A regional sales director wants to determine whether different customer service training programs lead to different levels of employee performance across three branches. Each branch uses one of the following training programs: Program A. Program B, or Program C. After one month, the director measures the performance score (out of 100) for 5 randomly selected employees from each branch. Using $\alpha =0.05$, perform a one-way ANOVA to determine whether there is a statistically significant difference in mean performance among the three training programs.

• A. Since the performance scores for all three programs (A, B, and C) range from 76 to 85, there is no statistically significant difference between the training programs.
• B. The P-value from the one-way ANOVA is greater than 0.05, so we fail to reject the null hypothesis and conclude that the type of training program does not affect employee performance.
• C. The P-value from the one-way ANOVA is less than 0.05, so we reject the $H_0$ and suggest $H_a$.
• D. The one-way ANOVA is inappropriate for this study because it compares two groups only, a different statistical test should be used.

Answer: (C)

Verified step by step guidance

1. Step 1: State the null hypothesis (H₀) and the alternative hypothesis (Hₐ). H₀: The mean performance scores are the same across all three training programs (Program A, Program B, and Program C). Hₐ: At least one training program has a different mean performance score.
2. Step 2: Calculate the group means for each training program. For Program A, B, and C, compute the average performance scores using the formula: Mean = (Sum of scores) / (Number of scores).
3. Step 3: Compute the overall mean (grand mean) of all performance scores across the three programs. Use the formula: Grand Mean = (Sum of all scores) / (Total number of scores).
4. Step 4: Calculate the between-group sum of squares (SSB) and within-group sum of squares (SSW). SSB measures the variation due to differences between group means, while SSW measures the variation within each group. Use the formulas: SSB = Σnᵢ(Meanᵢ - Grand Mean)² and SSW = ΣΣ(xᵢⱼ - Meanᵢ)².
5. Step 5: Compute the F-statistic using the formula: F = (SSB / df_between) / (SSW / df_within), where df_between = k - 1 (number of groups minus 1) and df_within = N - k (total number of observations minus number of groups). Compare the F-statistic to the critical value from the F-distribution table at α = 0.05 or use the p-value to determine whether to reject or fail to reject the null hypothesis.