In: Statistics and Probability
A sunscreen company is attempting to improve upon their formula so that it lasts in water longer. They have 4 lead scientists who each came up with a different formulas. In order to see if there is a difference in the time the sunscreen lasts the CEO collects a random sample of each of the four sunscreens the data is shown below. Test the claim that at least one sunscreen has a different lifespan in water at a 0.01 level of significance.
Sunscreen A | Sunscreen B | Sunscreen C | Sunscreen D |
64 | 33 | 57 | 69 |
80 | 58 | 38 | 73 |
61 | 44 | 47 | 57 |
55 | 62 | 47 | 71 |
44 | 37 | 61 | 47 |
69 | 35 | 46 | 54 |
The hypotheses for this ANOVA test would be:
H0:μA=μB=μC=μDH0:μA=μB=μC=μD
HA:HA: At least one mean is different. (claim)
α=0.01α=0.01
Complete the ANOVA table below: (round answers to 3 decimal places)
SS | df | MS | F | p-value | |
Between | |||||
Within |
The decision of the test is to:
The final conclusion is:
Null and alternative hypothesis:
H0: μA = μB = μC = μD
HA: At least one mean is different. (claim)
ANOVA:
Sunscreen A | Sunscreen B | Sunscreen C | Sunscreen D | Total | |
Sum | 373 | 269 | 296 | 371 | 1309 |
Count | 6 | 6 | 6 | 6 | 24 |
Sum of square, Ʃ(xᵢ-x̅)² | 750.833 | 766.833 | 345.333 | 564.833 |
Number of treatment, k = 4
Total sample Size, N = 24
df(between) = k-1 = 3
df(within) = k(n-1) = 20
df(total) = N-1 = 23
SS(between) = (Sum1)²/n1 + (Sum2)²/n2 + (Sum3)²/n3 + (Sum4)²/n4 - (Grand Sum)²/ N = 1396.13
SS(within) = SS1 + SS2 + SS3 + SS4 = 2427.83
SS(total) = SS(between) + SS(within) = 3823.96
MS(between) = SS(between)/df(between) = 465.375
MS(within) = SS(within)/df(within) = 121.392
F = MS(between)/MS(within) = 3.83367
p-value = F.DIST.RT(3.8337, 3, 20) = 0.0256
ANOVA | |||||
Source of Variation | SS | df | MS | F | P-value |
Between Groups | 1396.125 | 3 | 465.375 | 3.834 | 0.0256 |
Within Groups | 2427.833 | 20 | 121.392 | ||
Total | 3823.958 | 23 |
The decision of the test is to:
The final conclusion is: