Question

In a study of the physiological stress resulting from operating hand-held chain saws,
experimenters measured the kickback that occurs when a saw is used to cut a 3-inch thick
synthetic fiber board. The variable of interest was the angle (in degrees) to which the saw
is deflected when it begins to cut the board. Below are summary of the angles of deflection
recorded for 5 random saws from each of 4 manufacture’s models.
Chain Saw Model
     A         B                    C                 D
?1 = 165    ?2 = 215        ?3 = 245    ?4 = 155
?1 = 5         ?2 = 5            ?3 = 5            ?4 = 5
ΣΣ ?^2 ??= 33,120
(a) Write an ANOVA model for the study above, and state the underlying assumptions.
(b) Formulate the null and the alternative hypotheses.
(c) Construct an ANOVA table.
(d) State the value of the test statistic, and the critical value for a 5% level of
significance. Would you reject ?0 or fail to reject ?0 at 5% level of significance?
(e) What do you conclude about the mean kickbacks for the 4 types of saws?
(f) What percentage of the variability in kickbacks is is attributable to the ANOVA
model?
(g) Use Tukey’s procedure to identify the significant differences in the mean kickbacks
for the 4 types of saws, and state your conclusion.
(h) Use the least conservative LSD procedure to identify the differences in the mean
kickbacks for the 4 types of saws, and state your conclusion.
(i) The models A and D are lightweight chain saws for home use and that B and C are
heavy-duty industrial types. The investigator might want to determine if the
kickback from the home type is the same as the kickback from the industrial type.
Hence the contrast of interest is 0.5 ?? − 0.5 ?? − 0.5 ?? + 0.5 ?? . Test the
contrast using the Scheffe’s test for general contrasts, and write your conclusion.
Be sure to state the null and the alternative hypotheses.

Answer 1

Solution:

a. The ANOVA model for the study is

The assumption are:-

i.

ii. are independently normally distributed with a mean of zero and a common variance of .

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b. Null Hypothesis (Ho):

Alternative Hypothesis (Ha): At least one inequality or

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c. ANOVA table

Source	df	SS	MS
Among groups	a - 1 = 3	SSa = A - CF = 1080	MSa = SSa/(a-1) =360
Within groups	a (n-1) = 16	SSe = T - A = 1620	MSe = SSe/a(n-1) = 101.25

Calculations

T = = 33,120 (Given)

A = T1^2 + T2^2 + T3^2 + T4^2

A = (165)^2 + (215)^2 + (245)^2 + (155)^2

A = 27,225 + 46,225 + 60,025 + 24,025

A = 157,500

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d. Test Statistics

F = MSa/MSe

F = 360/101.25

F = 3.56

Using F-tables, the critical value at 0.05 level of significance and (3, 16) Degrees of freedom

F (0.05, 3, 16) = 3.239

Since test statistics is greater than the critical value, we reject the null hypothesis.

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e. Since null hypothesis is rejected, we can conclude that there is significant difference among the average kickbacks of the four types of saws.

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f. The percentage of the variability in kickbacks is is attributable to the ANOVA model is

R-square = 1 - SSe/SSt

R-square = 1 - ((1620)/(1620 + 1080))

R-square = 1 - 0.40

R-square = 0.60