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Chapter 14

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Chapter 14: Analysis of Variance
Analysis of variance
Determines whether differences exist between population means
Procedure works by analyzing the sample variance
Population means = treatment means
14.1: One-Way Analysis of Variance
Procedure that tests to determine whether differences exist between two or more population
means
Technique analyzes the variance of the data to determine whether we can infer that the
population means differ
Experimental design is a determinant in identifying the proper method to use
One-way analysis of variance
Procedure to apply when the samples are independently drawn
Figure 14.1: Sampling Scheme for Independent Samples page 526
Confirm that the data are interval and that the problem objective is to compare populations
Parameters are the four* population means: and
Null hypothesis will state that there are no differences between the population means
o
Analysis of variance determines whether there is enough statistical evidence to show that the
null hypothesis is false
Alternative hypothesis will always specify the following:
o
Next step is to determine the test statistic
Response variable
The variable X is called the response variable
Responses
The values of the response variable X
Experimental unit The unit we measure is called the experimental unit
Factor
The criterion by which we classify the population
Each population is called a factor level
Test Statistic
If null hypothesis is true, the population means would all be equal and we would then expect
the sample means to be close to one another
If the alternative hypothesis is true, there would be large differences between some of the
sample means
Between-treatments variation
Measures the proximity of the sample means to each other
Denoted SST (sum of squares for treatments)
Sum of squares for treatments (SST)
If the sample means are close to each other, all of the sample means would be close to the
grand mean; as a result SST would be small
SST achieves its smallest value(zero) when all the sample means are equal
A small value of SST supports the null hypothesis
Steps
Compute the sample means ( and the grand mean (X double bar)
Compute the sample sizes ( and the total of the sample sizes to get n
Then calculate SST
If large differences exist between the sample means, making SST a large value, it is reasonable
to reject the null hypothesis
Important question becomes: how large does the statistic have to be for us to justify rejecting
the null hypothesis? To answer question, important to know how much variation exists in response variables
Within-treatments variation
Provides a measure of the amount of variation in the response variable that is not caused by the
treatments
Denoted by SSE (sum of squares for error)
Sum of squares for error (SSE)
Each of the k components of SSE is a measure of the variability of that sample
If we divide each component by , we obtain the sample variances. We can express this be
rewriting SSE as:
SSE is the combined or pooled variation of the k samples
A condition for SSE requires that the population variances be equal
o
Steps
Calculate the sample variances (
Calculate SSE
Next step is to compute quantities called the mean squares Sampling Distribution of the Test Statistic
Test statistic is F-distributed with k-1 and n-k degrees of freedom provided that the response
variable is normally distributed
Steps
Calculate degrees of freedom
Calculate MSE and MST
Calculate test statistic F
Rejection Region and p-Value
Purpose of calculating the F-statistic is to determine whether the value of SST is large enough to
reject the null hypothesis
If SST is large, F will be large
We reject the null hypothesis only if:
o
Results of the analysis of variance are usually reported in an ANOVA table
Table 14.2: ANOVA Table for the One-Way Analysis of Variance page 532 If SST explains a significant portion of the total variation, we conclude that the population
means differ
Completely randomized design
When the data are obtained through a controlled experiment in the one-way analysis of
variance, we call the experimental design the completely randomized design of the analysis of
variance
Checking the Required Conditions
The F-test of the analysis of variance requires that the random variable be normally distributed
with equal variances
Normality requirement is easily checked graphically by producing the histograms for each
sample
The equality of variances is examined by printing the sample standard deviations or variances
The similarity of sample variances allows us to assume that the population variances are equal
Violation of the Required Conditions
If the data are not normally distributed, we can replace the one-way analysis of variance with its
nonparametric counterpart, which is the Kruskal-Wallis Test
If the population variances are unequal, we can use several methods to correct the problem
Can We Use the t-Test or the Difference between Two Means Instead of the Analysis of Variance?
Analysis of variance test determines whether there is evidence of differences between two or
more population means
The t-test of determines whether there is evidence of a difference between two
population means
The question arises, can we use t-tests instead of the analysis of variance? In other words,
instead of testing all the means in one test as in the analysis of variance, why not test each pair
of means?
There are two reasons why we don’t use multiple t-tests instead of one F-test:
o We would have to perform many more calculations
o Conducting multiple tests increases the probability of making Type I errors
When we want to compare more than two populations of interval data, we use the analysis of
variance
Can We Use the Analysis of Variance Instead of the t-Test of the Difference between Two Means?
If we want to determine whether is greater than we cannot use the analysis of variance
because this technique allows us to test for a difference only
If we want to test to determine whether one population mean exceed the other, we must use
the t-test Moreover, the analysis of variance requires that the population variance are equal and if they
are not, we must use the unequal variances test statistic
Relationship between the F-Statistic and the t-Statistic
If we square the quantity of the t-statistic, the result is the F-statistic
o
Developing an Understanding of Statistical Concepts
The F-test of the independent samples’ single-factor analysis of variance is an extension o

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