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Psychology 210 Spring, 2007 Statistics Lab #5: Confidence Intervals and One-Sample t-tests Welcome back to stats lab. Today we’ll be working with ways to analyze single (that is one at a time) interval or ratio variables. There are two appropriate tools for this: Confidence intervals and t-tests. Recall from class that a confidence interval specifies the range of possible population means (µ) that could reasonably have produced the sample of data being analyzed. On the other hand, a one-sample t-test asks whether a particular population mean (µ) could reasonably have produced the sample of data being analyzed. Consequently the output will look a little different. The former (a confidence interval) will provide two values: a lower limit and an upper limit, between which should fall the population mean. The latter (a t-test) will provide a “yes or no” answer (technically “reject or retain”) pertaining to one potential population mean. A quick example: After recently being cut off by some punk kids driving a fancy new sports car, I might shake my fist at them in anger, and ask one of two statistical questions: 1) Just how fast does the average teenager drive? 2) Do teenagers obey the speed limit? The first question above would call for a confidence interval, specifying the range of values that should contain the population mean of teenage drivers. The second calls for a t-test, with the null hypothesis that kids don’t drive at the speed limit: H0 : μ = 65. (The legal speed limit is the average driving speed of teenagers) H1 : μ ≠ 65. (Those darn kids don’t drive the speed limit!) More likely I’d phrase this as a one-tailed test: H0 : μ ≤ 65. (Teenagers drive the speed limit or less) H1 : μ > 65. (The world is populated with crazy, speed-demon kids who drive faster than 65 mph) Let’s work through the procedure for doing confidence intervals and t-tests in SPSS. Open the data file lab5.sav, located on the class web page: http://people.whitman.edu/~herbrawt/classes/210/psych210.html (remember that it doesn’t work well to open it right from your web browser. It’s best to save the file to your desktop, and then use SPSS to open it from there).The file contains data for three different variables. The first variable (subno) is simply an identifier for each participant, so I can keep their data straight while maintaining confidentiality. The second (wordlist) is the number of words from a list of 40 that participants were able to recall following a retention interval. The third (sleep) is the amount of sleep (in hours) participants reported getting on a nightly basis. Each row contains data from a different one of my 22 participants. Confidence Intervals in SPSS Confidence intervals for any scale (that is, interval or ratio) variable can be found using the Analyze Æ Descriptive Statistics Æ Explore command, which should produce a window like this: To get a 95% confidence interval for the number of words recalled, click on that variable and then on the right-facing arrow to move it into the Dependent List as shown. By the way, if you want a different confidence interval (say, a 90% or 99% confidence interval) you can change it by clicking on the button marked statistics at the bottom of the window. Leave it at 95% for now though, and click OK. The output you see should look a whole lot like this:In addition to some other familiar descriptive statistics (mean, variance, standard deviation), I can see the upper and lower bound for a 95% confidence interval. I would report the interval like this: “The 95% confidence interval for number of words recalled is 7.21 < µ < 10.25.” This means that I can be 95% certain that the population mean falls somewhere between 7.21 words and 10.25 words – even if it isn’t exactly equal to 8.73 (the sample mean). Some of you may be familiar with the idea that the capacity of short-term memory is alleged to be approximately 7 pieces of information (7 ± 2, as the Introduction to Psychology textbook tells us). The fact that the entire confidence interval is greater than 7 is important – it implies that my sample has better than average memory since there’s no reasonable way it could be as low as 7. However, if this particular comparison (with the number 7) was my primary goal, I might have instead opted to run a one-sample t-test…One-Sample t-test in SPSS For my word recall variable, 7 makes a nice null hypothesis. In effect, it’s saying that there’s nothing special about this sample – I’d expect them to remember the same number of words that anybody with a normal short-term memory would (that is, about 7). The research hypothesis would then be that the sample will remember significantly more or significantly fewer than 7 words: H0 : μ = 7. H1 : μ ≠ 7. To run the t-test, select Analyze Æ Compare means Æ One-Sample T Test. In the dialog box that appears, select the variable whose mean you would like to compare against a known population mean (number of words). Select it in the usual way, by highlighting the variable name and clicking it over to the test box as shown here. In the box conspicuously marked test value, include the number to which you want to compare the data. I’m sure you realize that’s where you’d enter the number 7. Go ahead and do that. This is μ0 according to my null hypothesis. Therefore, the test will allow me to determine if students consistently tend to over- or under perform relative to that magic number of 7. On the other hand, if the mean guess is relatively close to 7, I have to concede that the null distribution with μ0 = 7 could indeed have produced my sample data. Your results should include something like the following:Notice the t-statistic of 2.365. This is what we’ve been computing in class by hand. When doing it by hand, we would compare this statistic with cutoff values from the table in your textbook – in this case, with a two-tailed test and 21 degrees of freedom, I would have cutoff values at 2.080 and -2.080: 2.365 (my t-statistic computed by SPSS) is greater than the positive cutoff value of 2.080. This means that the null distribution would be very unlikely (< .05 probability) to produce a sample like the one we have, and so we should reject the null hypothesis. But this is where SPSS is more precise than we can be in class. In class where we calculate things by hand, all we know


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Whitman PSYCHOLOGY 210 - Statistics Lab

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