A random sample of five observations from three treatments produced the followin

Question

A random sample of five observations from three treatments produced the following data:

a. Calculate the grand mean.

grand mean = ________________

b. Calculate SSTR and MSTR.

SSTR = ________

MSTR = _________

c. Calculate SSE and MSE.

SSE = ________

MSE = ________

d. Specify the competing hypothesis in order to determine whether some differences exist between the population means.

H0: A B C; HA: Not all population means are equal. ?

H0: A B C; HA: Not all population means are equal. ?

H0: A = B = C; HA: Not all population means are equal. ?

e. Calculate the value of the F(df1,df2) test statistic.

test statistic = ___________

f. Using the critical value approach at the 5% significance level, what is the conclusion to the test?

A B C 29 25 22 19 16 27 26 16 16 32 20 18 30 16 16 Xa=27.2 Xb=18.6 Xc=19.8 Sa^2=25.7 Sb^2=15.8 Sc^2=22.2

Explanation / Answer

Solution

Back-up Theory

Suppose we have data of a 1-way classification ANOVA, with r rows and n observations per cell.

Let xij represent the jth observation in the ith row, k = 1,2,…,n; i = 1,2,……,r

Then the ANOVA model is: xij = µ + i + ij, where µ = common effect, i = effect of ith row, and ij is the error component which is assumed to be Normally Distributed with mean 0 and variance 2.

Now, to work out the solution,

Terminology:

Row total = xi.= sum over j of xij

Grand total = G = sum over i of xi.

Correction Factor = C = G2/N, where N = total number of observations = r x n =

Total Sum of Squares: SST = (sum over i,j of xij2) – C

Row Sum of Squares: SSR = {(sum over i of xi.2)/(n)} – C

Error Sum of Squares: SSE = SST – SSR

Mean Sum of Squares = Sum of squares/Degrees of Freedom

Degrees of Freedom:

Total: N (i.e., rn) – 1;

Error: DF for Total – DF for Row;

Rows: (r - 1);

Fobs: MSSR/MSSE;

Fcrit: upper % point of F-Distribution with degrees of freedom n1 and n2, where n1 is the DF for Row and n2 is the DF for Error

Significance: Fobs is significant if Fobs > Fcrit

Calculations [Details are given in the ANOVA table at the bottom]

Part (a)

Grand mean = _(27.2 + 18.6 + 19.8)/3 = 65.6/3 = 21.87 ANSWER

Part (b)

SSTR = 216.933 ANSWER 1

MSTR = 108.4667 ANSWER 2

Part (c)

SSE = 254.8 ANSWER 1

MSE = 21.23333 ANSWER 2

Part (d)

H0: A = B = C; HA: Not all population means are equal. ANSWER

Part (e)

Test statistic = 5.10832 ANSWER

Part (f)

Conclusion to the test

Reject H0 since the value of the test statistic exceeds 3.89.

ANOVA TABLE      

Source of

Variation

Degrees of Freedom

Sum of squares

Mean Sum

of squares

Fobs

Fcrit

Significance

Row

2

216.93

108.4667

5.10832

3.885294

significant

Error

12

254.8

21.23333

Total

14

471.73

Source of

Variation

Degrees of Freedom

Sum of squares

Mean Sum

of squares

Fobs

Fcrit

Significance

Row

2

216.93

108.4667

5.10832

3.885294

significant

Error

12

254.8

21.23333

Total

14

471.73

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