By NEAS - 7/2/2024 3:38:46 PM
MS Module 14 Two-factor ANOVA interaction practice exam questions
(The attached PDF file has better formatting.)
[The practice problems in the 24 modules explain the statistical procedures; the practice exam questions in this thread shows what you will be asked on the final exam.]
A two-factor classification table has two rows, two columns, and two observations in each cell.
Column 1 Column 2
Row 1 32; 33 32; 27
Row 2 14; 29 20; 22
We use analysis of variance to test
● whether the Row 1 mean differs from the Row 2 mean
○ the null hypothesis is that the row means are equal
● whether the Column 1 mean differs from the Column 2 mean
○ the null hypothesis is that the column means are equal
● whether the interaction effects are significant
○ the null hypothesis is that the interaction effects are zero
Question 14.1: Square of sum of observations
What is the square of the sum of all the observations, or x...2 ?
Answer 14.1: (32 + 33 + 32 + 27 + 14 + 29 + 20 + 22)2 = 43,681
Question 14.2: Correction factor
What is the correction factor used for the total sum of squares and the treatment sums of squares (for both rows and columns)?
Answer 14.2: 43,681 / 8 = 5,460.125
(correction factor = the square of the sum of the observations / the number of observations)
Question 14.3: Sum of squares of observations
What is the sum of the squares of all the observations, or i j k xijk2 ?
Answer 14.3: (322 + 332 + 322 + 272 + 142 + 292 + 202 + 222) = 5,787
Question 14.4: Sum of squares of totals by cell
What is the sum of the squares of the totals in each cell, or i j xij2 ?
Answer 14.4: (32 + 33)2 + (32 + 27)2 + (14 + 29)2 + (20 + 22)2 = 11,319
Question 14.5: Sum of squares of row totals
What is the sum of the squares of the row totals, or j xi..2
Answer 14.5: (32 + 33 + 32 + 27)2 + (14 + 29 + 20 + 22)2 = 22,601
Question 14.6: Sum of squares of column totals
What is the sum of the squares of the column totals, or j x.j.2
Answer 14.6: (32 + 33 + 14 + 29)2 + (32 + 27 + 20 + 22)2 = 21,865
Question 14.7: Total sum of squares
What is SST, the total sum of squared deviations?
Answer 14.7: 5,787 – 5,460.125 = 326.875
(total sum of squares = the sum of the squares of all the observations – the correction factor)
Question 14.8: SSA
What is SSA, the sum of squared deviations for the i dimension (the rows)?
Answer 14.8: 22,601 / 4 – 5,460.125 = 190.125
(SSA = the sum of the squares of the row totals / observations per row – the correction factor)
Question 14.9: SSB
What is SSB, the sum of squared deviations for the j dimension (the columns)?
Answer 14.9: 21,865 / 4 – 5,460.125 = 6.125
(SSB = the sum of the squares of the column totals / observations per column – the correction factor)
Question 14.10: Error sum of squares
What is SSE, the error sum of squared deviations?
Answer 14.10: 5,787 – 11,319 / 2 = 127.50
(error sum of squares = the sum of the squares of the observations – the sum of the squares of the totals in each cell / number of observations by cell)
Question 14.11: SSAB
What is SSAB, the sum of squared deviations for the interaction?
Answer 14.11: 326.875 – 190.125 – 6.125 – 127.50 = 3.125
Question 14.12: Degrees of freedom
What are the degrees of freedom for the rows (SSA)?
Answer 14.12: 2 – 1 = 1
(the degrees of freedom for the rows = number of rows – 1)
Question 14.13: Degrees of freedom
What are the degrees of freedom for the columns (SSB)?
Answer 14.13: 2 – 1 = 1
(the degrees of freedom for the columns = number of columns – 1)
Question 14.14: Degrees of freedom
What are the degrees of freedom for the interaction effects (SSAB)?
Answer 14.14: (2 – 1) × (2 – 1) = 1
(the degrees of freedom for the interaction effects = (number of rows – 1) × (number of columns – 1)
Question 14.15: Degrees of freedom
What are the degrees of freedom for the total sum of squares (SST)?
Answer 14.15: 8 – 1 = 7
(the degrees of freedom for the total sum of squares = number of observations – 1)
Question 14.16: Degrees of freedom
What are the degrees of freedom for the error sum of squares (SSE)?
Answer 14.16: 7 – 1 – 1 – 1 = 4
(degrees of freedom for SSE = degrees of freedom for SST – degrees of freedom for SSA – degrees of freedom for SSB – degrees of freedom for SSAB)
Question 14.17: Mean squared deviation for the rows
What is MSA, the mean squared deviation for the rows?
Answer 14.17: 190.125 / 1 = 190.125
(MSA = SSA / degrees of freedom)
Question 14.18: Mean squared deviation for the columns
What is MSB, the mean squared deviation for the columns?
Answer 14.18: 6.125 / 1 = 6.125
(MSB = SSB / degrees of freedom)
Question 14.19: Mean squared deviation for the interaction
What is MSAB, the mean squared deviation for the interaction?
Answer 14.19: 3.125 / 1 = 3.125
(MSAB = SSAB / degrees of freedom)
Question 14.20: Mean squared error
What is MSE, the mean squared error?
Answer 14.20: 127.50 / 4 = 31.875
(MSE = SSE / degrees of freedom)
Question 14.21: F value
What is fA, the f value for testing significance of the row differences?
Answer 14.21: 190.125 / 31.875 = 5.965
(fA, the f value for testing significance of the row differences, is MSA / MSE)
Question 14.22: F value
What is fB, the f value for testing significance of the column differences?
Answer 14.22: 6.125 / 31.875 = 0.192
(fB, the f value for testing significance of the column differences, is MSB / MSE)
Question 14.23: F value
What is fAB, the f value for testing significance of the interaction effect?
Answer 14.23: 3.125 / 31.875 = 0.098
(fAB, the f value for testing significance of the interaction effect, is MSAB / MSE)
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