1. 1 G89.2228 Lect 5b G89.2228 Lecture 5b A research question involving means The significance test approach »The problem of s 2 »Student’s t distribution A Confidence Interval Approach Interpretation A special case of testing a single mean: Mean Difference Score

2. 2 G89.2228 Lect 5b A research question involving means Suppose a colleague is interested in studying the perceived fairness of selection criteria for a variety of university selections and awards. Subjects use a nine point scale to rate ten scenarios regarding whether the system is biased: 1 2 3 4 5 6 7 8 9 Fifteen subjects are sampled from a population Is the mean of the ten ratings equal to 5 in the population that was sampled? Favors Men Favors Women Unbiased

3. 3 G89.2228 Lect 5b A formal analysis of the question Let X represent the average of the ten items for a single subject Suppose that the sample data are consistent with an assumption of approximate normality for X The sample mean is  and the sample standard deviation is s. The sample mean should be approximately normally distributed, with mean µ X and standard error,

4. 4 G89.2228 Lect 5b Normal theory: significance test approach Consider a null hypothesis, H 0 : µ = 5 Under H 0, the following test statistic, a function of is distributed approximately as N(0,1): If Z ~ N(0,1), then values with magnitudes larger than ±1.96 should not occur more often than 5 times in 100. If Z( ) has an unusual magnitude under H 0, then we question (reject) H 0. If Z( ) has a value that is not unusual under H 0, we don’t question (reject) H 0.

5. 5 G89.2228 Lect 5b Continuing the Example Suppose that we were able to calculate a value of Z=1.47 for our sample of raters. This is in the direction of the system favoring men, but is it an unusual value? We might note from Appendix Z on Howell that values >1.47 occur 7.08% of the time if a variable is distributed N(0,1). Would you call that unusual? If we did not have a good reason to look at one side of the distribution, we would have to note that N(0,1) values exceed |±1.47| about 14.16% of the time, which is around 1 in 7. Some might argue that the data show that the subjects perceive the system to be fair, since it does not significantly reject the H 0 that µ X =5.

6. 6 G89.2228 Lect 5b The problem of s 2 The function Z( ) requires a value for  2. When we do not know the population value, a reasonable approximation is the sample estimate, s 2. We know that the actual value of s 2 will vary from sample to sample. When s 2 is smaller than  2, then will be too large. The smaller the sample size, the more s 2 will be expected to vary. For small sample sizes, Z(  X ) will not be N(0,1) if s X 2 is substituted for  X 2.

7. 7 G89.2228 Lect 5b Student’s t distribution Gossett, an ale brewmaster, showed that the distribution of could be fully described mathematically when X~N(µ X,  X 2 ). Moreover, this distribution did not depend on knowing the value of  X 2. The so-called t distribution is a family of distributions that vary systematically with the degrees of freedom available for the estimate of  X 2. In our example, df=14. The critical value for a two tailed t statistic (from Howell’s Appendix t), is 2.145 instead of z=1.96 for  =.05. The critical value for a one tailed test is 1.761 instead of z=1.645. t and z converge for large df.

8. 8 G89.2228 Lect 5b Confidence Interval Approach Consider the shortcomings of the usual test of significance: silly H 0, no formal consideration of power, over- interpretation of negative result. Let’s address the ambiguity of the data by defining an interval that is likely to contain the true µ, whatever it is. 95% confidence interval is (4.63,6.97)

9. 9 G89.2228 Lect 5b Interpretation Our original question was, Is the mean of the ten ratings equal to 5 in the population sampled? We are really interested in whether there is evidence of perceived favoritism. The formal t test says that the null hypothesis can not be rejected. Reviewers of the literature may say, “We failed to find evidence of favoritism.” The CI, (4.63,6.97), says that there is not a lot of precision in the answer to the question, but that the data are not consistent with a general perception of female favoritism. 9 8 X 7 XXX 6 XXXX 5 XXX 4 XX 3 X 2 X 1

10. 10 G89.2228 Lect 5b Many psychological studies involve within subject-designs whereby a subject’s pretest score is compared to a post-treatment score The question is often, is there change? H 0 : µ D =0, where D=X 2 -X 1 Confidence interval for the difference is: A special case of testing a single mean: Mean Difference Score