Question

1. . Observation data for the three L1, L2 and L3  locations are presented in the table as below

L1	L2	L3
5 3 6 7 2	8 7 5 6	13 11 9 10

using α = 5%, test whether the conditions at the three locations are the same?

Answer 1

	L1	L2	L3	Total
Sum	23	26	43	92
Count	5	4	4	13
Mean, Sum/n	4.6	6.5	10.75
Sum of square, Ʃ(xᵢ-x̅)²	17.2	5	8.75

Null and Alternative Hypothesis:

Ho: µ1 = µ2 = µ3

H1: At least one mean is different.

Number of treatment, k = 3

Total sample Size, N = 13

df(between) = k-1 = 2

df(within) = N-k = 10

df(total) = N-1 = 12

SS(between) = (Sum1)²/n1 + (Sum2)²/n2 + (Sum3)²/n3 - (Grand Sum)²/ N = 85.97308

SS(within) = SS1 + SS2 + SS3 = 30.95

SS(total) = SS(between) + SS(within) = 116.9231

MS(between) = SS(between)/df(between) = 42.98654

MS(within) = SS(within)/df(within) = 3.095

F = MS(between)/MS(within) = 13.8890

p-value = F.DIST.RT(13.889, 2, 10) = 0.0013

Decision:

P-value < α, Reject the null hypothesis.

Conclusion:

There is not enough evidence to conclude that all the three locations are the same at 0.05 significance level.

ANOVA
Source of Variation	SS	df	MS	F	P-value
Between Groups	85.9731	2	42.9865	13.8890	0.0013
Within Groups	30.9500	10	3.0950
Total	116.9231	12