If the means for our groups on calories consumed a day were Group A = 2000, Group B = 2500, and Group C = 3000 could we just assume that all of the groups had statistically different means? How could we determine which groups were significantly different from each other?

Do a hypothesis test to determine if the groups are significantly different from each other.

To determine whether the means of the three groups (Group A, Group B, and Group C) are statistically different, you can use a statistical test called analysis of variance (ANOVA). ANOVA allows you to compare the means of multiple groups to determine if there is a significant difference between them.

To perform an ANOVA test, follow these steps:

Step 1: State the null hypothesis (H0) and the alternative hypothesis (Ha):
- Null hypothesis (H0): The means of all the groups are equal.
- Alternative hypothesis (Ha): At least one group mean is different from the others.

Step 2: Calculate the sum of squares (SS):
- Within-group sum of squares (SSW): This measures the variability within each group.
- Between-group sum of squares (SSB): This measures the variability between the group means.

Step 3: Compute the degrees of freedom (df) for both SSW and SSB:
- dfW: The degrees of freedom for SSW is the total number of observations minus the total number of groups.
- dfB: The degrees of freedom for SSB is the total number of groups minus 1.

Step 4: Calculate the mean square (MS) for both SSW and SSB:
- MSW: Divide the sum of squares within-group (SSW) by the degrees of freedom within-group (dfW).
- MSB: Divide the sum of squares between-group (SSB) by the degrees of freedom between-group (dfB).

Step 5: Compute the F-statistic:
- F-statistic: Divide the mean square between-group (MSB) by the mean square within-group (MSW).

Step 6: Determine the critical value or p-value threshold for your desired significance level (e.g., 0.05 or 0.01).

Step 7: Compare the F-statistic to the critical value:
- If the F-statistic is greater than the critical value, reject the null hypothesis and conclude that at least one group mean is different from the others.
- If the F-statistic is less than or equal to the critical value, fail to reject the null hypothesis and conclude that there is insufficient evidence to suggest a difference in means.

If the ANOVA test confirms that there is a significant difference among the group means, you can then conduct post-hoc tests, such as Tukey's Honestly Significant Difference (Tukey's HSD) or pairwise t-tests, to determine which specific groups differ significantly from each other.