1-

The following data represent the results from an independent-measures experiment comparing three treatment conditions. Conduct an analysis of variance with=0.05 to determine whether these data are sufficient to conclude that there are significant differences between the treatments.

Treatment A	Treatment B	Treatment C
21	23	25
19	21	26
19	19	27
21	20	24
20	22	23

F-ratio =

p-value =

Conclusion:

• These data do not provide evidence of a difference between the treatments
• There is a significant difference between treatments

The results obtained above were primarily due to the mean for the third treatment being noticeably different from the other two sample means. For the following data, the scores are the same as above except that the difference between treatments was reduced by moving the third treatment closer to the other two samples. In particular, 3 points have been subtracted from each score in the third sample.

Before you begin the calculation, predict how the changes in the data should influence the outcome of the analysis. That is, how will theF-ratio for these data compare with theF-ratio from above?

Treatment A	Treatment B	Treatment C
21	23	22
19	21	23
19	19	24
21	20	21
20	22	20

F-ratio =

p-value =

Conclusion:

• These data do not provide evidence of a difference between the treatments
• There is a significant difference between treatments

2-

The following data represent the results from an independent-measures experiment comparing three treatment conditions withn=4 in each sample. Conduct an analysis of variance with=0.05 to determine whether these data are sufficient to conclude that there are significant differences between the treatments.

Treatment A	Treatment B	Treatment C
23	24	27
22	24	27
22	22	22
25	26	24

F-ratio =

p-value =

Conclusion:

1. There is a significant difference between treatments
2. These data do not provide evidence of a difference between the treatments

2=

The results above were obtained because the sample means are close together. To construct the data set below, the same scores from above were used, then the size of the mean differences were increased. In particular, the first treatment scores were lowered by 2 points, and the third treatment scores were raised by 2 points. As a result, the three sample means are now much more spread out.

Before you begin the calculation, predict how the changes in the data should influence the outcome of the analysis. That is, how will theF-ratio for these data compare with theF-ratio from above?

Treatment A	Treatment B	Treatment C
21	24	29
20	24	29
20	22	24
23	26	26

F-ratio =

p-value =

Conclusion:

1. There is a significant difference between treatments
2. These data do not provide evidence of a difference between the treatments

2=

3-

The following data were obtained in a study using three separate samples to compare three different treatments. Conduct an analysis of variance with=0.05 to determine whether these data are sufficient to conclude that there are significant differences between the treatments.

Treatment A	Treatment B	Treatment C
18	15	18
17	17	20
19	15	22
18	17	20

F-ratio =

p-value =

Conclusion:

1. These data do not provide evidence of a difference between the treatments
2. There is a significant difference between treatments

2=

The above data was changed by increasing the variance within each sample (note below how the data sets have changed).

Before you begin the calculation, predict how the changes in the data should influence the outcome of the analysis. That is, how will theF-ratio for these data compare with theF-ratio from above?

Treatment A	Treatment B	Treatment C
18	15	17
16	18	20
20	14	23
18	17	20

F-ratio =

p-value =

Conclusion:

1. There is a significant difference between treatments
2. These data do not provide evidence of a difference between the treatments

2=

4-

Recent research indicates that the effectiveness of antidepressant medication is directly related to the severity of the depression (Khan, Brodhead, Kolts & Brown, 2005). Based on pretreatment depression scores, patients were divided into four groups based on their level of depression. After receiving the antidepressant medication, depression scores were measured again and the amount of improvement was recorded for each patient. The following data are similar to the results of the study.

Run the single-factor ANOVA for this data:

Low Moderate	High Moderate	Moderately Severe	Severe
0.2	3	1.8	4.3
0.8	3.2	3.1	3.4
1.6	1.1	4.9	5.5
2.9	2	3.3	1.8
2.3	2	1.6	3.8
2.3	4	4.5	5.3

Fill in the summary table for the ANOVA test:

	S.S.	d.f.	M.S.
Between
Within
TOTAL

From this table, obtain the necessary statistics for the ANOVA:

F-ratio:

p-value:

2=

What is your final conclusion? Use a significance level of=0.02.

1. These data do not provide evidence of a difference between the treatments
2. There is a significant difference between treatments

5-

In testing a new drug, we obtained the following results:

Placebo	Drug A	Drug B	Drug C
3	4	6	4
0	3	3	3
2	1	4	2
0	1	3	1
0	1	4	0

Run the ANOVA and fill in the summary table with the results obtained:

	SS	df	MS	F-ratio	p-value
Between		3
Within		16
TOTAL		19

(Report P-value & F-ratio accurate to 3 decimal places and all other values accurate to 2 decimal places.)

What conclusion can be drawn at the 0.1 significance level?

• The various drugs do not have results that are statistically different.
• The various drugs have results that are statistically different.

6-

Suppose we want to test whether or not three population means are equal. We can assume the population variances are equal and there is only one factor of difference. We want to perform this test with a 2% significance level.

If we perform an ANOVA test, what is the probability of the test producing accurate results (avoiding aType I error)?

Suppose we, instead, run three separate hypothesis tests (t-tests), each with 2% significance level.

• Mean 1 = Mean 2
• Mean 1 = Mean 3
• Mean 2 = Mean 3

What is the probability that all three tests would be accurate? Hint: useprinciples of probability to help your calculations:

P(accurate AND accurate AND accurate)

(Write your answer accurate without rounding.)

Why would we use ANOVA instead of three separate tests?

Why would we want to use three separate tests instead of ANOVA?

7-

Recent research indicates that the effectiveness of antidepressant medication is directly related to the severity of the depression (Khan, Brodhead, Kolts & Brown, 2005). Based on pretreatment depression scores, patients were divided into four groups based on their level of depression. After receiving the antidepressant medication, depression scores were measured again and the amount of improvement was recorded for each patient. The following data are similar to the results of the study.

Low Moderate	High Moderate	Moderately Severe	Severe
1.5	0	0.9	1.7
1.8	1.5	1.8	0.8
1.6	2.5	2.3	0.7
1.7	2.5	1.5	1.3
2.6	1.7	1.5	3.1
0.7	3	2.6	2.6
1.3	1	1.6	4.5
1.6	2.6	2	3.3
2.2	3.3	2.5	3.7
1.7	1.5	4	2.6
1.7	2.8	0.9	3.4
2.3	2.1	1	1.2
3.1	4.1	1.8	0
0.9	1.7	2.1	1.7
3.1	0.8	0.8	2.7
2.1	1.3	2.4	3.2
2.1	1.9	0.4	1.7
3.2	2	1.4	2.7
2.7	2.2	1.2	1.9
3.1	4	3	2.4
3.3	0.8	0	4.3
1.3	1.7	2	2.6
2.3	2.4	2.2	2.9
4	2.1	2.6	1.2
2.2	1.9	2.6	0.6
3.4	2.1	1.3	3.9
4	3.4	2	2.2
1.9	1.8	2.1	1.2
1.7	3	2	4.1
4.1	0.9	2.9	1.2
3.5	3.3	1.8	2.8
2.4	2.6	1.3	3.2
2.4	3	3.6	1.9
2.7	3.5	2	3.8
1.9	1.5	0.5	1.7
0.9	1.4	2	3.4
0	2.1	4.5	1.6
0.6	0.1	2.4	1.1
3.3	2.4	1	2.7
1.6	2.8	3.7	2.8
0.4	1.2	1.1	3.3
2.2	0.7	2.3	1
2	1.9	2.8	2.8
2.4	1	0.9	0.7
1.3	4.6	2	1.3
1.8	1.8	1.9	2.3
1.8	3.4	0.7	2.1
1.9	1.9	2.9	4.2
0.4	3.5	1.6	2.6
1.9	1.1	2.6	4.2
1.9	2.9	0.5	3.3
1.4	1.1	1.6	1.3
3.1	2.6	1.6	2.5
1.8	4.3	1.2	2
2.7	1.9	2.6	1.3
2.5	0.4	2.5	2.6
0.3	1.2	1.3	3.9
1.8	1.9	2	2.7
1.3	2.2	1	3
2.3	1.9	2.3	2.4
1.4	2.2	3.2	1.2
0.6	1.3	3.2	2.1
1.4	1.6	0.7	0.9
1.6	0.3	1.5	0.9
1.2	2.3	1.9	2
2.6	1.6	2.9	1.2
2.8	1.7	0.7	4
2.3	3.1	1.7	4.5
1.4	2.3	1.8	1.8
2.4	2.7	2.3	2.9
2.4	1.8	2.5	1.5
2	2.4	1.9	1.5
0.3	2.3	1.3	2.1
3.6	3	1.7	1
0	1.4	2.6	4.2
2.4	2	2.9	2
2.4	0.7	1.8	3.3
0.9	3.3	3.1	1.7
1.6	1.2	2.2	4.4
1.7	3.1	1.8	2.3
2.3	1.2	1.1	4.4
2.4	1.7	2.1	2.4
2.6	1.8	2.6	3.2
0.2	1.6	1.2	3.2
0.5	3.2	2.8	2.1
1.1	1.1	0.5	1.8
3.9	1.5	2.6	1.9
3.5	2.8	0.1	4.1
1.7	3.1	0.8	3.6
0.9	2.6	1.4	3
1.5	2.3	0.7	3.2
2.7	1.7	0.9	1.7
3.4	1.4	1.1	0.7
0.8	2.7	1.6	3.8
1.5	4.3	4.2	2

This is the summary table for the ANOVA test:

	S.S.	d.f.	M.S.
Between	16.007657894737	3	5.3358859649124
Within	361.04105263158	376	0.96021556550952
TOTAL	377.04871052632	379

From this table, you obtain the necessary statistics for the ANOVA:

F-ratio: 5.5569667443174

p-value: 0.00097

2= 0.042455145576263

What is your final conclusion? Use a significance level of=0.05.

1. There is a significant difference between treatments
2. These data do not provide evidence of a difference between the treatments

Explain what this tells us about the equality of means.