# In a study of the effects of stress on illness, a researcher taillied the number of colds people contracted during a 6-month period as a function of the amount of stress they reported during the...

In a study of the effects of stress on illness, a researcher taillied the number of colds people contracted during a 6-month period as a function of the amount of stress they reported during the same period. There were three stress levels: minimal, moderate, and high stress. The sums of squares appear in the following ANOVA summary table. The mean for each condition and the number of subjects per condition and the number of subjects per condition are also noted.

Source df SS MS f

Between groups 22.167

Within groups 14.750

Total 36.917

Stress level Mean N

minimal 3 4

moderate 4 4

maximum 6 4

A) Complete the ANOVA summary table

B) Is F obt significant at a=.05, or at a=.01

C) Perform post hoc comparisons if necessary.

D)What conclusions can be drawn from the F-ratio and the post hoc comparisons?

E) What is the effect size? What does it mean?

F) Graph the means.

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### 1 Answer

Tally of the number of colds people contracted in a specific time period with the amount of their stress level:

Stress level Mean n

--------------- ------- -------

Minimum 3 4

Moderate 4 4

Maximum 6 4

-------------------------------------------

(a) ANOVA summary table of the number of colds people contracted in a specific time period with the amount of their stress level:

Source df (formula) SS MS F

--------- -------------- -------- -------- ---------

Between 2 (k-1) 22.167 11.083 6.76

Within 9 (N-k) 14.750 1.639

Total 11 (N-1) 36.917

-----------------------------------------------------------------------------

(b) F_(obt) =6.76

Critical values of F_(2,9): 4.26 (p=0.05); 8.02 (p=0.01)

Therefore, F_(obt) is significant at p<0.05, but not significant at p<0.01.

(c) Post hoc comparison by Tukey’s HSD test:

q(3,9)=3.95(.05)

HSD_(.05)=q*sqrt(MSwithin/n)=3.95*sqrt(1.639/4)=2.53

Thus, the mean difference between any two samples must be at least 2.53 to be significant. Using this value, we can make the following conclusions:

- Minimum stress is not significantly different from moderate stress (
*M*A -*M*B =1.00) - Moderate stress is not significantly different from maximum stress (
*MB-MC*=2.00)

3. Minimum stress is significantly different from maximum stress (*M*A - *M*C =3.00)

(d) The amount of stress had a significant effect on catching a cold for those who experienced maximum stress, compared to those who experienced minimum stress. It is however not significant for intermediate comparisons.

(e) The effect size (`eta^2` =SSbetween/SStotal) is 60%. Thus, knowing the stress level can explain 60% of the frequency of catching a cold in the maximum stress level, when compared to the minimum stress level.

(f) The plot is shown in the attached figure.

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