Intro to Business Statistics: Estimation Confidence Intervals for Population Mean σ Known Z σ Unknown tdistribution

What This Is

Confidence intervals for the population mean are used to estimate the average value of a population based on a sample of data. A retail chain wants to know if the average daily sales exceed $10,000. By constructing a confidence interval, they can determine if the sample mean of $12,000 is significantly higher than the population mean.

Key Formulas & Symbols

• Z = (x̄ – μ) / (σ/√n) where x̄ = sample mean, μ = population mean, σ = population standard deviation, n = sample size.
• t = (x̄ – μ) / (s/√n) where x̄ = sample mean, μ = population mean, s = sample standard deviation, n = sample size.
• E = (Z * σ) / √n where E = margin of error, Z = Z-score, σ = population standard deviation, n = sample size.
• CI = x̄ ± E where CI = confidence interval, x̄ = sample mean, E = margin of error.
• df = n - 1 where df = degrees of freedom, n = sample size.
• t-critical = t(df, α/2) where t-critical = critical t-value, df = degrees of freedom, α = significance level.
• Z-critical = Z(α/2) where Z-critical = critical Z-value, α = significance level.

Step-by-Step Procedure

1. State hypotheses: Define the null and alternative hypotheses (e.g., H₀: μ = 10,000 vs. H₁: μ > 10,000).
2. Choose test: Select the appropriate test (Z-test or t-test) based on the known or unknown population standard deviation.
3. Compute test statistic: Calculate the Z-score or t-statistic using the sample mean, population standard deviation (or sample standard deviation), and sample size.
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 = 0.05).
6. Conclude: Make a decision based on the comparison (e.g., reject H₀ if p-value < α).

Common Mistakes

• Mistake: Using Z when σ is unknown.
• Correction: Use t-test when σ is unknown, and calculate the sample standard deviation (s).
• 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.
• Mistake: Not checking assumptions (normality, independence).
• Correction: Verify normality using a Q-Q plot or Shapiro-Wilk test, and check for independence using a scatterplot or correlation analysis.

Quick Practice Problems

1. A company wants to estimate the average weight of a new product. A sample of 36 units has a mean weight of 250 grams with a population standard deviation of 10 grams. Construct a 95% confidence interval for the population mean.

Answer: 245.31, 254.69. The calculation involves using the Z-score formula and the population standard deviation.

1. A marketing firm wants to determine if the average response rate to a new ad campaign exceeds 5%. A sample of 100 responses has a mean response rate of 6% with a sample standard deviation of 2%. What is the p-value for a one-tailed test at α = 0.05?

Answer: 0.012. The calculation involves using the t-test formula and the sample standard deviation.

1. A quality control team wants to estimate the average defect rate in a manufacturing process. A sample of 25 units has a mean defect rate of 2% with a sample standard deviation of 1%. Construct a 90% confidence interval for the population mean.

Answer: 1.63, 2.37. The calculation involves using the t-test formula and the sample standard deviation.

Last-Minute Cram Sheet

1. Z-test: Use when σ is known, and the population is normally distributed.
2. t-test: Use when σ is unknown, and the population is normally distributed.
3. CI = x̄ ± (Z * σ) / √n: Formula for confidence interval when σ is known.
4. CI = x̄ ± (t * s) / √n: Formula for confidence interval when σ is unknown.
5. df = n - 1: Calculate degrees of freedom for t-test.
6. t-critical = t(df, α/2): Find critical t-value for t-test.
7. Z-critical = Z(α/2): Find critical Z-value for Z-test.
8. ⚠️ p-value is NOT the probability that H₀ is true.
9. ⚠️ Use t-test when σ is unknown.
10. ⚠️ Check assumptions (normality, independence).