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In hypothesis testing, we are trying to determine if a specific statement about a population parameter is true. In this exercise, we are comparing the mean gasoline mileage for three different brands of automobiles.

The initial step involves setting up a Null Hypothesis (\( H_0 \)) and an Alternative Hypothesis (\( H_a \)).

• The Null Hypothesis (\( H_0 \)) states that there is no difference in the mean miles per gallon for all automobile brands. In mathematical terms: \( \mu_A = \mu_B = \mu_C \).
• The Alternative Hypothesis (\( H_a \)) claims that at least one brand has a different mean miles per gallon. This suggests that not all means are equal.

Hypothesis testing is crucial to determine if observed differences are due to sampling variability or if there are actual differences among groups.

The Sum of Squares (SS) helps us understand the variability present in our data. Within ANOVA, there are two main types of sum of squares: Between-groups Sum of Squares (SSB) and Within-groups Sum of Squares (SSW).

• **Between-groups Sum of Squares (SSB):** This measures the variability due to the differences between group means and the overall mean. It tells us how much the group means deviate from the grand mean. In this problem: \( SSB = \sum n((\bar{x}_i - \bar{x})^2) \), which calculates to 80.
• **Within-groups Sum of Squares (SSW):** This measures the variability within each group. It reflects how scores within each group differ from their group mean. For this exercise, \( SSW = \sum (x - \bar{x})^2 \) sums up to 34.

Understanding sum of squares is essential to analyze how much of the total variability can be attributed to differences within groups versus differences between groups.

Mean Squares (MS) are derived from sums of squares, and they represent the average variability per degree of freedom. They are a vital part of ANOVA calculations.

• **Mean Square Between (MSB):** This is calculated by dividing the between-groups sum of squares (SSB) by the degrees of freedom for the groups. Here, \( MSB = \frac{SSB}{k-1} = \frac{80}{2} = 40 \).
• **Mean Square Within (MSW):** Similarly, this is computed by dividing the within-groups sum of squares (SSW) by its degrees of freedom. In this case, \( MSW = \frac{SSW}{N-k} = \frac{34}{12} \approx 2.83 \).

Mean squares help us understand the variation at each level of analysis, setting the stage for further hypothesis testing with the F-test.

The F-test is a statistical test used in ANOVA to determine if there are any statistically significant differences between the means of three or more groups. It compares the variability between the groups to the variability within groups.

To perform the F-test, we calculate the F statistic:

• **F-statistic Formula:** This is given by \( F = \frac{MSB}{MSW} \). For this exercise, it becomes \( F = \frac{40}{2.83} \approx 14.14 \).

Once the F value is computed, it is compared against a critical value from the F-distribution table at a given significance level (\( \alpha \)).

• The degrees of freedom for the numerator (between-groups) are \( k-1 \) and for the denominator (within-groups) are \( N-k \).

If the calculated F statistic is greater than the critical value, we reject the null hypothesis, suggesting at least one group mean is significantly different. Here, since \( 14.14 > 3.88 \), the null hypothesis is rejected, indicating a significant difference in gasoline mileage among the automobile brands.