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Whitman PSYCHOLOGY 210 - Binomial and Sign Tests

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Psychology 210 February 8, 2007 Lab #4: Binomial and Sign Tests We’ll be running two kinds of statistical tests in SPSS today, both of which have been calculated in class. Begin by logging onto your computer and launching SPSS. There’s a data set for today’s class located on the class web page (go to and click on classes). The data file is called lab4.sav. Most of what we’ll do from here will be constrained to the drop-down menu labeled Analyze. First, notice that there are several dozen options within the menu. Each of these statistical tests has different features appropriate for different sorts of data, and most of them will be considered in turn during the semester. Today we’ll limit ourselves to analyses of nominal (also known as categorical) data. The data in the file represent scores on 4 exams from a mythical class. For good measure, I’ve included the sex, but not the identity of each student. The structure of the data file is pretty simple… I’ll assume you can figure out what’s what. Exams are out of 50 possible points, and males are coded as “1”, females as “2”. Let’s get right to it then! The Binomial Test Select a binomial test by choosing Analyze Æ Nonparametric Tests Æ Binomial from the drop-down menu. When you do so, a window like the one below will appear. Recall that in order to produce a meaningful result, you need to use a binomial variable – a variable that has only two possible values (such as sex). You can select variables by clicking their names in the variable list on the left and then clicking the right-facing arrow. It is possible to select more than one variable. If you do so, the software will run a separate binomial test on each. You can also define the test proportion (p) as you see fit by modifying the value in the box marked Test Proportion. The default value is .50, corresponding to a fair coin toss; each category is equally likely. When you’ve selected a variable and test proportion and can wait no longer to see the results, click OK.An analysis of sex produces a result that should look something like this: The table shows the number of students of each sex (under N), the proportion of all cases in each group (Observed Prop.), the tested proportion (Test Prop., in case you forgot), and the significance level (Asymp. Sig.). The latter value is the most important part of the output. This tells you the probability of getting a distribution at least this extreme by chance alone. It’s also a two-tailed value, meaning that it includes the extreme values from both ends of the binomial distribution. Half of the .041 is in each tail (.0205 each). In other words, SPSS reports the probability of getting 15 or more out of 20 students in one group or the other: 15 or more females, or 15 or more males. The smaller this is, the less likely it is that the true proportion is actually .50 (or whatever your test proportion was). In this case, .0205.0205there’s only about a 4% chance that you could get a distribution this skewed by chance if there was no gender bias in enrolments at all. Thus, the notion that females and males are equally likely to enroll doesn’t fit the observed data very well. I should also point out that a value of .000 in the output window would not actually mean zero. The probability value gets rounded to three digits, so you can assume a value of zero designates that the likelihood is vanishingly small (less than .0005), but not truly impossible. Naturally, If you want to know the probability for a one-tailed test (15 or more females, but not 15 or more males), you can simply divide the significance level by two, since the binomial distribution is perfectly symmetrical. This would be the case if I decided ahead of time that I was only interested in seeing if females were overrepresented and simply didn’t care if things were skewed the other way. The difference is captured in the following sentences: The probability of getting 15 or more students of the same sex in a class of 20 is .04 (2 tailed). The probability of getting 15 or more females in a class of 20 is .02 (1 tailed). In this case, it looks like females are indeed overrepresented in my class. But also recall that this test began with the assumption that men and women are equally likely to enroll (test proportion = .50). A recent study shows that more women are majoring in psychology; specifically, 60% of recent psychology graduates have been female (with the remaining 40% male, of course). Re-run the analysis using this proportion. Is the 60% bias toward female psychology majors enough to account for the skewed results?... (decide before reading on) My test came out at .126, meaning that there’s a higher (12%) chance of getting results this skewed if the proportion of females enrolling in psychology classes is 60%. Psychologists often use 5% as the cutoff for when something is so unlikely that mere chance variability can’t be considered a good explanation. Pertaining to our results, it’s possible that randomly drawing a class of 20 students from a population consisting of 50% women and 50% men could produce a skewed sex distribution like that seen in our data file. However, it would only do so only about 4% of the time, which is rare enough that we should be skeptical that the mix truly is even. On the other hand, if we assume that the sample consists of 60% women and 40% men, we’d get a distribution this extreme (or more extreme) over 12% of the time – enough that this 60% / 40% population could be considered a real possibility.The Sign Test Occasionally, one can use the binomial distribution to test data that are not binomial in nature. For instance, if an educator wanted to tell whether a new education program worked, he might wish to look at individuals’ test scores from before and after the program to see if they increased, decreased, or stayed about the same. This would be the equivalent of subtracting one from the other and seeing if the resulting values were more likely to be positive or negative. One could then use the binomial distribution to calculate the likelihood of getting the observed number of positive and negative differences. To run a sign test in SPSS, select Analyze Æ Nonparametric Tests Æ 2 Related Samples. The following window will appear: First, select Sign and deselect Wilcoxon among the

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