ANOVA EXERCISES

 

Example 1.  You have conducted an experiment on the relationship between group size and helping behavior.  Essentially, what you did was randomly assign 60 people to groups of 1, 2, or 3 other people and then measured how helpful the subject was to someone who dropped their books.  Here are the data:

 

Group-Size Condition

 

M

 

s2-hat

 

n

1 other person

5

1

20

2 other persons

7

.8

20

3 other persons

9

1.2

20

 

What is the F-ratio corresponding to the ANOVA for these data?  Note:  According to my tables, the F-ratio needs to be larger than 3.17 for the p-value to be <.05.  Also indicate whether the F-ratio you observe is “statistically significant,” and whether you should accept or reject the null hypothesis.

 

Example 2.  You have conducted a study on the relationship between age groups and happiness.  You studied the happiness of groups of people whose ages ranged between 10 and 20, 21 and 30, and 31 and 40..  Here are the data:

 

Age group

 

M

 

s2-hat

 

n

10-20

5

1

20

21-30

5

.8

20

31-40

5

1.2

20

 

What is the F-ratio corresponding to the ANOVA for these data?  Note:  According to my tables, the F-ratio needs to be larger than 3.17 for the p-value to be <.05.  Also indicate whether the F-ratio you observe is “statistically significant,” and whether you should accept or reject the null hypothesis.

 

 

 

 

SOLUTIONS

 

Example 1.  You have conducted an experiment on the relationship between group size and helping behavior.  Essentially, what you did was randomly assign 60 people to groups of 1, 2, or 3 other people and then measured how helpful the subject was to someone who dropped their books.  Here are the data:

 

Group-Size Condition

 

M

 

s2-hat

 

n

1 other person

5

1

20

2 other persons

7

.8

20

3 other persons

9

1.2

20

 

What is the F-ratio corresponding to the ANOVA for these data?  Note:  According to my tables, the F-ratio needs to be larger than 3.17 for the p-value to be <.05.  Also indicate whether the F-ratio you observe is “statistically significant,” and whether you should accept or reject the null hypothesis.

 

To find the F-ratio, we need to estimate the population variance within each condition and between conditions (assuming the null hypothesis is true: all samples come from populations with identical means and variances). 

 

To estimate the population variance within conditions, we can pool the three variance estimates we have from each condition:

 

(1 + .8 + 1.2)/3 = 1

 

Thus, 1 is our MS within estimate of the population variance.

 

Next, we need to estimate the population variance across or between conditions.  To do this, we study the variances between the condition means as an approximation of the variance of sampling distribution of means.

 

Find the grand mean: (5 + 7 + 9)/3 = 7

 

Find the average squared deviation of the condition means from this grand mean:

 

sum(M – GM)2/(Ngroups-1)

 

(5 - 7) 2                            (-2) 2                4

(7 - 7) 2                        (0) 2            0

(8 - 7) 2                        (1) 2            1

                                    sum(4, 0, 1) = 5

This sum, divided by Ngroups –1:   5/2 = 2.5

 

Now, we multiply this by the sample size of each condition (20) to get 2.5 * 20 = 50.  Thus, our between estimate of the population variance is 50.

 

Our F-ratio is  MS between over MS within or 50/1 or 50.   This is considerably larger than 3.17, so we should reject the null hypothesis.  The differences we observed between conditions is much larger than what we would expect if the differences were due to sampling error alone.

 

 

 

Example 2.  You have conducted a study on the relationship between age groups and happiness.  You studied the happiness of groups of people whose ages ranged between 10 and 20, 21 and 30, and 31 and 40..  Here are the data:

 

Age group

 

M

 

s2-hat

 

n

10-20

5

1

20

21-30

5

.8

20

31-40

5

1.2

20

 

What is the F-ratio corresponding to the ANOVA for these data?  Note:  According to my tables, the F-ratio needs to be larger than 3.17 for the p-value to be <.05.  Also indicate whether the F-ratio you observe is “statistically significant,” and whether you should accept or reject the null hypothesis.

 

To find the F-ratio, we need to estimate the population variance within each condition and between conditions (assuming the null hypothesis is true: all samples come from populations with identical means and variances). 

 

To estimate the population variance within conditions, we can pool the three variance estimates we have from each condition:

 

(1 + .8 + 1.2)/3 = 1

 

Thus, 1 is our MS within estimate of the population variance.

 

Next, we need to estimate the population variance across or between conditions.  To do this, we study the variances between the condition means as an approximation of the variance of sampling distribution of means.

 

Find the grand mean: (5 + 5 + 5)/3 = 5

 

Find the average squared deviation of the condition means from this grand mean:

 

sum(M – GM)2/(Ngroups-1)

 

(5 - 5) 2                            (0) 2                0

(5 - 5) 2                        (0) 2            0

(5 - 5) 2                        (0) 2            0

                                    sum(0, 0, 0) = 0

0/2 = 0

 

Now, we multiply this by the sample size of each condition (20) to get 0 * 20 = 0.  Thus, our between estimate of the population variance is 0.

 

Our F-ratio is  MS between over MS within or 0/1 or 0. This is not larger than 3.17, so we should accept the null hypothesis.  The differences we observed between conditions (actually, we observed no differences) were about the size we would expect if the differences were due to sampling error alone and the null hypothesis was true.