Question

In: Statistics and Probability

A sunscreen company is attempting to improve upon their formula so that it lasts in water...

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:

  • reject H0H0
  • do not reject H0H0

The final conclusion is:

  • There is enough evidence to reject the claim that at least one sunscreen lasts a different amount of time.
  • There is enough evidence to support the claim that at least one sunscreen lasts a different amount of time.
  • There is not enough evidence to reject the claim that at least one sunscreen lasts a different amount of time.
  • There is not enough evidence to support the claim that at least one sunscreen lasts a different amount of time.

Solutions

Expert Solution

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:

  • As P-value = 0.0256 > 0.01, do not reject H0.

The final conclusion is:

  • There is not enough evidence to support the claim that at least one sunscreen lasts a different amount of time.

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