Module/Week 7 PPOL 505 P.E.5

13 TWO GROUPS TOO MANY? TRY ANALYSIS OF VARIANCE

13: MEDIA LIBRARY

Premium Videos

Core Concepts in Stats Video

· One-way ANOVA

Lightboard Lecture Video

· The F Test

Time to Practice Video

· Chapter 13: Problem 4

Difficulty Scale

(longer and harder than the others but a very interesting and useful procedure—worth the work!)

WHAT YOU WILL LEARN IN THIS CHAPTER

· Deciding when it is appropriate to use analysis of variance

· Computing and interpreting the F statistic

· Using SPSS to complete an analysis of variance

· Computing the effect size for a one-way analysis of variance

INTRODUCTION TO ANALYSIS OF VARIANCE

One of the more popular fields in the area of psychology is the psychology of sports. Although the field focuses mostly on enhancing performance, it pays attention to many aspects of sports. One area of interest is which psychological skills are necessary to be a successful athlete. With this question in mind, Marious Goudas, Yiannis Theodorakis, and Georgios Karamousalidis have tested the usefulness of the Athletic Coping Skills Inventory.

As part of their research, they used a simple  analysis of variance  (or ANOVA) to test the hypothesis that number of years of experience in sports is related to coping skill (or an athlete’s score on the Athletic Coping Skills Inventory). ANOVA was used because more than two levels of the same variable were being tested, and these groups were compared on their average performance. (When you compare means for more than two groups, analysis of variance is the procedure to use.) In particular, Group 1 included athletes with 6 years of experience or fewer, Group 2 included athletes with 7 to 10 years of experience, and Group 3 included athletes with more than 10 years of experience.

The test statistic for ANOVA is the F test (named for R. A. Fisher, the creator of the statistic), and the results of this study on the Peaking Under Pressure subscale of the test showed that F(2, 110) = 13.08, p < .01. The means of the three groups’ scores on this subscale differed from one another. In other words, any difference in test scores was probably related to which group they were in (number of years of experience in athletics) rather than some chance occurrence.

Want to know more? Go online or to the library and find …

Goudas, M., Theodorakis, Y., & Karamousalidis, G. (1998). Psychological skills in basketball: Preliminary study for development of a Greek form of the Athletic Coping Skills Inventory-28. Perceptual and Motor Skills, 86, 59–65.

THE PATH TO WISDOM AND KNOWLEDGE

Here’s how you can use the flowchart shown in Figure 13.1 to select ANOVA as the appropriate test statistic. Follow along the highlighted sequence of steps.

1. We are testing for a difference between scores of different groups, in this case, the difference between the pressure felt by athletes.

2. The athletes are being tested just once, not being tested more than once.

3. There are three groups (fewer than 6 years, 7–10 years, and more than 10 years of experience).

4. The appropriate test statistic is simple analysis of variance. (By the way, we call one-way analysis of variance “simple” because there is only one way in which the participants are grouped and compared.)

DIFFERENT FLAVORS OF ANALYSIS OF VARIANCE

ANOVA comes in many flavors. The simplest kind, and the focus of this chapter, is the  simple analysis of variance , used when one factor or one independent variable (such as group membership) is being explored and this factor has more than two levels. Simple ANOVA is also called  one-way analysis of variance  because there is only one grouping factor. The technique is called analysis of variance because the variance due to differences in performance is separated into (a) variance that’s due to differences between individuals between groups and (b) variance due to differences within groups. The between-groups variance is assumed to be due to treatment differences, while the within-group variance is due to differences between individuals within each group. This information is used to figure out how much the groups would be expected to differ just by chance. Then, the observed difference between groups is compared to the difference that would be expected by chance, and a statistical conclusion is reached.

ANOVA is, in many ways, similar to a t test. (An ANOVA with two groups is, in effect, a t test. In fact, the t computed in an independent t test is the square root of the F computed in a one-way analysis of variance!) In both ANOVA and t tests, differences between means are computed. But with ANOVA, there are more than two means, and you are interested in the average difference between means.

For example, let’s say we were investigating the effects on language development of being in preschool for 5, 10, or 20 hours per week. The group to which the children belong is the independent variable (or the treatment variable), or the grouping or between-groups factor. Language development is the dependent variable (or outcome variable). The experimental design will look something like this, with three levels of the one variable (hours of participation).

Group 1 (5 hours per week)	Group 2 (10 hours per week)	Group 3 (20 hours per week)
Language development test score	Language development test score	Language development test score

The more complex type of ANOVA,  factorial design , is used to explore more than one independent variable. Here’s an example where the effect of number of hours of preschool participation is being examined, but the effects of gender differences are being examined as well. The experimental design may look like this:

	Number of Hours of Preschool Participation
Gender	Group 1 (5 hours per week)	Group 2 (10 hours per week)	Group 3 (20 hours per week)
Male	Language development test score	Language development test score	Language development test score
Female	Language development test score	Language development test score	Language development test score

This factorial design is described as a 3 × 2 factorial design. The 3 indicates that there are three levels of one grouping factor (Group 1, Group 2, and Group 3). The 2 indicates that there are two levels of the other grouping factor (male and female). In combination, there are six possibilities (males who spend 5 hours per week in preschool, females who spend 5 hours per week in preschool, males who spend 10 hours per week in preschool, etc.).

These factorial designs follow the same basic logic and principles of simple ANOVA; they are just more ambitious in that they can test the influence of more than one factor at a time as well as a combination of factors. Don’t worry—you’ll learn all about factorial designs in the next chapter.

COMPUTING THE F TEST STATISTIC

Simple ANOVA involves testing the difference between the means of more than two groups on one factor or dimension. For example, you might want to know whether four groups of people (20, 25, 30, and 35 years of age) differ in their attitude toward public support of private schools. Or you might be interested in determining whether five groups of children from different grades (2nd, 4th, 6th, 8th, and 10th) differ in the level of parental participation in school activities.

Simple ANOVA must be used in any analysis where

· there is only one dimension or treatment,

· there are more than two levels of the grouping factor, and

· one is looking at mean differences across groups.

LIGHTBOARD LECTURE VIDEO

The F Test

The formula for the computation of the F value, which is the test statistic needed to evaluate the hypothesis that there are overall differences between groups, is shown in Formula 13.1. It is simple at this level, but it takes a bit more arithmetic to compute than some of the test statistics you have worked with in earlier chapters.

(13.1)

F=MSbetweenMSwithin,F=MSbetweenMSwithin,

where

· MSbetween is the variance between groups and

· MSwithin is the variance within groups.

The logic behind this ratio goes something like this. If there were absolutely no variability within each group (all the scores were the same), then any difference between groups would be meaningful, right? Probably so. The ANOVA formula (which is a ratio) compares the amount of variability between groups (which is due to the grouping factor) with the amount of variability within groups (which is due to chance). If that ratio is 1, then the amount of variability due to within-group differences is equal to the amount of variability due to between-groups differences, and any difference between groups is not significant. As the average difference between groups gets larger (and the numerator of the ratio increases in value), the F value increases as well. As the F value increases, it becomes more extreme in relation to the expected distribution of all F values and is more likely due to something other than chance. Whew!

Here are some data and some preliminary calculations to illustrate how the F value is computed. For our example, let’s assume these are three groups of preschoolers and their language scores.

Group 1 Scores	Group 2 Scores	Group 3 Scores
87	87	89
86	85	91
76	99	96
56	85	87
78	79	89
98	81	90
77	82	89
66	78	96
75	85	96
67	91	93

Here are the famous eight steps and the computation of the F test statistic:

1. State the null and research hypotheses.

The null hypothesis, shown in Formula 13.2, states that there is no difference among the means of the three different groups. ANOVA, also called the F test (because it produces an F statistic or an F ratio), looks for an overall difference among groups.

Note that this test does not look at specific pairs of means (pairwise differences), such as the difference between Group 1 and Group 2. For that, we have to use another technique, which we will discuss later in the chapter.

(13.2)

H0:μ1=μ2=μ3.H0:μ1=μ2=μ3.

The research hypothesis, shown in Formula 13.3, states that there is an overall difference among the means of the three groups. Note that there is no direction to the difference because all F tests are nondirectional.

(13.3)

H1:X1≠X2≠X3.H1:X1≠X2≠X3.

2. Set the level of risk (or the level of significance or Type I error) associated with the null hypothesis.

The level of risk or chance of Type I error or level of significance is .05 here. Once again, the level of significance used is totally at the discretion of the researcher.

3. Select the appropriate test statistic.

Using the flowchart shown in Figure 13.1, we determine that the appropriate test is a simple ANOVA.

4. Compute the test statistic value (called the obtained value).

Now’s your chance to plug in values and do some computation. There’s a good deal of computation to do.

· The F ratio is a ratio of the variability between groups to the variability within groups. To compute these values, we first have to compute what is called the sum of squares for each source of variability—between groups, within groups, and the total.

· The between-groups sum of squares is equal to the sum of the differences between the mean of all scores and the mean of each group’s score, which is then squared. This gives us an idea of how different each group’s mean is from the overall mean.

· The within-group sum of squares is equal to the sum of the differences between each individual score in a group and the mean of each group, which is then squared. This gives us an idea how different each score in a group is from the mean of that group.

· The total sum of squares is equal to the sum of the between-groups and within-group sum of squares.

Okay—let’s figure out these values.

Up to now, we’ve talked about one- and two-tailed tests. There’s no such thing when talking about ANOVA! Because more than two groups are being tested, 

and because the F test is an omnibus test (how’s that for a cool word?), meaning that ANOVA of any flavor tests for an overall difference between means, talking about the specific direction of differences does not make any sense.

Figure 13.2 shows the practice data you saw previously with all the calculations you need to compute the between-group, within-group, and total sum of squares. First, let’s look at what we have in this expanded table. We’ll start with the left-hand column:

· n is the number of participants in each group (such as 10).

· ∑X is the sum of the scores in each group (such as 766).

· ¯¯¯XX¯ is the mean of each group (such as 76.60).

· ∑(X  2) is the sum of each score squared (such as 59,964).

· (∑X)2/n is the sum of the scores in each group squared and then divided by the size of the group (such as 58,675.60).

Second, let’s look at the right-most column:

· n is the total number of participants (such as 30).

· ∑∑ X is the sum of all the scores across groups.

· (∑∑ X)2/n is the sum of all the scores across groups squared and divided by n.

· ∑∑(X  2) is the sum of all the sums of squared scores.

· ∑(∑ X )2/n is the sum of the sum of each group’s scores squared divided by n.

That is a load of computation to carry out, but we are almost finished!

First, we compute the sum of squares for each source of variability. Here are the calculations:

Between sum of squares	∑(∑X)2/n − (∑∑X)2/n or 215,171.60 − 214,038.53	1,133.07
Within sum of squares	∑∑(X2) − ∑(∑X)2/n or 216,910 − 215,171.6	1,738.40
Total sum of squares	∑∑(X2) − (∑∑X)2/n or 216,910 − 214,038.53	2,871.47

Second, we need to compute the mean sum of squares, which is simply an average sum of squares. These are the variance estimates that we need to eventually compute the all-important F ratio.

Figure 13.2 ⬢ Computing the important values for a one-way ANOVA

We do that by dividing each sum of squares by the appropriate number of degrees of freedom (df). Remember, degrees of freedom is an approximation of the sample or group size. We need two sets of degrees of freedom for ANOVA. For the between-groups estimate, it is k − 1, where k equals the number of groups (in this case, there are three groups and 2 degrees of freedom). For the within-group estimate, we need N − k, where N equals the total sample size (which means that the number of degrees of freedom is 30 − 3, or 27). Then the F ratio is simply a ratio of the mean sum of squares due to between-groups differences to the mean sum of squares due to within-group differences, or 566.54/64.39 or 8.799. This is the obtained F value.

Here’s a  source table  of the variance estimates used to compute the F ratio. This is how most F tables appear in professional journals and manuscripts.

Source	Sum of Squares	df	Mean Sum of Squares	F
Between groups	1,133.07	2	566.54	8.799
Within groups	1,738.40	27	64.39
Total	2,871.47	29

All that trouble for one little F ratio? Yes, but as we have said earlier, it’s essential to do these procedures at least once by hand. It gives you the important appreciation of where the numbers come from and some insight into what they mean.

Because you already know about t tests, you might be wondering how a t value (which is always used for the test between the difference of the means for two groups) and an F value (which is always used for more than two groups) are related. As we mentioned earlier, an F value for two groups is equal to a t value for two groups squared, or F = t2. Handy trivia question, right? But also useful if you know one and need to know the other. And it shows that all of these inferential statistical procedures use the same strategy—compare an observed difference to a difference expected by chance alone.

· 5. Determine the value needed for rejection of the null hypothesis using the appropriate table of critical values for the particular statistic.

As we have done before, we have to compare the obtained and critical values. We now need to turn to the table that lists the critical values for the F test, Table B.3 in Appendix B. Our first task is to determine the degrees of freedom for the numerator, which is k − 1, or 3 − 1 = 2. Then we determine the degrees of freedom for the denominator, which is N − k, or 30 − 3 = 27. Together, they are represented as F(2, 27).The obtained value is 8.80, or F(2, 27) = 8.80. The critical value at the .05 level with 2 degrees of freedom in the numerator (represented by columns in Table B.3) 

·  and 27 degrees of freedom in the denominator (represented by rows in Table B.3) is 3.36. So at the .05 level, with 2 and 27 degrees of freedom for an omnibus test among the means of the three groups, the value needed for rejection of the null hypothesis is 3.36.

· 6. Compare the obtained value and the critical value.

The obtained value is 8.80, and the critical value for rejection of the null hypothesis at the .05 level that the three groups are different from one another (without concern for where the difference lies) is 3.36.

· 7. and 8. Decision time.

Now comes our decision. If the obtained value is more extreme than the critical value, the null hypothesis should not be accepted. If the obtained value does not exceed the critical value, the null hypothesis is the most attractive explanation. In this case, the obtained value does exceed the critical value—it is extreme enough for us to say that the difference among the three groups is not due to chance. And if we did our experiment correctly, then what factor do we think affected the outcome? Easy—the number of hours of preschool. We know the difference is due to a particular factor because the difference among the groups could not have occurred by chance but instead is probably due to the treatment.

So How Do I Interpret F(2, 27) = 8.80, p < .05?

· F represents the test statistic that was used.

· 2 and 27 are the numbers of degrees of freedom for the between-groups and within-group estimates, respectively.

· 8.80 is the obtained value using the formula we showed you earlier in the chapter.

· p < .05 (the really important part of this little phrase) indicates that the probability is less than 5% on any one test of the null hypothesis that the average scores of each group’s language skills differ due to chance alone rather than due to the effect of the treatment. Because we defined .05 as our criterion for the research hypothesis being more attractive than the null hypothesis, our conclusion is that there is a significant difference among the three sets of scores.

Imagine this scenario. You’re a high-powered researcher at an advertising company, and you want to see whether the use of color in advertising communications makes a difference to sales. You’ve decided to test this at the .05 level. So you put together a brochure that is all black-and-white, one that is 25% color, the next 50%, then 75%, and finally 100% color—for five levels. You do an ANOVA and find out that a difference in sales exists. But because ANOVA is an omnibus test, you don’t know where the source of the significant difference lies. What if you’re interested in that, though? And you will almost certainly be. Well, you could take two groups (or pairs) at a time (such as 25% color and 75% color) and test them against each other. In fact, you could test every combination of two against each other. Kosher? Maybe not. Performing multiple t tests like this without a well-thought-out hypothesis for each comparison is sometimes called fishing, and fishing is actually against the law in some jurisdictions. When you do this, the Type I error rate (which you set at .05) balloons, depending on the number of tests you want to conduct. There are 10 possible comparisons (no color vs. 25%, no color vs. 50%, no color vs. 75%, etc.), and the real Type I error rate is

1−(1−α)k,1−(1−α)k,

where

· α is the Type I error rate (.05 in this example) and

· k is the number of comparisons.

So, instead of .05, the actual error rate that each comparison is being tested at is

1−(1−.05)10=.40(!!!!!).1−(1−.05)10=.40(!!!!!).

This is a far cry from .05. Later in this chapter, we will learn how to do a bunch of these pairwise comparisons safely.

USING SPSS TO COMPUTE THE F RATIO

The F ratio is a tedious value to compute by hand—not difficult, just time-consuming. That’s all there is to it. Using the computer is much easier and more accurate because it eliminates any computational errors. That said, you should be glad you have seen the value computed manually because it’s an important skill to have and helps you understand the concepts behind the process. But also be glad that there are tools such as SPSS.

We’ll use the data found in Chapter 13 Data Set 1 (the same data used in the earlier preschool example).

1. Enter the data in the Data Editor. Be sure that there is a column for group and that you have three groups represented in that column. In Figure 13.3, you can see that the cell entries are labeled Group and Language_Score.

Understanding the SPSS Output

This SPSS output is straightforward and looks just like the table that we created earlier to show you how to compute the F ratio along with some descriptive statistics. Here’s what we have:

1. The descriptive values are reported (sample sizes, means, standard deviations) for each group and overall.

2. The source of the variance as between groups, within groups, and the total is identified.

3. The respective sum of squares is given for each source.

4. The degrees of freedom follows. This is followed by the mean square, which is the sum of squares divided by the degrees of freedom.

5. Finally, there’s the obtained value and the associated level of significance.

Keep in mind that this hypothesis was tested at the .05 level. The SPSS output provides the exact probability of an outcome that is this large or larger is .001. This is a much smaller probability than .05.

THE EFFECT SIZE FOR ONE-WAY ANOVA

In previous chapters, you saw how we used Cohen’s d as a measure of effect size. Here, we change directions and use a value called eta squared or η2. As with Cohen’s d, η2 has a scale of how large the effect size is:

· A small effect size is about .01.

· A medium effect size is about .06.

· A large effect size is about .14.

Now to the actual effect size.

The formula for η2 is as follows:

η2=Between−groupssumsofsquaresTotalsumsofsquares,η2=Between-groupssumsofsquaresTotalsumsofsquares,

and you get that information right from the source table that SPSS (or doing it by hand) generates.

In the example of the three groups we used earlier, the between-groups sums of squares equals 1,133.07, and the total sums of squares equals 2,871.47. The easy computation is

η2=1,133.072,871.47=.39,η2=1,133.072,871.47=.39,

and according to your small–medium–large guidelines for η2, .39 is a large effect. The effect size is a terrific tool to aid in the evaluation of F ratios, as indeed it is with almost any test statistic. Instead of interpreting this effect size as the size of the difference between groups as we did for Cohen’s d, we interpret eta squared as the proportion of the variance in the dependent variable explained by the independent variable. In this case, 39% of the variability in language scores is explained by the number of hours of preschool. And that’s a strong relationship (in our made-up data)!

Okay, so you’ve run an ANOVA and you know that an overall difference exists among the means of three or four or more groups. But where does that difference lie?

You already know not to perform multiple t tests. You need to perform what are called  post hoc , or after-the-fact, comparisons. You’ll compare each mean with every other mean and see where the difference lies, but what’s most important is that the Type I error for each comparison is controlled at the same level as you set. There are a bunch of ways to do these comparisons, among them being the Bonferroni (your dear authors’ favorite statistical term). To complete this specific analysis using SPSS, click the Post Hoc option you see in the ANOVA dialog box (Figure 13.4), then click Bonferroni, then Continue, and so on, and you’ll see output something like that shown in Figure 13.6. It essentially shows the results of a bunch of independent t tests between all possible pairs. Looking at the “Sig.” column, you can see that the significant pairwise differences between the groups contributing to the overall significant difference among all three groups lie between Groups 1 and 3; there is no pairwise difference between Groups 1 and 2 or between Groups 2 and 3. This pairwise stuff is very important because it allows you to understand the source of the difference among more than two groups. And in this example, it makes sense that the groups that differ the most in terms of the independent variable (preschool hours) drive the significant omnibus F.

Figure 13.6 ⬢ Post hoc comparisons following a significant F value

Real-World Stats

How interesting is it when researchers combine expertise from different disciplines to answer different questions? Just take a look at the journal in which this appeared: Music and Medicine! In this study, in which the authors examined anxiety among performing musicians, they tested five professional singers’ and four flute players’ arousal levels. In addition, they used a 5-point Likert-type scale (you know, those strongly disagree to strongly agree attitude questions) to assess the subjects’ nervousness. Every musician performed a relaxed and a strenuous piece with an audience (as if playing a concert) and without an audience (as though in rehearsal). Then the researchers used a one-way analysis of variance measuring heart rate (HR), which showed a significant difference across the four conditions (easy/rehearsal, strenuous/rehearsal, easy/concert, and strenuous/concert) within subjects. Moreover, there was no difference due to age, sex, or the instrument (song/flute) when those factors were examined.

Want to know more? Go online or to the library and find …

Harmat, L., & Theorell, T. (2010). Heart rate variability during singing and flute playing. Music and Medicine, 2, 10–17.

ummary

Analysis of variance (ANOVA) is the most complex of all the inferential tests you will learn in Statistics for People Who (Think They) Hate Statistics. It takes a good deal of concentration to perform the manual calculations, and even when you use SPSS, you have to be on your toes to understand that this is an overall test—one part will not give you information about differences between pairs of treatments. If you choose to go on and do post hoc analysis, you’re only then completing all the tasks that go along with the powerful tool. We’ll learn about just one more test between averages, and that’s a factorial ANOVA. This, the holy grail of ANOVAs, can involve two or more factors or independent variables, but we’ll stick with two and SPSS will show us the way.

Time to Practice

1. When is analysis of variance a more appropriate statistical technique to use than a t test between a pair of means?

2. What is the difference between a one-way analysis of variance and a factorial analysis of variance?

3. Using the following table, provide three examples of a simple one-way ANOVA, two examples of a two-factor two-way ANOVA, and one example of a three-factor three-way ANOVA. We show you some examples—you provide the others. Be sure to identify the grouping and the test variable as we have done here.

Design	Grouping Variable(s)	Test Variable
Simple ANOVA	Four levels of hours of training—2, 4, 6, and 8 hours	Typing accuracy
Enter your example here.	Enter your example here.
Enter your example here.	Enter your example here.
Enter your example here.	Enter your example here.
Two-factor ANOVA	Two levels of training and gender (2 × 2 design)	Typing accuracy
Enter your example here.	Enter your example here.
Enter your example here.	Enter your example here.
Three-factor ANOVA	Two levels of training and two of gender and three of income	Voting attitudes
Enter your example here.	Enter your example here.

4. Time to Practice Video

5. Chapter 13: Problem 4

1. Chapter 13 problem 4 will require you to open up Chapter 13 Data Set 2, and we're going to do an ANOVA. The F ratio will indicate that it's an ANOVA test. We're really wanting to examine how the average time of swimmers is associated with how much time they practice weekly. Do they practice fewer than 15 hours, 15 to 25 hours, or more than 25 hours? When we look at our data set, there's something that we need to change. Practice, when you look at that, is considered right now, or categorized, as a scale. When we look at Variable view, it's a scale. But notice that it's really just the three levels of hours. In this case, that would be nominal. So we want to go back and change this from scale to nominal. And then when we look here, you notice a little Venn diagram will show that it's changed. Then to answer this, under Analyze, go to Compare Means to a one-way ANOVA. In this situation, our factors, like our independent variable-- in a t-test, it was called a grouping variable. So let's put Practice, which is the number of times. And then under the Dependent list is their actual time here. Under T-tests, it would ask us to indicate what they are. This is an ANOVA, so it'll allow more than two levels. So we'll compare all the levels that are in there. Under Options, you want to select-- you want the descriptive statistics. It's actually really important that you have that. Under Contrast, we don't need anything here, because we're just going to do a regular ANOVA. We're not doing a polynomial. And under a post hoc test, we could say what do we want to do. This question doesn't call for it, but I'm going to throw it in anyway. So select Tukey here, and I'll show you why we did that. Hit Continue, and hit OK. And here is our one-way ANOVA. In this situation, you'll see here, under the ANOVA, the significance is a In other words, there was no difference, in terms of their speed, based on the number of hours that they practiced. If you look here at the mean score, you'll see 58, 57, 59-- even in seconds, they're still really pretty close there. So there's not a big difference. But we want to look at the post hoc test to see, if there had been a difference, where it would be. So I'm just showing you what you would look for. Under post hoc, it compares. Notice fewer than 15 being compared to 15 to 25 then to above 25, and it would show you the significance. Because in the ANOVA, it just tells you there is a difference. It doesn't tell you where that difference is. So if this had been statistically significant, we would have to look here to know what that difference was in terms of the time and then the outcome. And that's how we answer our problem about an ANOVA. Using the data in Chapter 13 Data Set 2 and SPSS, compute the F ratio for a comparison among the three levels representing the average amount of time that swimmers practice weekly (<15, 15–25, and >25 hours), with the outcome variable being their time for the 100-yard freestyle. Answer the question of whether practice time makes a difference. Don’t forget to use the Options feature to get the means for the groups.

2. The data in Chapter 13 Data Set 3 were collected by a researcher who wants to know whether the amount of stress is different for three groups of employees. Group 1 employees work the morning/day shift, Group 2 employees work the day/evening shift, and Group 3 employees work the night shift. The null hypothesis is that there is no difference in the amount of stress between groups. Test this in SPSS and provide your conclusion.

3. The Noodle company wants to know what thickness of noodle consumers find most pleasing to their palate (on a scale of 1 to 5, with 1 being most pleasing), so the food manufacturer put it to the test. The data are found in Chapter 13 Data Set 4. Turns out that there is a significant difference (F(2, 57) = 19.398, p < .001), and thin noodles are most preferred. But what about differences among thin, medium, and thick noodles? Post hoc analysis to the rescue!

4. Why is it only appropriate to do a post hoc analysis if the F ratio is significant?

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