After random assignment of ten loads to each brand, amount of dirt (measured in

Question

After random assignment of ten loads to each brand, amount of dirt (measured in the milligrams) removed was determined. Test the hypothesis there is a significant difference in the population means of the four brands? a) Specify the dependent variable, the independent variable and the treatments b) State the null and alternative hypotheses for the test of interest. c) Find Mean of the squares among groups (MSA) d) Find Mean of the squares within groups (MSW)? Also called Mean of the squares due to error (MSE).

Explanation / Answer

              The dependent Variable is-The amount of dirt removed.

              The treatments are-The different brands i.e A,B,C,D

H0: There is no significant difference in the mean amount of dirt removed amongst the four brands i.e µA=µB=µC=µD

Against the alternative Hypothesis

H1: There is a significant difference in the mean amount of dirt removed amongst the four brands i.e µAµBµCµD

MSA= SSA/(k-1)

    Where SSA=ki=1ni(Yi. –Y..)2 or   SSA =ki=1yi.2/ni -y..2/n

    And k is the number of treatments.

    ni is the number of observation in the ith treatment.

Now calculating using the given values we have:

y.. =643

n=40 (Total number of observations)

ki=1yi.2/ni =10459.5

y..2/n = 10336.225

Therefore, SSA=123.275

And MSA = 123.275/3 =41.091667

MSW = SSW/(n-k)

We know that the total sum of squares is given by

SST = KI=1nij=1(yij – Y..)2 or SST = KI=1nij=1yij2 - y..2/n

Calculating using the above formula we have

KI=1nij=1yij2 =10673

y..2/n = 10336.225

SST=336.225

And the residual sum of squares is given by SSR =SST -SSB

Putting the values from above we have

SSR = 336.225-123.275=212.95

Therefore, the mean of the squares due to errors is given by

MSE = SSR/(n-k)

i.e MSE = 5.9152

**The symbols used denote the following

Yi. – The mean of the observation of the ith class.

Y.. – The overall mean.

yi. –The sum of the observations of the ith class.

y..- The overall sum.

yij – The jth observation of the ith class.

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