Your textbook:

• Gravetter, F. J., Wallnau, L. B. (1996). Statistics for the Behavioral Sciences:
  A First Course for Students of Psychology and Education, 4th Edition. New York: West Publishing.
  

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  Chapter 11: Hypothesis tests with related samples

  In the prior chapter we examined how to use a t-test with two independent samples. For example, comparing men vesus women. However, that kind of design is not always possible and/or desirable.

  Consider the following examples:

  1) suppose that you want to compare married couples opinions about what makes a relationship work. So you decide to ask the husbands and wives to rate, on a scale from 1 to 10, how important communication is. In this senario, even though you've got two groups (husbands and wives), the two groups are not independent.. The members of each group are related to each other ("related" with respect to statistical selection issues, not religous or legal issues). So the t-test that we discussed in the last chapter is not the appropriate test.

  2) suppose that you want to find out whether viagra impairs vision. Instead of comparing two separate groups, you decide to test the same set of individuals. In the first stage of the experiment you give your participants a placebo (a sugar pill that should have no effect on vision), and then test their vision. In the second stage, you give them viagra and then test their vision. So now you have the same people in both conditions. Clearly your samples are related, so again the t-test from the last chapter isn't appropriate.

  3) suppose that you are interested in the effect of studying on test performance. So you decide to use two groups of people for your study. However, you also decide that you want the two groups of people to be as similar as possible, so you match each individual in the two groups on as many important characteristics as you can. Again, the two samples are related, so the t-test from the last chapter isn't appropriate.

  In the first case, the situation has been decided for you, there is a pre-existing relationship between the two samples.

  In the second and third case, you, as the experimenter make a decision to make the two samples related. Why would you ever want to do that? To control for individual differences that might add more noise (error) to your data. In situation 2, each individual acts as their own control. In situation 3, the control group is made up of people as similar to the people in the experimental group as you could get them. Both of these designs are used to try to reduce error resulting from individual differences.

  A repeated-measures study is one in which a single sample of subjects is used to compare two (or more) different treatment conditions. Each individual is measured in one treatment, and then the same individual is measured again in the second treatment. Thus, a repeatted-measures study produces two (or more) sets of scores, but each set is obtained from the same sample of subjects. Sometimes this type of study is called a within-subjects design.

  In a matched-subjects study, each individual in one sample is matched with a subject in the other sample. The matching is done so that the two individuals are equivalent (or nearly equivalent) with respect to a specific variable that the researcher would like to control. Sometimes this type of styd is called a related-samples design.

  Okay, so now we know that for repeated-measures and matched-subject designs we need a new t-test. So, what is the t statistic for related samples?

  Again, the logic of the hypothesis test is pretty much the same as it has been in the last few chapters. Once again we'll go through the same steps. What changes are the nature of the hypothesis, and how the tobs is computed.
  All of the tests that we've looked at are examining differences. In the previous chapter we were interested in overall differences between the populations (as estimated by the differences between the samples). The t-test for this chapter is also interested in the differences, but because the two samples are related, the differences are based on differences between each individual.

  Consider the following example: An instructor asks his statistics class, on the first day of classes, to rate how much they like statistics, on a scale of 1 to 10 (1 hate it, 10 love it). Then, at the end of the semester, the instructor asks the same students, the same question. The instructor wants to know if taking the stats course had an impact on the students feelings about statistics.

  The results of the two ratings are presented below. D stands for the difference between the pre- and post-ratings for each individual.

  	Student	Pre-test (first day)	Post-test (end of semester)	D	D2
  	     1			1			4			3	9
  	     2			3			5			2	4
  	     3			4			6			2	4
  	     4			7			8			1	1
  	     5			2			3			1	1
  	     6			2			2			0	0
  	     7			4			6			2	4
  	     8			3			4			1	1
  	     9			6			6			0	0
  	    10			8			6		           -2	4 
  				40			50			10       28
  

  mean difference = = 10/10 = 1.0

  Okay, so let's start our 7 step process of hypothesis testing.

  step 1: state your H0 and H1 & figure out your critera: a = ?

  Okay, for this example let's assume that a = 0.05 What is our H0?

  Conceptually it is similar to last chapter, but instead of having two separate populations of individuals, we've got a single population of differences.
  In other words, the distribution that we're intersted in is the distribution of D, the distribution of the pre-test scores subtracted from the post-test scores.
  So our H0 will be something like this, taking stats has no effect on a persons preference for statistics.
  H0: mD = 0 H1 = mD ­ 0
  step 2: figure out if your test is one-tailed or two-tailed

  Does taking stats have an impact on feelings about statistics? This just asks about a general difference, so it is a two-tailed test.

  step 3: figure out the df for your test

  Now we only a single sample, a sample of the differences. With only one sample, our df = n - 1.

  step 4: find the critical t-score from the table

  Finding tcrit is the same as ususal, look at the table.

  a = 0.05, two-tailed, df = 10 - 1 = 9
  tcrit = ± 2.262

  step 5: compute your t-score for your sample

  Okay, as was the case last chapter, the overall form of the t statistic equation is the same, but the details are different.

  tobs =
  So we already computed our , and we know mD = 0 (for the H0), so we just need to figure out what is equal to. This is the estimated standard error of the difference distribution.

  So first we need to figure out the variance.

  SSD = (D2 - ((D)2 = 28 - 102 = 28 - 10 = 18 n 10

  sD2 = = = = 2.0

  Now we can figure out the estimated standard error

  = = = 0.447

  Now we are read to compute our tobs

  tobs = = = 2.24

  step 6: compare your t-score with the critical t-score

  tcrit = 2.262 tobs = 2.24

  step 7: make your conclusions about the H0

  our tobs does not fit in the critical region, so we fail to reject the H0.

  However, if we had made a directional hypothesis, that the stats class would increase preference of stats. What would happen?

  We'd increase the power of our test to detect a difference (because we are looking at 0.05 in only one tail, instead of 0.025 in two tails). Our tcrit = 1.833. Our tobs would still be the same (2.24), so now in step 7 we would end up rejecting the H0.

  Okay, what about Hypothesis testing with a matched-subject design?

  Basically we do things exactly as we did in the previous example, except now we subtract the matched control person's score from the experimental group person. For an example, see 11.3 in the book (starting pg 323).

  So, as an experimenter, how do we know when to use related sample designs or independent sample designs?

  Related samples designs are used when large individual differences are expected and considered to be "normal". Why? Because individual differences can contribute to sampling error. So by using related samples designs, one can reduce sampling error and have a better chance of finding a difference if there really is one.

  Note: point out statistics organizer on page a-60 (near end of text) nice decision maps