Intro to Business Statistics: Analysis of Variance ANOVA - One-Way ANOVA, Purpose Compare 2 Means Assumptions FTest

What This Is

One-Way ANOVA (Analysis of Variance) is a statistical method used to compare the means of three or more groups to determine if there is a significant difference between them. A retail chain wants to know if the average daily sales of its stores in three different locations (New York, Los Angeles, and Chicago) exceed $10,000. By using One-Way ANOVA, the chain can determine if there is a significant difference in daily sales among the three locations.

Key Formulas & Symbols

• F = (MSB / MSE) where MSB = Mean Square Between, MSE = Mean Square Error, k = number of groups, n = total sample size, and dfB = k - 1, dfE = N - k.
• MSB = ?(k-1) * (x?i - x?)² / (k - 1) where x?i = group mean, x? = overall mean, and k = number of groups.
• MSE = ?(xij - x?i)² / (N - k) where xij = individual data point, x?i = group mean, and N = total sample size.
• dfB = k - 1 where k = number of groups.
• dfE = N - k where N = total sample size and k = number of groups.
• F-critical = F(?, dfB, dfE) where-= 0.05, dfB = k - 1, and dfE = N - k.
• p-value = P(F > F-test statistic) where F-test statistic = F = (MSB / MSE).

Step-by-Step Procedure

1. State hypotheses: H?: = = … = (no significant difference among groups) vs. H?: not all means are equal (at least one group mean is different).
2. Choose test: One-Way ANOVA is used to compare means of three or more groups.
3. Compute test statistic: Calculate F = (MSB / MSE) and determine dfB and dfE.
4. Find p-value or critical value: Use an F-distribution table or calculator to find the p-value or critical value.
5. Compare to ?: Compare the p-value to? (0.05) or compare the F-test statistic to the F-critical value.
6. Conclude: If p-value <-or F-test statistic > F-critical, reject H? and conclude that at least one group mean is different.

Common Mistakes

• Mistake: Misinterpreting p-value as the probability that H? is true.
• Correction: The p-value is the probability of observing the data (or more extreme) if H? is true. It does not directly tell us the probability that H? is true.
• Mistake: Failing to check assumptions (normality, equal variances).
• Correction: Check assumptions before conducting One-Way ANOVA. If assumptions are not met, consider alternative methods.
• Mistake: Using F-test statistic without considering dfB and dfE.
• Correction: Calculate F-test statistic using dfB and dfE, and consider the degrees of freedom when interpreting results.

Quick Practice Problems

1. A company wants to compare the average salaries of its employees in three different departments (Sales, Marketing, and Finance). The sample means are $50,000, $60,000, and $70,000, respectively. The overall mean is $60,000, and the sample size is 10 in each department. What is the F-test statistic?

F-test statistic = (MSB / MSE) = (?(k-1) * (x?i - x?)² / (k - 1)) / (?(xij - x?i)² / (N - k)) = (2 * (50,000 - 60,000)² + 2 * (60,000 - 60,000)² + 2 * (70,000 - 60,000)²) / (10 - 3) / (?(xij - x?i)² / (10 - 3)) = 3.33

1. A researcher wants to compare the average exam scores of students in three different classes (Math, Science, and English). The sample means are 80, 90, and 70, respectively. The overall mean is 80, and the sample size is 10 in each class. What is the p-value?

p-value = P(F > F-test statistic) = P(F > 3.33) = 0.05

1. A company wants to compare the average sales of its products in three different regions (North, South, and East). The sample means are $10,000, $15,000, and $20,000, respectively. The overall mean is $15,000, and the sample size is 10 in each region. What is the F-critical value?

F-critical = F(0.05, 2, 24) = 3.49

Last-Minute Cram Sheet

• One-Way ANOVA is used to compare means of three or more groups.
• F = (MSB / MSE) where MSB = Mean Square Between, MSE = Mean Square Error, k = number of groups, n = total sample size, and dfB = k - 1, dfE = N - k.
• MSB = ?(k-1) * (x?i - x?)² / (k - 1) where x?i = group mean, x? = overall mean, and k = number of groups.
• MSE = ?(xij - x?i)² / (N - k) where xij = individual data point, x?i = group mean, and N = total sample size.
• dfB = k - 1 where k = number of groups.
• dfE = N - k where N = total sample size and k = number of groups.
• F-critical = F(?, dfB, dfE) where-= 0.05, dfB = k - 1, and dfE = N - k.
• p-value is the probability of observing the data (or more extreme) if H? is true.
• p-value is NOT the probability that H? is true.
• Check assumptions (normality, equal variances) before conducting One-Way ANOVA.
• Failing to check assumptions can lead to incorrect conclusions.
• Calculate F-test statistic using dfB and dfE, and consider the degrees of freedom when interpreting results.