Intro to Business Statistics: Statistical Software and Applications Conducting Hypothesis Tests ttests ANOVA in Software

Conducting Hypothesis Tests (t-tests, ANOVA) in Software

A retail chain wants to know if average daily sales exceed $10,000. They collect data from 16 stores and find the sample mean is $11,500 with a sample standard deviation of $2,000. They want to test the hypothesis that the population mean is greater than $10,000.

Key Formulas & Symbols

• t = (x̄ – μ) / (s/√n) where x̄ = sample mean, μ = population mean, s = sample standard deviation, n = sample size.
• t = (x̄ – μ) / (s/√n) is used for small samples (n < 30) when σ is unknown.
• F = (MSB / MSE) where MSB = mean square between, MSE = mean square error.
• F = (MSB / MSE) is used for ANOVA when comparing means of three or more groups.
• df = n - 1 where n = sample size, used for t-tests and ANOVA.
• α = 0.05 is the default significance level for hypothesis testing.
• p-value is the probability of observing the data (or more extreme) if H₀ is true.
• H₀ is the null hypothesis, a statement of no effect or no difference.
• H₁ is the alternative hypothesis, a statement of an effect or difference.

Step-by-Step Procedure

1. State hypotheses: Clearly define H₀ and H₁.
2. Choose test: Select the appropriate test (t-test or ANOVA) based on the research question and data.
3. Compute test statistic: Use the formula to calculate the test statistic (t or F).
4. Find p-value or critical value: Use software or a t-distribution table to find the p-value or critical value.
5. Compare to α: Compare the p-value to α (0.05) or the test statistic to the critical value.
6. Conclude: Based on the comparison, reject H₀ or fail to reject H₀.

Common Mistakes

• Mistake: Using Z when σ is unknown.
• Correction: Use t when σ is unknown, as it is more robust and accounts for sample size.
• 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, not the probability that H₀ is true.
• Mistake: Failing to check assumptions (normality, equal variances).
• Correction: Check assumptions before conducting the test, as they affect the validity of the results.

Quick Practice Problems

1. A company wants to know if the average salary of employees is greater than $50,000. They collect data from 25 employees and find the sample mean is $52,000 with a sample standard deviation of $5,000. What is the p-value? Answer: 0.02, This is the p-value for a one-tailed t-test with a sample size of 25 and a significance level of 0.05.
2. A marketing firm wants to compare the average sales of three different products. They collect data from 10 customers for each product and find the sample means are $100, $120, and $150. What is the F-statistic? Answer: 6.67, This is the F-statistic for an ANOVA test with three groups and a significance level of 0.05.
3. A quality control team wants to know if the average defect rate is greater than 5%. They collect data from 15 samples and find the sample mean is 6% with a sample standard deviation of 2%. What is the t-statistic? Answer: 2.45, This is the t-statistic for a one-tailed t-test with a sample size of 15 and a significance level of 0.05.

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 t when σ is unknown and n < 30.
• Use F for ANOVA when comparing means of three or more groups.
• df = n - 1 for t-tests and ANOVA.
• α = 0.05 is the default significance level.
• H₀ is the null hypothesis, a statement of no effect or no difference.
• H₁ is the alternative hypothesis, a statement of an effect or difference.
• Normality assumption is checked using a Q-Q plot or Shapiro-Wilk test.
• Equal variances assumption is checked using Levene's test or F-test.
• Critical values for t-distribution are found using a t-distribution table.
• Critical values for F-distribution are found using an F-distribution table.