Intro to Business Statistics: Hypothesis Testing Paired Dependent Samples tTest

What This Is

A paired (dependent) samples t-test is used to compare the means of two related groups. For example, a company wants to evaluate the effectiveness of a new sales training program by comparing the average sales of its sales representatives before and after the training. This test is essential in business decisions as it helps to determine if there is a significant difference between the two groups, which can inform future business strategies.

Key Formulas & Symbols

• t = (x̄d - μd) / (s_d / √n) where x̄d = sample mean of differences, μd = population mean of differences, s_d = sample standard deviation of differences, n = sample size.
• s_d = √[(Σ(xi - x̄)(yi - ȳ)) / (n - 1)] where xi and yi are individual data points, x̄ and ȳ are sample means.
• t-statistic: calculated using the formula above.
• degrees of freedom (df): n - 1, where n is the sample size.
• p-value: the probability of observing the data (or more extreme) if the null hypothesis is true.
• α: the significance level (default = 0.05).
• H₀: the null hypothesis (e.g., μd = 0).
• H_a: the alternative hypothesis (e.g., μd ≠ 0).

Step-by-Step Procedure

1. State hypotheses: Define the null and alternative hypotheses. For example, H₀: μd = 0 (no difference) and H_a: μd ≠ 0 (difference exists).
2. Choose test: Select the paired samples t-test as the appropriate statistical test.
3. Compute test statistic: Calculate the t-statistic using the formula t = (x̄d - μd) / (s_d / √n).
4. Find p-value or critical value: Determine the p-value or critical value using a t-distribution table or calculator.
5. Compare to α: Compare the p-value or critical value to the significance level α (default = 0.05).
6. Conclude: If the p-value < α or the critical value is exceeded, reject the null hypothesis and conclude that there is a significant difference between the two groups.

Common Mistakes

• 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. It does not provide information about the probability of the null hypothesis being true.
• Mistake: Failing to check the assumptions of the paired samples t-test (e.g., normality, equal variances).
• Correction: Check the assumptions before conducting the test. If the assumptions are not met, consider alternative tests or transformations.
• Mistake: Using the wrong degrees of freedom (e.g., using n instead of n - 1).
• Correction: Use the correct degrees of freedom (n - 1) when calculating the t-statistic and p-value.

Quick Practice Problems

1. A company wants to evaluate the effectiveness of a new marketing campaign by comparing the average sales of its customers before and after the campaign. The sample mean of differences is 500, the sample standard deviation of differences is 200, and the sample size is 25. Calculate the confidence interval for the population mean of differences.

Answer: (300, 700) Explanation: The confidence interval is calculated using the formula x̄d ± (t * (s_d / √n)), where t is the critical value from the t-distribution table.

1. A retail chain wants to know if the average daily sales of its stores exceed $10,000 after implementing a new sales strategy. The sample mean of differences is 1500, the sample standard deviation of differences is 500, and the sample size is 30. What is the p-value?

Answer: 0.02 Explanation: The p-value is calculated using a t-distribution table or calculator, given the t-statistic and degrees of freedom.

1. A manufacturing company wants to compare the average quality of its products before and after implementing a new quality control process. The sample mean of differences is -2, the sample standard deviation of differences is 1, and the sample size is 20. What is the decision?

Answer: Reject the null hypothesis Explanation: The p-value is less than α (0.05), so the null hypothesis is rejected, indicating a significant difference between the two groups.

Last-Minute Cram Sheet

1. t = (x̄d - μd) / (s_d / √n): the formula for the t-statistic.
2. s_d = √[(Σ(xi - x̄)(yi - ȳ)) / (n - 1)]: the formula for the sample standard deviation of differences.
3. df = n - 1: the degrees of freedom for the paired samples t-test.
4. p-value ≠ probability of H₀: the p-value is the probability of observing the data (or more extreme) if H₀ is true.
5. α = 0.05: the default significance level.
6. H₀: μd = 0: the null hypothesis for the paired samples t-test.
7. H_a: μd ≠ 0: the alternative hypothesis for the paired samples t-test.
8. ⚠️ p-value is NOT the probability that H₀ is true: the p-value is the probability of observing the data (or more extreme) if H₀ is true.
9. ⚠️ use correct degrees of freedom (n - 1): the degrees of freedom for the paired samples t-test is n - 1.
10. ⚠️ check assumptions (normality, equal variances): the paired samples t-test assumes normality and equal variances.