In an analysis of variance, if the null hypothesis is true then:



- The research hypothesis is also true

-You will need fewer participants for the experiment

- There is less variance among the sample means than if the null hypothesis was not true

- The within-group estimate of population variance is smaller than the between-group estimate

the within-groups variance is smaller than the between-groups variance.

In an analysis of variance (ANOVA), if the null hypothesis is true, it means that there is no significant difference among the means of the groups being compared. In this case:

- The research hypothesis, which suggests that there is a significant difference among the means, would be false.
- The number of participants required for the experiment does not change based on the null hypothesis. The sample size is determined by various factors, such as desired power, effect size, and statistical significance level.
- With the null hypothesis being true, there would be less variance among the sample means compared to if the null hypothesis were false. This is because if there is no difference among the means, the individual sample means would be closer to each other.
- The within-group estimate of population variance is typically larger than the between-group estimate, regardless of whether the null hypothesis is true or false. This is because the within-group estimate accounts for the variability within each group, while the between-group estimate compares the means of the groups.

In an analysis of variance (ANOVA), the null hypothesis states that there is no significant difference between the means of the groups being compared. Therefore, if the null hypothesis is true, it means that there is no real difference between the groups, and any observed differences are due to random chance.

Now let's go through each of the given options:

- The research hypothesis is also true: This statement is incorrect. The null hypothesis and the research hypothesis are mutually exclusive. If the null hypothesis is true, it means that the research hypothesis is not true.

- You will need fewer participants for the experiment: This statement is incorrect. The sample size required for an experiment is determined based on the desired level of statistical power, not on the validity of the null hypothesis.

- There is less variance among the sample means than if the null hypothesis was not true: This statement is incorrect. If the null hypothesis is true, it means that any observed differences in sample means are due to random chance. In such cases, there would be more variance among the sample means compared to when the null hypothesis is not true and there is a true difference between the groups.

- The within-group estimate of population variance is smaller than the between-group estimate: This statement is correct. In ANOVA, the total variance is divided into two components: within-group variance and between-group variance. If the null hypothesis is true, it means that there is no true difference between the groups, so the observed differences are due to random variation within the groups. Therefore, the within-group estimate of population variance (representing the random variation within the groups) would be smaller than the between-group estimate (representing the systematic differences between the groups).

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