Question

1. Problem 1: To compare the effectiveness of three different types of phosphorescent coatings of airplane instrument dials, eight dials each are coated with the three types. Then the dials are illuminated by an ultraviolet light, and the following are the number of minutes each glowed after the light source was shut off: Туре 3 Type 1 Type 2 52.9 58.4 71.3 62.1 55 66.6 57.4 59.8 63.4 50 62.5 64.7 59.3 64.7 75.8 59.9 61.2 65.6 60.8 54.7 72.9 53.1 58.4 67.3 a) Is there a difference in the effectiveness of the three coatings? Use a -0.01 State and check assumptions used in a). b) c) Use Tuckey method with a-0.01 to analyze the means for the three different coating types

Answer 1

	Type 1	Type 2	Type 3	Total
Sum	456.8	473.4	547.6	1477.8
Count	8	8	8	24
Mean, Sum/n	57.1	59.175	68.45
Sum of square, Ʃ(xᵢ-x̅)²	144.48	81.155	134.38

Number of treatment, k = 3

Total sample Size, N = 24

df(between) = k-1 = 2

df(within) = N-k = 21

df(total) = N-1 = 23

SS(between) = (Sum1)²/n1 + (Sum2)²/n2 + (Sum3)²/n3 - (Grand Sum)²/ N = 584.41

SS(within) = SS1 + SS2 + SS3 = 360.015

SS(total) = SS(between) + SS(within) = 944.425

MS(between) = SS(between)/df(between) = 292.205

MS(within) = SS(within)/df(within) = 17.1436

a) Null and Alternative Hypothesis:

Ho: µ1 = µ2 = µ3

H1: At least one mean is different.

F = MS(between)/MS(within) = 17.0446

p-value = F.DIST.RT(17.0446, 2, 21) = 0.0000

As p-value = 0.0000 < 0.01, we reject the null hypothesis.

ANOVA
Source of Variation	SS	df	MS	F	P-value
Between Groups	584.4100	2	292.2050	17.0446	0.0000
Within Groups	360.0150	21	17.1436
Total	944.4250	23

b) Assumptions:

• The groups are independent of each other.
• The data in each cell should be approximately normally distributed.
• The variance in each cell should be similar.

c) Tukey's method:

Q statistic at α = 0.01, k = 3, N-k = 21, Q = 4.62

Samples in each treatment, n =8

Comparison	Absolute Diff. = |xi - xj|	Critical Range, CV = Q* √(MS(Within)/n)	Results
µ1-µ2	2.075	6.7631	Means are not different
µ1-µ3	11.35	6.7631	Means are different
µ2-µ3	9.275	6.7631	Means are different