Chapter 11

Question 1

List the steps involved in constructing the F distribution.

Answer 1

The steps to construct the F distribution include determining the degrees of freedom for the numerator and denominator, calculating the mean squares, and plotting the distribution based on these values.

Question 2

Describe the purpose of ANOVA in statistical analysis.

Answer 2

ANOVA, or Analysis of Variance, is used to evaluate the differences between the means of two or more groups, making it suitable for comparing multiple populations.

Question 3

How does ANOVA compare to using multiple t-tests for group comparisons?

Answer 3

Using ANOVA is more efficient than conducting multiple t-tests because it avoids the increased risk of Type I errors that arise from performing several tests, as well as reducing the overall workload.

Question 4

Describe the process of constructing an empirical F distribution.

Answer 4

To construct an empirical F distribution, randomly select two samples of fixed sizes from a population, calculate the unbiased variance estimates for each sample, divide the first variance estimate by the second to obtain the F statistic, and repeat this process until all possible pairs of samples have been drawn, then place the F ratios in a relative frequency distribution.

Question 5

How does the F statistic relate to making inferences about population means?

Answer 5

The F statistic, derived from the ratio of variance estimates from two samples, has properties that allow researchers to make inferences about population means, particularly in the context of ANOVA.

Question 6

Describe the purpose of ANOVA in research.

Answer 6

ANOVA is used to make inferences about the means of populations from which samples have been drawn, particularly to determine if different treatments or independent variables have an effect on the dependent variable.

Question 7

How does ANOVA determine if a treatment has an effect on sample means?

Answer 7

ANOVA compares the means of different samples; if the treatment has no effect, the means will be similar due to chance, but if the treatment does have an effect, the means will differ significantly, indicating that the samples come from populations with different means.

Question 8

Define the null hypothesis in the context of ANOVA using the F distribution.

Answer 8

The null hypothesis in ANOVA states that all means are equal, represented as H0: μ1 = μ2 = μ3 = … = μk, where k is the number of samples.

Question 9

Describe the role of the alternative hypothesis in ANOVA.

Answer 9

The alternative hypothesis in ANOVA indicates that at least one mean is different from the others, represented as H1, but does not specify which means are different.

Question 10

Describe the relationship between one-way ANOVA and t-tests when comparing two samples.

Answer 10

The outcome of a one-way ANOVA done on two samples is identical to the result of a t-test conducted on those same two samples.

Question 11

How does the variability of individual scores relate to group means in a one-way ANOVA?

Answer 11

In a one-way ANOVA, individual scores within each group vary around the group mean, which indicates that while the means of the groups may be similar, they are not identical.

Question 12

Describe the concept of error variance in the context of treatment effects.

Answer 12

Error variance refers to the variability among participants that is not influenced by the treatment effects. It represents the inherent differences among individuals within a group.

Question 13

How does treatment effect influence sample means in a study?

Answer 13

If a treatment has an effect, the samples will come from populations with different means, leading to larger variation between the sample means compared to when there is no treatment effect.

Question 14

Define total variability in the context of an experiment.

Answer 14

Total variability refers to the overall variation of all scores from the combined mean of all groups in an experiment.

Question 15

How is variability in an experiment partitioned according to ANOVA?

Answer 15

Variability in an experiment is partitioned into two main components: variability of participants within groups and variability between groups.

Question 16

Describe the process of calculating total variability in an experiment.

Answer 16

Total variability is calculated by subtracting the combined mean from each score in the experiment, squaring these differences, and then summing them.

Question 17

How is the between-group sum of squares defined in statistical analysis?

Answer 17

The between-group sum of squares is defined by subtracting the combined mean from each group mean, squaring each difference, multiplying by the number of observations in each group (n), and summing these values.

Question 18

Describe the steps to calculate the total sum of squares in a behavioral and social sciences experiment.

Answer 18

1. Sum the raw scores in each group. 2. Square each group sum and divide by its group size. 3. Sum the values obtained in Step 2. 4. Sum the scores in the entire experiment, square this sum, and divide by the total number of observations. 5. Subtract the result of Step 4 from that of Step 3.

Question 19

Describe the purpose of comparing mean squares in a one-way ANOVA.

Answer 19

The purpose of comparing mean squares in a one-way ANOVA is to determine the effect of the treatment by evaluating the variance estimates between groups and within groups.

Question 20

How are mean squares calculated in a one-way ANOVA?

Answer 20

Mean squares in a one-way ANOVA are calculated by dividing the sum of squares by the degrees of freedom for both between-groups and within-groups variability.

Question 21

Describe the significance of the F ratio values in statistical analysis.

Answer 21

The F ratio compares the between-groups variance estimate to the within-groups variance estimate. A value of 1 suggests no treatment effect, while a value greater than 1 indicates a significant treatment effect.

Question 22

How do you determine whether to reject the null hypothesis using the F ratio?

Answer 22

To reject the null hypothesis, the obtained F value must be equal to or larger than the critical value found in the F distribution table, based on the degrees of freedom for both the numerator and denominator.

Question 23

How is the data organized for a one-way ANOVA analysis?

Answer 23

In a one-way ANOVA analysis, data is typically organized into groups, where each group represents a different treatment or condition. Each group has its own set of scores, and the analysis compares the means of these groups to determine if there are statistically significant differences among them.

Question 24

Why not just do multiple t-test comparisons since an ANOVA is basically multiple t-test?

Answer 24

Each pairwise comparison (i.e., t-test) has possibility of false positive (Type I error, or α) • Probability of Type I error is greater with multiple pairwise comparisons

Question 25

What does the area under the curve associated with the f statistic indicate?

Answer 25

Area under curve associated with F-statistic indicates the likelihood of getting that F-statistic by chance if the null hypothesis was true (i.e., p value)

Question 26

what is total variability?

Answer 26

Within-group variability + between-group variability • Difference between all the individual scores and the overall means.

Question 27

What are the two types of variance?

Answer 27

Within-group variability is also called error variance, and group variability is also called treatment variance.

Question 28

Differentiate between the within group variability and between group.

Answer 28

Within group
- Variation in scores within each group due to individual differences or random error.
- Caused by random noise, measurement error, or individual differences.

Between group
- Variation in group means due to differences between groups.
- Caused by the effect of the independent variable or treatment.

Question 29

What is Tukey’s Honestly Significant Difference (HSD) Test?

Answer 29

Tukey’s HSD value is benchmark for differences between means • Mean differences above HSD are significantly different, mean differences below HSD are not.

Question 30

What is the q value in HSD test?

Answer 30

q = value from Table B.5 • Need 𝜶 (.05), k (number of groups, which is 3), and dfWG (18) • q = 3.61

Question 31

Describe the characteristics of the f distribution

Answer 31

The F statistic is a ratio between two variance estimates. Because an estimate of the variance cannot be negative (zero variability is as low as it goes), the frequency distribution is limited at zero. The other end of the distribution, however, is not limited, and so the distribution is positively skewed.

Question 32

What is error variance?

Answer 32

Inherent variation or error variance is free of the effects of the treatment. It is the variation we see because of individual differences among our participants

Question 33

The subgroup means vary around the combined mean, Why?

Answer 33

(1) inherent variation of the participants
(2) the effect of the treatment, if indeed it had an effect. The variation between subgroup means is due to inherent variation plus variation due to treatment.