MS Modules 9 and 10 Single-Factor ANOVA and Levene’s test practice exam questions
(The attached PDF file has better formatting.)
A experiment has three groups and four observations in each group.
obsv → 1 2 3 4
group 1 14 22 16 14
group 2 12 15 14 25
group 3 7 17 24 32
The groups are normally distributed with the same variance.
The null hypothesis is that the means of the groups are the same: H0: μ1 = μ2 = μ3
Question 10.1: Square of the sum of the observations
What is the square of the sum of all the observations, or x..2 ?
Answer 10.1: (14 + 22 + 16 + 14 + 12 + 15 + 14 + 25 + 7 + 17 + 24 + 32)2 = 44,944
Question 10.2: Sum of the squares of the observations
What is the sum of the squares of all the observations, or i j xij2 ?
Answer 10.2: 142 + 222 + 162 + 142 + 122 + 152 + 142 + 252 + 72 + 172 + 242 + 322 = 4,260
Question 10.3: Total sum of squares
What is SST, the total sum of squares?
Answer 10.3: 4,260 – 44,944 / 12 = 514.67
(the total sum of squares = the sum of the squares of all the observations – the square of the sum of all the observations / the number of observations)
Question 10.4: Sums of squares of group totals
What is the sum of squares of the group totals?
Answer 10.4: (14 + 22 + 16 + 14)2 + (12 + 15 + 14 + 25)2 + (7 + 17 + 24 + 32)2 = 15,112
Question 10.5: Treatment sums of squares
What is SSTr, the treatment sum of squares?
Answer 10.5: 15,112 / 4 – 44,944 / 12 = 32.67
(treatment sums of squares = the sum of squares of the group totals / the number of observations per group – the square of the sum of all the observations / the total number of observations)
Question 10.6: Error sum of squares
What is SSE, the error sum of squares?
Answer 10.6: 514.67 – 32.67 = 482.00
(error sum of squares = total sum of squares – treatment sums of squares)
Question 10.7: Total degrees of freedom
What are the total degrees of freedom?
Answer 10.7: 12 – 1 = 11
(total degrees of freedom = number of observations – 1)
Question 10.8: Degrees of freedom for the groups
What are the degrees of freedom for the groups?
Answer 10.8: 3 – 1 = 2
(degrees of freedom for the groups = number of groups – 1)
Question 10.9: Degrees of freedom for the error sum of squares
What are the degrees of freedom for the error sum of squares?
Answer 10.9: 11 – 2 = 9
(degrees of freedom for the error sum of squares = total degrees of freedom – degrees of freedom for the groups)
Question 10.10: Mean squared deviation for the groups
What is MSTr, the mean squared deviation for the groups?
Answer 10.10: 32.667 / 2 = 16.33
(mean squared deviation for the groups = treatment sums of squares / degrees of freedom for the groups)
Question 10.11: Mean squared error
What is MSE, the mean squared error?
Answer 10.11: 482 / 9 = 53.556
(mean squared error = error sum of squares / degrees of freedom for the error sum of squares)
Question 10.12: F value
What is the F value for testing the null hypothesis?
Answer 10.12: 16.333 / 53.556 = 0.305
( F value = treatment mean square / mean squared error)
Levene’s method
Levene’s method tests whether the group variances are the same. The groups are normally distributed, and the null hypothesis is that the variances are the same: H0: σ21 = σ22 = σ23 [σ2j is the variance of Group j ].
Question 10.13: Absolute deviations
What are the absolute deviations of the observations in each group?
Answer 10.13: absolute deviation = absolute value of cell value – group mean
obsv → 1 2 3 4 Mean
group 1 14.0 22.0 16.0 14.0 16.5
group 2 12.0 15.0 14.0 25.0 16.5
group 3 7.0 17.0 24.0 32.0 20.0
absolute deviations sample variance
group 1 2.5 5.5 0.5 2.5 14.333
group 2 4.5 1.5 2.5 8.5 33.667
group 3 13.0 3.0 4.0 12.0 112.667
Question 10.14: Sample variance
What is the sample variance in each group?
Answer 10.14: the sample variances indicate whether the variances appear to differ significantly; the group means do not differ much here, but the sample variances differ greatly
Question 10.15: Square of sum of absolute deviations
What is the square of the sum of the absolute deviations?
Answer 10.15: (2.5 + 5.5 + 0.5 + 2.5 + 4.5 + 1.5 + 2.5 + 8.5 + 13 + 3 + 4 + 12)2 = 3,600
Question 10.16: Sum of squares of absolute deviations
What is the sum of the squares of the absolute deviations?
Answer 10.16: (2.52 + 5.52 + 0.52 + 2.52 + 4.52 + 1.52 + 2.52 + 8.52 + 132 + 32 + 42 + 122) = 482
Question 10.17: Total sum of squares (SST) for Levene’s test
What is the total sum of squares (SST) for Levene’s test?
Answer 10.17: 482 – 3,600 / 12 = 182
(the total sum of squares = the sum of the squares of all the observations – the square of the sum of all the observations / the number of observations)
Question 10.18: Sums of squares of group totals
What is the sum of squares of the group totals for Levene’s test?
Answer 10.18: (2.5 + 5.5 + 0.5 + 2.5)2 + (4.5 + 1.5 + 2.5 + 8.5)2 + (13 + 3 + 4 + 12)2 = 1,434
Question 10.19: Treatment sums of squares
What is SSTr, the treatment sum of squares for Levene’s test?
Answer 10.19: 1,434 / 4 – 3,600 / 12 = 58.50
(treatment sums of squares = the sum of squares of the group totals / the number of observations per group – the square of the sum of all the observations / the total number of observations)
Question 10.20: Error sum of squares
What is SSE, the error sum of squares for Levene’s test?
Answer 10.20: 182 – 58.50 = 123.50
(error sum of squares = total sum of squares – treatment sums of squares)
Question 10.21: Total degrees of freedom
What are the total degrees of freedom for Levene’s test?
Answer 10.21: 12 – 1 = 11
(total degrees of freedom = number of observations – 1)
Question 10.22: Degrees of freedom for the groups
What are the degrees of freedom for the groups for Levene’s test?
Answer 10.22: 3 – 1 = 2
(degrees of freedom for the groups = number of groups – 1)
Question 10.23: Degrees of freedom for the error sum of squares
What are the degrees of freedom for the error sum of squares for Levene’s test?
Answer 10.23: 11 – 2 = 9
(degrees of freedom for the error sum of squares = total degrees of freedom – degrees of freedom for the groups)
Question 10.24: Mean squared deviation for the groups
What is MSTr, the mean squared deviation for the groups for Levene’s test?
Answer 10.24: 58.50 / 2 = 29.25
(mean squared deviation for the groups = treatment sums of squares / degrees of freedom for the groups)
Question 10.25: Mean squared error
What is MSE, the mean squared error for Levene’s test?
Answer 10.25: 123.50 / 9 = 13.722
(mean squared error = error sum of squares / degrees of freedom for the error sum of squares)
Question 10.26: F value
What is the F value for testing the null hypothesis for Levene’s test?
Answer 10.26: 29.25 / 13.722 = 2.132
( F value = treatment mean square / mean squared error)
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