Question

 

The physician wanted to know if conception periods were different due to the degrees of coffee drinking habit. Thus she collected data on the number of month to conception for 3 light coffee drinkers (1-2 cups per day), 4 moderate coffee drinkers (3-4 cups per day), and 3 heavy coffee drinkers (more than 4 cups per day). The table below summarizes the month to conception as a function of the three groups (light, moderate, and heavy coffee drinkers).

Months to conception for people with different coffee drinking
Light coffee drinkers	Moderate coffee drinkers	Heavy coffee drinkers
4	5	8
5	6	9
6	6	10
	7

You are asked to analyze the above data with one-way ANOVA. Please answer the following questions with regards to one-way ANOVA on the above data.

1. State the null and alternative hypotheses (2pts)

2. Identify the degrees of freedom(s). (2pts)

3. Identify the F critical based on the degrees of freedoms above by consulting with F table (2pts)

4. Here is the summary table presenting SSwithin, SS between, and SStotal. Please compute F value based on SSs and Degrees of Freedoms that you identified in the above. This should require computing MSwithin, MSbetween, and F. (2pts each for these three values)

I have these answers to 1-4

1. The hypothesis being tested is:

 

H0: µ1 = µ2 = µ3

Ha: Not all means are equal

2. df = 9

3. F critical = 4.74

4.

Source	SS	df	MS	F	p-value
Treatment	26.40	2	13.200	15.40	.0027
Error	6.00	7	0.857

	SS	DF	MS	F
Between	26.4	Use answers for 2 above	??	??
Within	6.00	Use answers for 2 above	??
Total	32.4

5. State your decision on the hypothesis based on the calculation above that led to the F value (2pt).

6. You tried to replicate above study with the exact same number of participants, and you got the following results:

Months to conception for people with different coffee drinking
Light coffee drinkers	Moderate coffee drinkers	Heavy coffee drinkers
2	2	7
5	5	9
8	7	11
	10

You may notice that the means of the three groups are the same as the previous results with three means for L, M, H conditions being 5, 6, and 9 respectively. Please fill out the blank sections of the summary table below based on your understanding/knowledge of how One-way ANOVA works. If you get the correct F value, you will get 9pts. If you do not get the correct F value, 1 pt is given for each of the 6 values (SSs, DFs, and MSs) that is correct.

	SS	DF	MS	F
Between	??	??	??	??
Within	??	??	??
Total	86.4

7. Please compare the two results (and two summary tables), and explain why this difference in F values occurred across the two studies. The explanation should involve the concept of variance between and variance within. (3pt)

Answer 1

For the first case

Anova: Single Factor
SUMMARY
Groups	Count	Sum	Average	Variance
Column 1	3	15	5	1
Column 2	4	24	6	0.666667
Column 3	3	27	9	1
ANOVA
Source of Variation	SS	df	MS	F	P-value	F crit
Between Groups	26.4	2	13.2	15.4	0.002733	4.737414
Within Groups	6	7	0.857143
Total	32.4	9

for the second case

Anova: Single Factor
SUMMARY
Groups	Count	Sum	Average	Variance
Column 1	3	15	5	9
Column 2	4	24	6	11.33333
Column 3	3	27	9	4
ANOVA
Source of Variation	SS	df	MS	F	P-value	F crit
Between Groups	26.4	2	13.2	1.54	0.279082	4.737414
Within Groups	60	7	8.571429
Total	86.4	9

7)the results differ because of the difference in case of within variability.For the second case this is quite high leading to larger MSE and hence smaller F-value.As a result the null hypothesis is not rejected. But the opposite scenario is seen for the first case.