Intro to Business Statistics: Sampling and Sampling Distributions Sampling Distribution of the Sample Proportion

What This Is

The sampling distribution of the sample proportion is a statistical concept used to make inferences about a population proportion. A retail chain wants to know if the proportion of customers who prefer their new online shopping platform exceeds 0.5. They collect a random sample of 1,000 customers and find that 550 prefer the new platform. By analyzing the sampling distribution of the sample proportion, the retail chain can make a decision about whether to invest more in the platform.

Key Formulas & Symbols

• p̂ = X / n where X = number of successes, n = sample size.
• p̂ = (number of successes) / (sample size).
• σp̂ = √(p(1-p)/n) where p = population proportion, n = sample size.
• Z = (p̂ - p) / (σp̂) where p̂ = sample proportion, p = population proportion, σp̂ = standard error of the proportion.
• p̂ = (X/n) ± Z * (σp̂) where X = number of successes, n = sample size, Z = critical value, σp̂ = standard error of the proportion.
• p̂ = (X/n) ± (Z * √(p(1-p)/n)) where X = number of successes, n = sample size, Z = critical value, p = population proportion.
• p̂ = (X/n) ± (Z * √((X/n) * (1-X/n)/n)) where X = number of successes, n = sample size, Z = critical value.

Step-by-Step Procedure

1. State hypotheses: Write the null and alternative hypotheses in terms of the population proportion (e.g., H₀: p = 0.5, H₁: p > 0.5).
2. Choose test: Select the appropriate test (e.g., one-proportion z-test, two-proportion z-test).
3. Compute test statistic: Calculate the sample proportion (p̂) and the standard error of the proportion (σp̂).
4. Find p-value or critical value: Determine the p-value or critical value using a standard normal distribution (Z-distribution).
5. Compare to α: Compare the p-value or critical value to the significance level (α = 0.05).
6. Conclude: Make a decision about the null hypothesis based on the comparison.

Common Mistakes

• Mistake: Using a two-proportion z-test when the samples are not independent.
• Correction: Use a two-sample t-test or a chi-squared test instead.
• Mistake: Misinterpreting the p-value as the probability that the null hypothesis is true.
• Correction: The p-value is the probability of observing the data (or more extreme) if the null hypothesis is true.
• Mistake: Failing to check the assumptions of the test (e.g., normality, independence).
• Correction: Check the assumptions before selecting the test and interpreting the results.

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 500 customers and find that 320 prefer the new product. Calculate the confidence interval for the population proportion.

p̂ = 320/500 = 0.64 σp̂ = √(0.6(1-0.6)/500) = 0.024 Z = 1.645 (critical value for 95% confidence interval) CI: 0.64 ± 1.645 * 0.024 = (0.576, 0.704)

1. A marketing firm wants to know if the proportion of customers who prefer their new ad is greater than 0.4. They collect a random sample of 1,000 customers and find that 420 prefer the new ad. What is the p-value?

p̂ = 420/1000 = 0.42 σp̂ = √(0.4(1-0.4)/1000) = 0.015 Z = (0.42 - 0.4) / 0.015 = 1.33 p-value ≈ 0.091

1. A quality control team wants to know if the proportion of defective products is less than 0.1. They collect a random sample of 200 products and find that 10 are defective. Calculate the confidence interval for the population proportion.

p̂ = 10/200 = 0.05 σp̂ = √(0.1(1-0.1)/200) = 0.031 Z = 1.645 (critical value for 95% confidence interval) CI: 0.05 ± 1.645 * 0.031 = (0.013, 0.087)

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 = (p̂ - p) / (σp̂) for one-proportion z-test.
• Use p̂ = (X/n) ± Z * (σp̂) for confidence interval.
• σp̂ = √(p(1-p)/n) is the standard error of the proportion.
• p̂ = (X/n) is the sample proportion.
• Use t-distribution for two-sample t-test.
• α = 0.05 is the default significance level.
• df = n - 1 is the degrees of freedom for t-test.
• Z-distribution is used for one-proportion z-test.
• p̂ = (X/n) ± (Z * √(p(1-p)/n)) is the confidence interval formula.