥ 10, n(1-p) ≥ 10): check that the sample size is large enough before conducting the test.
8. Critical Value: Zα/2 = ±1.96: critical value for a two-tailed test with α = 0.05.
9. ⚠️ 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: do not misinterpret the p-value.
10. ⚠️ Use a one-sample t-test for proportion when the population proportion is unknown: do not use a Z-test when the population proportion is unknown.