Hypothesis Testing - Difference Between Means
Hypothesis Testing - Difference Between Means
To illustrate the hypothesis test appropriate for the difference between sample means, consider the following pricing experiment. Three prices are under consideration for a new product: 39 cents, 44 cents, and 49 cents. To determine the influence that the various price levels will have on sales, three samples of five supermarkets are randomly selected from the geographic area of interest. Each sample is assigned one of the three price levels. Figure 14-3 shows the resulting sales levels in both graphic and tabular form. The 390 stores, the first row, had sales of 8, 12, 10, 9, and 11, averaging 10 units. The 44<f stores, the second row, averaged 8 units; and the 49<t stores, the third row, averaged 7 units. Obviously, the determination of the optimal price will require an extensive analysis involving a host of considerations. However, before the analysis begins, it is appropriate to consider the hypothesis that price levels have no effect on sales. Under the null hypothesis, differences between sample means could be caused by the fact that only a sample of five was employed for each price level. The alternate hypothesis is that there is a price effect, that sales would not be the same for each of the price levels if they were applied to all stores.
Variation Between Stores
Two other estimates of the population also are available and provide the basis for the test of the hypothesis that price levels have no effect on sales. The first is to focus on the variation between stores at different price levels [Xl = 10, X2 = 8, and X3 = 7). The second is to consider the variation between stores at the same price level (for example, the stores with the 39 cents price level had sales of 8, 12, 10, 9, and 11). Under the null hypothesis that price levels have no effect on sales, each of these estimates should be similar. If the estimate based on variation between stores of different price levels is inflated, then doubt will be cast on the null hypothesis.
Variation Between Stores at Different Price Levels
The first variance estimate is based on the variation between the sample mean values of each row (price level), which is calculated as follows:
SSr = sum of squares between rows (price level) = variation explained by the row (price level)
= n
2 (Xp - X)2
5[(10 - 8i)2 + (8 - 8§P/(7 - 8i)2] = 23.3
where
Xp = mean sales at price level p (i. e., Xx = 10) X = overall mean (i. e., X = 85)
n = number of observations at each price level (n = 5) p = price level (p = 1, 2, 3) r = number of rows (price levels) i. e., r = 3
Clearlyj_as the difference between means gets larger so will the sum of terms
(Xp - xp.
The first variance estimate is termed MSS, and is obtained by dividing the SSrby its associated degrees of freedom (d. f.), which here is the number of rows less one:
MSSr = mean sum of squares between rows (price levels) = an estimate of the variance among stores = SSr/d. f. = (23.3)/2 = 11.65
Again, the determination of the degrees of freedom need not concern us here. In fact, the actual calculation formula for SSr and the other terms is also of little consequence. As a practical matter, the computer program or a reference statistics text can provide the appropriate formula for degrees of freedom, sum of squares, and other terms.8 The intent here is to provide a feel for the terms and expressions that will be encountered in this rather commonly used statistical hypothesis test and to provide a conceptual understanding of the test itself, rather than to teach the reader to actually make the raw calculations.
Variation Within Price Levels
The second variance estimate is based on the variation within each row (price level), which is
SSU = sum of squares unexplained by the row (price level) = variation within the rows (price levels)
n r
= 2 2 tx(p - xp)2
i=lp=l
= (8 - 10)2 + (12 - lO)2 + . . . + (9 - 7)2 + (7 - 7)2 = 34
where Xlp equals the sales of observations i at price level p. Note that SSU is based on the squared deviation from the row mean.
The second variance estimate is termed MSSU and is generated by dividing SSU by its associated degrees of freedom, which is here equal to r(n - 1) or 12:
MSSU = mean sum of squares unexplained by the row (price level) = an estimate of the variance among stores = SSu/d. f. = 34/12 = 2.8
ANOVA Table
These expressions are summarized in Table 14-3, which presents an analysis of variance and is termed an ANOVA table. The ANOVA table is a conventional way to present a hypothesis test regarding the difference between several means. The table indicates, at the left, the source of the variation. The first row summarizes the determination of MSSr, which is based on the variation between rows (the explained variation, or the variation explained by the price level).
The second row summarizes the determination of MSSU, which is based on the within-row variation (variation unexplained by the price levels). The third row represents the total variation based on the deviations of the individual sales results from the overall mean. Notice that the total variation (53.3) is equal to the sum of SSr and SSU. All the variation is accounted for.9
F-Statistic and p-Value
We now consider the ratio of the two estimates of the (population) variance of the store sales.10 This ratio is termed an F-ratio or F-statistic:
^he reader also might note that the total degrees of freedom, which is the total sample size of 15 less one, or 14, is equal to the sum of the degrees of freedom associated with the first two rows of the ANOVA table.
10The estimate of population variance introduced at the outset was the total variation divided by its degrees of freedom. In this case, it was 3.8, which is in between the other two estimates.
MSSr = 11.65 MSSU 2.8
= 4.16
If the null hypothesis that price levels have no effect on sales is true, then our variance estimates using the difference between the sample means, MSSr, should be the same as that based upon the within-row (price level) variations. The F-ratio should then be close to one. If, however, the hypothesis is not true and the different price levels generate different sales levels, the MSSr term will have two components. One component will reflect the variance among stores; the other will reflect the different price effects. As a result, the F-ratio will tend to become large.
The p-value is the probability that the F-ratio would be larger than 4.66, given the null hypothesis. To generate the p-value, the F-probability distribution is used. Associated with each F-ratio are the numerator (MSSr) degrees of freedom (2), and the denominator (MSSU) degrees of freedom (12). Knowing this pair of degrees of freedom, a table of the F-distribution (Table A-3 at the back of the book) can be used to determine, at least approximately, the p-value. The F-distribution table provides the following p-val-ues for our case, in which the degrees of freedom are 2 and 12.
F-Statistic
1.46 2.81 3.89 6.93 p-Value
0.25 0.10 0.05 0.01
Thus, the p-value associated with 4.66 is not in the table but would be about 0.04. If the null hypothesis were true, there would be a 0.04 probability of getting an F-statistic of 4.66 or larger. Therefore, the evidence that the null hypothesis is not true is fairly substantial. The observed difference between sample means could have occurred by accident even if the null hypothesis were true, but the probability is low (1 chance in 25). Since the p-value is less than 0.05, we can say that the F-statistic is significant at the .05 level.
When possible, it is always useful to plot the data and get a graphic perspective. Figure 14-3 showed such a plot. The null hypothesis is that all points come from the same population and that there is no difference between the three price levels. The p-value denotes the probability of getting the difference between the three sample means, given the null hypothesis.
Significance Level
Sometimes it is convenient to report only the significance level. Recall Table 13-1, which reported the difference between the transit user and nonuser.
One footnote indicated that the difference between sample means for question 3 was significant at the 0.01 level.11 That means that the p-value was below 0.01. Therefore, under the hypothesis that the population means are the same on this question, the probability of getting sample means differing by that much or more is less than 0.01. Another Table 13-1 footnote indicated that the difference found in question 5 is significant at the 0.10 level This means that under the null hypothesis that the population percentages are the same, the probability of getting sample percentages that differ by .25 or more is less than 0.10. The evidence thus suggests—but of course does not prove—that the null hypotheses are not true for questions 3 and 5.
The reader should note that the evidence similarly does not prove that the null hypothesis is true for questions 1, 2, and 4 in Table 13-1. The evidence merely suggests that the hypothesis could be true, that it is consistent with the data. Thus, the hypothesis test is a very limited although useful preliminary step in the analysis. It really identifies those pieces c: evidence that are so unimpressive (in that they could be due just to the luck of the sample) that more detailed analysis is simply not worthwhile.
Expanding the ANOVA Table
In Chapter 10 we saw how an experiment involving a treatment variable such as price could be expanded. It is possible to control experimentally for one or more variables, such as store size or city, by adding one or more "block effects." In essence, the experiment is repeated for each block (i.e.. large stores and small stores). It also is possible to add more treatment variables. In either case, more than one nominally scaled variable is introduced, and there exist several "difference between sample means" relationships. To handle such experiments, the ANOVA table is expanded.
To illustrate, consider the experiment conducted by Keith Hunt on corrective advertising, which is advertising required by the FTC to "correct" a previous advertisement deemed deceptive.12 The advertisement in question was one introducing F-310, a gasoline additive. The FTC claimed, in part, that the product did not significantly reduce pollution as claimed and that the demonstration involving a balloon attached to the exhaust emissions of two cars was rigged, in that one car had a dirty engine and the other emitted invisible pollutants. The effect of various "corrective advertise-
11 Actually, when only two groups are involved, an equivalent test uses the t-statistic instead of the F-statistic. For further details see an introductory statistics text.
12H. Keith Hunt, "Effects of Corrective Advertising," Journal of Advertising Research, 15 (October 1973), 15-22.
merits" had policy implications for the FTC, which wanted a fair and effective remedy for deceptive advertising but did not want to be harsh and punitive.
Three types of corrective advertisements were tested:
1. Explicit A specific statement explicitly pointing out the deceptive characteristics of the advertisement in question.
2. General A general statement about the deception of the advertisement.
3. No corrective advertisement A bland statement by the FTC on gasoline additives with no mention of the company.
Prior to being exposed to one of the corrective advertisements, the respondents were exposed to one of three "inoculation" advertisements. An inoculation advertisement is hypothesized to mitigate the effect of the corrective advertising, either by giving high levels of support (supportive inoculation) or by giving weak doses of the corrective advertisement, which are refuted (refutational inoculation). Three inoculation treatments were used:
1. Refutation This advertisement warned of the upcoming corrective ad and refuted it. "If every motorist used F-310 for 2000 miles, air pollutants would be reduced by thousands of tons per day. The FTC doesn't seem to think that is significant. We think it is."
2. Supportive This contains no mention of the FTC or the upcoming corrective advertisement but does restate the positive arguments.
3. No inoculation This advertisement makes no mention of the positive arguments.
A 3 x 3 factorial design was used, as outlined in Figure 14-4. Each of the 9 cells had 22 respondents, each of which was exposed to the "deceptive" advertisement, the inoculation treatment, and the corrective advertisement. The criterion measure was the degree of agreement or disagreement, on a 28-point scale (where larger numbers indicate greater agreement), with the statement, "I like Chevron with F-310." Figure 14-4 shows the sample means for each of the nine cells. The hypothesis test involves determining the probability of obtaining the observed differences between the sample means, under the hypothesis that the population means were the same.
The expanded ANOVA table is shown in Table 14-4. Notice that each of the two treatments, inoculation and corrective advertising, now has an associated variation (sum of squares) and variance estimate (mean sum of
Cell size = 22
FIGURE 14-4
A factorial design.
squares). Consider first the inoculation treatment. The F-ratio for inoculation is:
F-ratio = MSSJnocufatton = 35^2 = im
"^^Unexplained 34.4
The associated p-value would be approximately 0.36. Thus, the evidence against the null hypothesis of no inoculation effect is not impressive. The evidence is the fact that the three sample means found in Figure 14-4 (16.7. 16.9, and 15.4) are not equal. However, although they are not equal, they
Interaction
are close enough so that we cannot reject the null hypothesis that the population means are equal.
Consider next the corrective advertising. The F-ratio is calculated for the corrective advertising treatment in the same manner:
F-ratiO = ?Corrective Advertising = igtLf = 56.43
"^•^Unexplained 34.4
The associated p-value is less than 0.001. Thus, the evidence is extremely impressive against the null hypothesis that there is no corrective advertising effect. The evidence is the fact that the three sample means found in Figure 14-4 (10.1, 18.2, and 20.4) are not equal. Thus, the null hypothesis that there is no corrective advertising effect can be rejected at the 0.001 level. A closer look at the three sample means in Figure 14-4 reveals that it is the explicit corrective advertising that is effective at changing attitudes. This finding is potentially important in designing remedies for deceptive advertising.
There is an advantage to analyzing the two treatments in the same analysis of variance table. By including both, the unexplained variation is reduced, as is the associated mean sum of squares (MSSU). As a result, there will be less "noise" in the data, the results will be more sensitive, and the F-ratios will be larger.
There is a third term in Table 14-4, the interaction between the two treatments. An interaction effect means that the impact of one treatment, such as inoculation, will not be the same for each condition of the other treatment. Figure 14-5 shows the results graphically. Note that inoculation affects the attitude created by explicit corrective advertising, but it really has little effect under the "no corrective advertising" and the "general corrective advertising" conditions. There is thus an interaction effect present. If there were no interaction, the shape of the three lines shown in Figure 14-5 would be the same. Their levels would differ, but their shapes would be the same.
The hypothesis of no interaction can be tested in the ANOVA table given in Table 14-4 by determining the appropriate F-ratio for interaction:
The associated p-value is approximately .007. Thus, the evidence against the null hypothesis of no interaction is very impressive.
To illustrate the hypothesis test appropriate for the difference between sample means, consider the following pricing experiment. Three prices are under consideration for a new product: 39 cents, 44 cents, and 49 cents. To determine the influence that the various price levels will have on sales, three samples of five supermarkets are randomly selected from the geographic area of interest. Each sample is assigned one of the three price levels. Figure 14-3 shows the resulting sales levels in both graphic and tabular form. The 390 stores, the first row, had sales of 8, 12, 10, 9, and 11, averaging 10 units. The 44<f stores, the second row, averaged 8 units; and the 49<t stores, the third row, averaged 7 units. Obviously, the determination of the optimal price will require an extensive analysis involving a host of considerations. However, before the analysis begins, it is appropriate to consider the hypothesis that price levels have no effect on sales. Under the null hypothesis, differences between sample means could be caused by the fact that only a sample of five was employed for each price level. The alternate hypothesis is that there is a price effect, that sales would not be the same for each of the price levels if they were applied to all stores.
Variation Between Stores
Two other estimates of the population also are available and provide the basis for the test of the hypothesis that price levels have no effect on sales. The first is to focus on the variation between stores at different price levels [Xl = 10, X2 = 8, and X3 = 7). The second is to consider the variation between stores at the same price level (for example, the stores with the 39 cents price level had sales of 8, 12, 10, 9, and 11). Under the null hypothesis that price levels have no effect on sales, each of these estimates should be similar. If the estimate based on variation between stores of different price levels is inflated, then doubt will be cast on the null hypothesis.
Variation Between Stores at Different Price Levels
The first variance estimate is based on the variation between the sample mean values of each row (price level), which is calculated as follows:
SSr = sum of squares between rows (price level) = variation explained by the row (price level)
= n
2 (Xp - X)2
5[(10 - 8i)2 + (8 - 8§P/(7 - 8i)2] = 23.3
where
Xp = mean sales at price level p (i. e., Xx = 10) X = overall mean (i. e., X = 85)
n = number of observations at each price level (n = 5) p = price level (p = 1, 2, 3) r = number of rows (price levels) i. e., r = 3
Clearlyj_as the difference between means gets larger so will the sum of terms
(Xp - xp.
The first variance estimate is termed MSS, and is obtained by dividing the SSrby its associated degrees of freedom (d. f.), which here is the number of rows less one:
MSSr = mean sum of squares between rows (price levels) = an estimate of the variance among stores = SSr/d. f. = (23.3)/2 = 11.65
Again, the determination of the degrees of freedom need not concern us here. In fact, the actual calculation formula for SSr and the other terms is also of little consequence. As a practical matter, the computer program or a reference statistics text can provide the appropriate formula for degrees of freedom, sum of squares, and other terms.8 The intent here is to provide a feel for the terms and expressions that will be encountered in this rather commonly used statistical hypothesis test and to provide a conceptual understanding of the test itself, rather than to teach the reader to actually make the raw calculations.
Variation Within Price Levels
The second variance estimate is based on the variation within each row (price level), which is
SSU = sum of squares unexplained by the row (price level) = variation within the rows (price levels)
n r
= 2 2 tx(p - xp)2
i=lp=l
= (8 - 10)2 + (12 - lO)2 + . . . + (9 - 7)2 + (7 - 7)2 = 34
where Xlp equals the sales of observations i at price level p. Note that SSU is based on the squared deviation from the row mean.
The second variance estimate is termed MSSU and is generated by dividing SSU by its associated degrees of freedom, which is here equal to r(n - 1) or 12:
MSSU = mean sum of squares unexplained by the row (price level) = an estimate of the variance among stores = SSu/d. f. = 34/12 = 2.8
ANOVA Table
These expressions are summarized in Table 14-3, which presents an analysis of variance and is termed an ANOVA table. The ANOVA table is a conventional way to present a hypothesis test regarding the difference between several means. The table indicates, at the left, the source of the variation. The first row summarizes the determination of MSSr, which is based on the variation between rows (the explained variation, or the variation explained by the price level).
The second row summarizes the determination of MSSU, which is based on the within-row variation (variation unexplained by the price levels). The third row represents the total variation based on the deviations of the individual sales results from the overall mean. Notice that the total variation (53.3) is equal to the sum of SSr and SSU. All the variation is accounted for.9
F-Statistic and p-Value
We now consider the ratio of the two estimates of the (population) variance of the store sales.10 This ratio is termed an F-ratio or F-statistic:
^he reader also might note that the total degrees of freedom, which is the total sample size of 15 less one, or 14, is equal to the sum of the degrees of freedom associated with the first two rows of the ANOVA table.
10The estimate of population variance introduced at the outset was the total variation divided by its degrees of freedom. In this case, it was 3.8, which is in between the other two estimates.
MSSr = 11.65 MSSU 2.8
= 4.16
If the null hypothesis that price levels have no effect on sales is true, then our variance estimates using the difference between the sample means, MSSr, should be the same as that based upon the within-row (price level) variations. The F-ratio should then be close to one. If, however, the hypothesis is not true and the different price levels generate different sales levels, the MSSr term will have two components. One component will reflect the variance among stores; the other will reflect the different price effects. As a result, the F-ratio will tend to become large.
The p-value is the probability that the F-ratio would be larger than 4.66, given the null hypothesis. To generate the p-value, the F-probability distribution is used. Associated with each F-ratio are the numerator (MSSr) degrees of freedom (2), and the denominator (MSSU) degrees of freedom (12). Knowing this pair of degrees of freedom, a table of the F-distribution (Table A-3 at the back of the book) can be used to determine, at least approximately, the p-value. The F-distribution table provides the following p-val-ues for our case, in which the degrees of freedom are 2 and 12.
F-Statistic
1.46 2.81 3.89 6.93 p-Value
0.25 0.10 0.05 0.01
Thus, the p-value associated with 4.66 is not in the table but would be about 0.04. If the null hypothesis were true, there would be a 0.04 probability of getting an F-statistic of 4.66 or larger. Therefore, the evidence that the null hypothesis is not true is fairly substantial. The observed difference between sample means could have occurred by accident even if the null hypothesis were true, but the probability is low (1 chance in 25). Since the p-value is less than 0.05, we can say that the F-statistic is significant at the .05 level.
When possible, it is always useful to plot the data and get a graphic perspective. Figure 14-3 showed such a plot. The null hypothesis is that all points come from the same population and that there is no difference between the three price levels. The p-value denotes the probability of getting the difference between the three sample means, given the null hypothesis.
Significance Level
Sometimes it is convenient to report only the significance level. Recall Table 13-1, which reported the difference between the transit user and nonuser.
One footnote indicated that the difference between sample means for question 3 was significant at the 0.01 level.11 That means that the p-value was below 0.01. Therefore, under the hypothesis that the population means are the same on this question, the probability of getting sample means differing by that much or more is less than 0.01. Another Table 13-1 footnote indicated that the difference found in question 5 is significant at the 0.10 level This means that under the null hypothesis that the population percentages are the same, the probability of getting sample percentages that differ by .25 or more is less than 0.10. The evidence thus suggests—but of course does not prove—that the null hypotheses are not true for questions 3 and 5.
The reader should note that the evidence similarly does not prove that the null hypothesis is true for questions 1, 2, and 4 in Table 13-1. The evidence merely suggests that the hypothesis could be true, that it is consistent with the data. Thus, the hypothesis test is a very limited although useful preliminary step in the analysis. It really identifies those pieces c: evidence that are so unimpressive (in that they could be due just to the luck of the sample) that more detailed analysis is simply not worthwhile.
Expanding the ANOVA Table
In Chapter 10 we saw how an experiment involving a treatment variable such as price could be expanded. It is possible to control experimentally for one or more variables, such as store size or city, by adding one or more "block effects." In essence, the experiment is repeated for each block (i.e.. large stores and small stores). It also is possible to add more treatment variables. In either case, more than one nominally scaled variable is introduced, and there exist several "difference between sample means" relationships. To handle such experiments, the ANOVA table is expanded.
To illustrate, consider the experiment conducted by Keith Hunt on corrective advertising, which is advertising required by the FTC to "correct" a previous advertisement deemed deceptive.12 The advertisement in question was one introducing F-310, a gasoline additive. The FTC claimed, in part, that the product did not significantly reduce pollution as claimed and that the demonstration involving a balloon attached to the exhaust emissions of two cars was rigged, in that one car had a dirty engine and the other emitted invisible pollutants. The effect of various "corrective advertise-
11 Actually, when only two groups are involved, an equivalent test uses the t-statistic instead of the F-statistic. For further details see an introductory statistics text.
12H. Keith Hunt, "Effects of Corrective Advertising," Journal of Advertising Research, 15 (October 1973), 15-22.
merits" had policy implications for the FTC, which wanted a fair and effective remedy for deceptive advertising but did not want to be harsh and punitive.
Three types of corrective advertisements were tested:
1. Explicit A specific statement explicitly pointing out the deceptive characteristics of the advertisement in question.
2. General A general statement about the deception of the advertisement.
3. No corrective advertisement A bland statement by the FTC on gasoline additives with no mention of the company.
Prior to being exposed to one of the corrective advertisements, the respondents were exposed to one of three "inoculation" advertisements. An inoculation advertisement is hypothesized to mitigate the effect of the corrective advertising, either by giving high levels of support (supportive inoculation) or by giving weak doses of the corrective advertisement, which are refuted (refutational inoculation). Three inoculation treatments were used:
1. Refutation This advertisement warned of the upcoming corrective ad and refuted it. "If every motorist used F-310 for 2000 miles, air pollutants would be reduced by thousands of tons per day. The FTC doesn't seem to think that is significant. We think it is."
2. Supportive This contains no mention of the FTC or the upcoming corrective advertisement but does restate the positive arguments.
3. No inoculation This advertisement makes no mention of the positive arguments.
A 3 x 3 factorial design was used, as outlined in Figure 14-4. Each of the 9 cells had 22 respondents, each of which was exposed to the "deceptive" advertisement, the inoculation treatment, and the corrective advertisement. The criterion measure was the degree of agreement or disagreement, on a 28-point scale (where larger numbers indicate greater agreement), with the statement, "I like Chevron with F-310." Figure 14-4 shows the sample means for each of the nine cells. The hypothesis test involves determining the probability of obtaining the observed differences between the sample means, under the hypothesis that the population means were the same.
The expanded ANOVA table is shown in Table 14-4. Notice that each of the two treatments, inoculation and corrective advertising, now has an associated variation (sum of squares) and variance estimate (mean sum of
Cell size = 22
FIGURE 14-4
A factorial design.
squares). Consider first the inoculation treatment. The F-ratio for inoculation is:
F-ratio = MSSJnocufatton = 35^2 = im
"^^Unexplained 34.4
The associated p-value would be approximately 0.36. Thus, the evidence against the null hypothesis of no inoculation effect is not impressive. The evidence is the fact that the three sample means found in Figure 14-4 (16.7. 16.9, and 15.4) are not equal. However, although they are not equal, they
Interaction
are close enough so that we cannot reject the null hypothesis that the population means are equal.
Consider next the corrective advertising. The F-ratio is calculated for the corrective advertising treatment in the same manner:
F-ratiO = ?Corrective Advertising = igtLf = 56.43
"^•^Unexplained 34.4
The associated p-value is less than 0.001. Thus, the evidence is extremely impressive against the null hypothesis that there is no corrective advertising effect. The evidence is the fact that the three sample means found in Figure 14-4 (10.1, 18.2, and 20.4) are not equal. Thus, the null hypothesis that there is no corrective advertising effect can be rejected at the 0.001 level. A closer look at the three sample means in Figure 14-4 reveals that it is the explicit corrective advertising that is effective at changing attitudes. This finding is potentially important in designing remedies for deceptive advertising.
There is an advantage to analyzing the two treatments in the same analysis of variance table. By including both, the unexplained variation is reduced, as is the associated mean sum of squares (MSSU). As a result, there will be less "noise" in the data, the results will be more sensitive, and the F-ratios will be larger.
There is a third term in Table 14-4, the interaction between the two treatments. An interaction effect means that the impact of one treatment, such as inoculation, will not be the same for each condition of the other treatment. Figure 14-5 shows the results graphically. Note that inoculation affects the attitude created by explicit corrective advertising, but it really has little effect under the "no corrective advertising" and the "general corrective advertising" conditions. There is thus an interaction effect present. If there were no interaction, the shape of the three lines shown in Figure 14-5 would be the same. Their levels would differ, but their shapes would be the same.
The hypothesis of no interaction can be tested in the ANOVA table given in Table 14-4 by determining the appropriate F-ratio for interaction:
The associated p-value is approximately .007. Thus, the evidence against the null hypothesis of no interaction is very impressive.
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