Question

In: Statistics and Probability

Seasonal affective disorder (SAD) is a type of depression during seasons with less daylight (e.g., winter...

Seasonal affective disorder (SAD) is a type of depression during seasons with less daylight (e.g., winter months). One therapy for SAD is phototherapy, which is increased exposure to light used to improve mood. A researcher tests this therapy by exposing a sample of patients with SAD to different intensities of light (low, medium, high) in a light box, either in the morning or at night (these are the times thought to be most effective for light therapy). All participants rated their mood following this therapy on a scale from 1 (poor mood) to 9 (improved mood). The hypothetical results are given in the following table.

   Light Intensity
Low Medium High
Time of
Day
Morning 5 5 7
6 6 8
3 4 6
7 7 8
5 9 5
6 8 8
Night 5 6 8
8 8 7
6 7 6
7 5 8
4 9 7
3 8 6

(a) Complete the F-table and make a decision to retain or reject the null hypothesis for each hypothesis test. (Round your answers to two decimal places. Assume experimentwise alpha equal to 0.05.)

Source of
Variation
SS df MS F
Time of day
Intensity
Time of
day ×
Intensity
Error   
Total


State the decision for the main effect of the time of day.

Retain the null hypothesis.Reject the null hypothesis.     


State the decision for the main effect of intensity.

Retain the null hypothesis.Reject the null hypothesis.     


State the decision for the interaction effect.

Retain the null hypothesis.Reject the null hypothesis.     


(b) Compute Tukey's HSD to analyze the significant main effect.

The critical value is  for each pairwise comparison.

Solutions

Expert Solution

Output using excel:

Anova: Two-Factor With Replication
SUMMARY Low Medium High Total
Morning
Count 6 6 6 18
Sum 32 39 42 113
Average 5.333333 6.5 7 6.277778
Variance 1.866667 3.5 1.6 2.565359
Night
Count 6 6 6 18
Sum 33 43 42 118
Average 5.5 7.166667 7 6.555556
Variance 3.5 2.166667 0.8 2.496732
Total
Count 12 12 12
Sum 65 82 84
Average 5.416667 6.833333 7
Variance 2.44697 2.69697 1.090909
ANOVA
Source of Variation SS df MS F P-value
Time of day 0.69 1 0.69 0.31 0.5817
Intensity 18.17 2 9.08 4.06 0.0276
Time of day x Intensity 0.72 2 0.36 0.16 0.8518
Error 67.17 30 2.24
Total 86.75 35

Decision for the main effect of the time of day:

Retain the null hypothesis.

---

Decision for the main effect of intensity:

Reject the null hypothesis.     

---

Decision for the interaction effect:

Retain the null hypothesis.

---

(b) Compute Tukey's HSD to analyze the significant main effect.

At α = 0.05, k = 3, N-K = 30, Q value = 3.49

Critical value, CV = Q*√(MSW/n) = 3.49*√(2.24/12) = 1.51


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