Concept explainers
In single-factor ANOVA with I treatments and J observations per treatment, let
a. Express E(
b. Determine E(
c. Determine E(
d. Determine E(SSTr) and then show that
e. Using the result of part (d), what is E(MSTr) when H0 is true? When H0 is false, how does E(MSTr) compare to σ 2?
a.
Find the expression for
Answer to Problem 10E
The expression for
Explanation of Solution
Given info:
A single factor ANOVA consists of I treatments and J observations per treatment, with
Calculation:
Denote the sample mean corresponding to the ith treatment as
Taking expectation on both sides,
Now,
Thus,
Hence,
Thus, the expression for
b.
Find the value of
Answer to Problem 10E
The value of
Explanation of Solution
Calculation:
Let
Thus, the sample mean for the ith treatment,
Now, variance of a certain quantity is
Thus,
Thus,
Thus, value of
c.
Find the value of
Answer to Problem 10E
The value of
Explanation of Solution
Calculation:
From part a,
The grand sample mean,
Thus,
Thus, value of
d.
Find the expression for
Show that
Answer to Problem 10E
The expression for
Explanation of Solution
Calculation:
For equal sample sizes, the treatment sum of squares, SSTr is:
Now,
Thus,
As a result,
Taking expectation of SSTr,
From part b,
From part c,
Substitute these values in the expression for
From the expansion of SSTr, it is seen that
Thus, the expression for
It is known that the mean square treatment (MSTr) is,
Now, divide both sides of
e.
Find the value of
Compare this value to the value of
Answer to Problem 10E
The value of
The value of
Explanation of Solution
Calculation:
The null hypothesis
When
Given that
Thus, when
Replace
When
In such a situation,
As a result,
Hence,
Thus, when
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Chapter 10 Solutions
Probability and Statistics for Engineering and the Sciences
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