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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
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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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 shows the important sample statistics for each book.
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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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5. 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 is called one-way analysis of variance (one-way 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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7. 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 one-way analysis of variance is the ratio of those two variances:
The actual calculation of this test statistic is tedious, so these days it is almost always done with statistical software.
test statistic F (for one-way ANOVA) =
variance between samples
variance within samples
Page See page 437 for discussion of using technology.
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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 require-ments 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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Notice that the display includes columns for F and for the P-value. 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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(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 go in the expected order.
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The End
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