Intro to Business Statistics: Hypothesis Testing Type I and Type II Errors α β Power 1 β

What This Is

Type I and Type II errors are critical concepts in statistical hypothesis testing that can significantly impact business decisions. A retail chain wants to know if average daily sales exceed $10,000 to determine whether to open a new store. If they reject the null hypothesis (H₀) that average daily sales are less than or equal to $10,000 when, in fact, they are, they will make a Type I error. Conversely, if they fail to reject H₀ when average daily sales are indeed greater than $10,000, they will make a Type II error.

Key Formulas & Symbols

• α (alpha): The maximum probability of a Type I error, typically set at 0.05.
• β (beta): The probability of a Type II error, which is equal to 1 – Power.
• Power: The probability of correctly rejecting H₀ when it is false, calculated as 1 – β.
• Type I Error: Rejecting H₀ when it is true.
• Type II Error: Failing to reject H₀ when it is false.
• p-value: The probability of observing the data (or more extreme) if H₀ is true.
• Z-score: Z = (x̄ – μ) / (σ/√n) where x̄ = sample mean, μ = population mean, σ = population standard deviation, n = sample size.
• t-score: t = (x̄ – μ) / (s/√n) where x̄ = sample mean, μ = population mean, s = sample standard deviation, n = sample size.
• Degrees of Freedom (df): The number of values in the final calculation of a statistic that are free to vary.

Step-by-Step Procedure

1. State hypotheses: Clearly define the null (H₀) and alternative (H₁) hypotheses.
2. Choose test: Select the appropriate statistical test based on the research question and data type.
3. Compute test statistic: Calculate the test statistic (e.g., Z-score or t-score) using the sample data.
4. Find p-value or critical value: Determine the p-value or critical value associated with the test statistic and degrees of freedom.
5. Compare to α: Compare the p-value to the significance level (α) to make a decision.
6. Conclude: Based on the comparison, reject or fail to reject H₀.

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, not the probability that H₀ is true.
• Mistake: Failing to check assumptions before selecting a test.
• Correction: Verify that the data meet the assumptions of the test (e.g., normality, independence) before proceeding.
• Mistake: Ignoring the concept of power and β.
• Correction: Consider the probability of a Type II error (β) and the power of the test when designing the study.

Quick Practice Problems

1. A marketing firm wants to know if the average response rate to a new ad campaign exceeds 5%. They collect a sample of 100 responses with a mean of 6% and a standard deviation of 2%. What is the p-value? Answer: 0.012 (The p-value is calculated using a Z-test with a sample mean of 6%, a population mean of 5%, a sample standard deviation of 2%, and a sample size of 100.)
2. A quality control team wants to determine if the average defect rate in a manufacturing process is less than 2%. They collect a sample of 50 defects with a mean of 1.8 and a standard deviation of 0.5. What is the p-value? Answer: 0.23 (The p-value is calculated using a t-test with a sample mean of 1.8, a population mean of 2%, a sample standard deviation of 0.5, and a sample size of 50.)
3. A company wants to know if the average customer satisfaction rating exceeds 4.5 on a 5-point scale. They collect a sample of 200 ratings with a mean of 4.7 and a standard deviation of 0.8. What is the p-value? Answer: 0.01 (The p-value is calculated using a Z-test with a sample mean of 4.7, a population mean of 4.5, a sample standard deviation of 0.8, and a sample size of 200.)

Last-Minute Cram Sheet

1. α (alpha): Maximum probability of a Type I error, typically 0.05.
2. β (beta): Probability of a Type II error, equal to 1 – Power.
3. Power: Probability of correctly rejecting H₀ when it is false, calculated as 1 – β.
4. Type I Error: Rejecting H₀ when it is true.
5. Type II Error: Failing to reject H₀ when it is false.
6. p-value: Probability of observing the data (or more extreme) if H₀ is true.
7. Z-score: Z = (x̄ – μ) / (σ/√n) where x̄ = sample mean, μ = population mean, σ = population standard deviation, n = sample size.
8. t-score: t = (x̄ – μ) / (s/√n) where x̄ = sample mean, μ = population mean, s = sample standard deviation, n = sample size.
9. Degrees of Freedom (df): Number of values in the final calculation of a statistic that are free to vary.
10. ⚠️ 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.