You report your ANOVA and conclude that there is a statistically significant difference between the three eye-colour groups, F (2, 500) = 7.72, p = .007. Which of the following statements is true?

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First, let's restate the question in plain terms and lay out the options clearly so we can evaluate them one by one.
Question: You report your ANOVA and conclude that there is a statistically significant difference between the three eye-colour groups, F (2, 500) = 7.72, p = .007. Which of the following statements is true?
Answer options:
1) Because the ANOVA is statistically significant we can proceed to consider pairwise comparisons between groups, so we find out where the statistically significant difference(s) lie. If the ANOVA was non-significant there would be no point doing pairwise comparisons because they would all be non-significant anyway.
2) The reader can conclude that all the group means must be very different from one another.
3) The test statistic is in the tail (rightmost part) of the F-distribution.
4) We have reported a measure of effect size.

Option 1: This statement is about following up a significant ANOVA with post hoc pairwise comparisons. It is a common and correct practice: when an ANOVA shows a significant F, researchers often conduct post hoc tests to identify which specific group means differ. However, the option also includes an absolute claim that if the ANOVA were non-significant, there would be no point doing pairwise comparisons because they would all be non-significant. While it is true that non-significant ANOVA reduces the likelihood of meaningful pairwise differences, researchers sometimes still examine planned comparisons, but the general emphasis is that post hoc tests are warranted when the overall test is significant. The main drawback here is that the statement is somewhat overconfident about the necessity and universality of post hoc procedures in all contexts, but it is broadly aligned with standard practice.
Option 2: This says all group means must be very different from one another. A significant F from ANOVA indicates at least one pair differs, but does not imply that every pair differs or that all means are very different. The typical interpretation is that there is some difference among means, not necessarily widespread or large differences across all pairs. Hence this claim is incorrect because it overstates the conclusion.
Option 3: This states that the test statistic is in the tail (rightmost part) of the F-distribution. In classical hypothesis testing, a statistically significant result with p = .007 means the observed F value is in the tail region of the F-distribution under the null hypothesis. An F of 7.72 with df1 = 2 and df2 = 500 corresponds to a tail probability of about 0.007, which places the statistic in the upper tail. This option correctly characterizes what a significant F means in terms of the sampling distribution and p-value, making it a true statement.
Option 4: This says we have reported a measure of effect size. While many studies do report effect size, this particular statement is not something that is entailed by the given information. The F statistic and p-value do not automatically confirm that an effect size was reported, and the sentence does not reference any effect-size metric (e.g., eta-squared or omega-squared). Therefore, as a standalone claim about the given scenario, it is not necessarily true.

In summary, option 3 correctly describes the statistical meaning of a significant F statistic: the test statistic lies in the tail of the F-distribution corresponding to the observed p-value. The other options contain partial truths or overstatements and are not universally correct given the information provided. The nuance in option 1 about post hoc comparisons is relevant but not definitive as an absolute rule, option 2 misstates the interpretation, and option 4 makes an assumption not guaranteed by the reported statistics.

PSYC20007_2025_SM2 ANOVA, II

You report your ANOVA and conclude that there is a statistically significant difference between the three eye-colour groups, F (2, 500) = 7.72, p = .007. Which of the following statements is true?

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