Transcript Section 10-3 Analysis of Variance

10.3 Analysis of Variance
(One-Way ANOVA)
LEARNING GOAL
Interpret and carry out hypothesis tests using the method of
one-way analysis of variance.
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Hypothesis Testing for Variance
A simple random sample of 12 pages was obtained from each of
three different books: Tom Clancy’s The Bear and the Dragon, J.
K. Rowling’s Harry Potter and the Sorcerer’s Stone, and Leo
Tolstoy’s War and Peace.
The Flesch Reading Ease score was
obtained for each of those pages, and the
results are listed in Table 10.14.
The Flesch Reading Ease scoring system
results in higher scores for text that is
easier to read. Low scores are associated
with works that are difficult to read.
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Slide 10.3- 2
Our goal in this section is to use these sample data from just
12 pages of each book to make inferences about the
readability of the population of all pages in each book.
Table 10.15 shows the important sample statistics for each
book.
Do these sample data provide sufficient evidence for us to
conclude that the books by Clancy, Rowling, and Tolstoy
really do have different mean Flesch scores?
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We follow the same general principles laid out for hypothesis
testing in Section 9.1.
To begin with, we identify the null hypothesis; the mean Flesch
scores for all three books are equal. The alternative hypothesis,
then, is that the three population means are different.
The hypothesis test must tell us whether to reject or not reject
the null hypothesis.
Rejecting the null hypothesis would allow us to conclude that
the books really do have different mean Flesch scores, as we
expect.
Not rejecting the null hypothesis would tell us that the data do
not provide sufficient evidence for concluding that the mean
Flesch scores are different.
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We write the null hypothesis as
H0: μClancy = μRowling = μTolstoy
We need a hypothesis test that will allow us to determine
whether three different populations have the same mean.
The method we use is called analysis of variance,
commonly abbreviated ANOVA.
The name comes from the formal statistic known as the
variance of a set of sample values; as we noted briefly in
Section 4.3, variance is defined as the square of the sample
standard deviation, or s2.
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Definition
Analysis of variance (ANOVA) is a method of testing the
equality of three or more population means by analyzing
sample variances.
More specifically, the method used to analyze data like those
from Table 10.14 is called one-way analysis of variance (oneway ANOVA), because the sample data are separated into
groups according to just one characteristic (or factor).
In this example, the characteristic is the author (Clancy,
Rowling, or Tolstoy).
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Conducting the Test
Analysis of variance is based on this fundamental concept:
We assume that the populations all have the same variance, and
we then compare the variance between the samples to the
variance within the samples.
More specifically, the test statistic (usually called F) for oneway analysis of variance is the ratio of those two variances:
variance between samples
test statistic F (for one-way ANOVA) =
variance within samples
The actual calculation of this test statistic is tedious, so these
days it is almost always done with statistical software.
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We can interpret the F statistic as follows, using our example
of the readability of the three books:
• The variance between samples is a measure of how much
the three sample means (from Table 10.15, slide 3) differ
from one another.
• The variance within samples is a measure of how much
the Flesch Reading Ease scores for the 12 pages in each
individual sample (from Table 10.14, slide 2) differ from
one another.
• If the three population means were really all equal—as the
null hypothesis claims—then we would expect the sample
mean from any one individual sample to fall well within
the range of variation for any other individual sample.
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The test statistic (F = variance between samples/variance
within samples) tells us whether that is the case:
A large test statistic tells us that the sample means differ
more than the data within the individual samples, which
would be unlikely if the populations means really were equal
(as the null hypothesis claims). That is, a large test statistic
provides evidence for rejecting the null hypothesis that the
population means are equal.
A small test statistic tells us that the sample means differ less
than the data within the individual samples, suggesting that
the difference among the sample means could easily have
arisen by chance. Therefore, a small test statistic does not
provide evidence for rejecting the null hypothesis that the
population means are equal.
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We can quantify the interpretation of the test statistic by
finding its P-value, which tells us the probability of getting
sample results at least as extreme as those obtained, assuming
that the null hypothesis is true (the population means are all
equal).
A small P-value shows that it is unlikely that we would get
the sample results by chance with equal population means. A
large P-value shows that we could easily get the sample
results by chance with equal population means.
Like the test statistic itself, the P-value calculation is
generally done with the aid of software.
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One-Way ANOVA for Testing H0: μ1 = μ2 = μ3
=...
Step 1. Enter sample data into a statistical software
package, and use the software to determine the
test statistic (F = variance between samples /
variance within samples) and the P-value of the
test statistic.
Step 2. Make a decision to reject or not reject the null
hypothesis based on the P-value of the test
statistic:
• If the P-value is less than or equal to the
significance level, reject the null hypothesis
of equal means and conclude that at least
one of the means is different from the others.
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One-Way ANOVA for Testing H0: μ1 = μ2 = μ3
=...
Step 2. (cont.)
• If the P-value is greater than the significance
level, do not reject the null hypothesis of
equal means.
This method is valid as long as the following requirements are met: The populations have distributions that
are approximately normal with the same variance, and
the samples from each population are simple random
samples that are independent of each other.
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EXAMPLE 1 Readability of Clancy, Rowling,
Tolstoy
Given the readability scores listed in Table
10.14 and a significance level of 0.05, test
the null hypothesis that the three samples
come from populations with means that are
all the same.
Solution: We begin by checking the
requirements for using one-way analysis
of variance. As noted earlier, close
examination of the data suggests that each
sample comes from a distribution
that is approximately normal. The sample standard deviations are
not dramatically different, so it is reasonable to assume that the
three populations have the same variance.
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EXAMPLE 1 Readability of Clancy, Rowling,
Tolstoy
Solution: (cont.)
The samples are simple random samples
and they are all independent. The
requirements are therefore satisfied.
We now test the null hypothesis that the
population means are all equal (H0: μ1 = μ2
= μ3).
The table on the next slide shows the
resulting display from Excel; other
software packages will give similar
displays.
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Slide 10.3- 14
Notice that the display includes columns for F and for the Pvalue. These are the two items of interest to us here, which
we interpret as follows:
• F is the test statistic for the one-way analysis of variance (F =
variance between samples/variance within samples). Notice
that it is much greater than 1, indicating that the sample means
differ more than we would expect if all the population means
were equal.
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Slide 10.3- 15
(cont.)
• The P-value tells us the probability of having obtained such an
extreme result by chance if the null hypothesis is true. Notice
that the P-value is extremely small—much less than the value
of 0.05 necessary to reject the null hypothesis at the 0.05 level
of significance (and also much less than the 0.01 necessary to
reject at the 0.01 level of significance).
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We conclude that there is sufficient evidence to reject the
null hypothesis, which means the sample data support the
claim that the three population means are not all the same.
Note that we have not concluded that the three books have
the readability order that we expect—Rowling as easiest
and Tolstoy as hardest—because the hypothesis test shows
only that the readabilities are unequal.
Nevertheless, our expectation seems reasonable since the
sample means in Table 10.15 go in the expected order.
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The End
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