Math 143 C/E, Spring 2001
IPS Reading Questions
Chapter 7, Section 1, Part 2: pp. 507-517 (“power of the t test”) and pp. 519 (“sign test”)-523



  1. In the blue box on p. 508 that summarizes the “One-Sample t Test”, the authors say that the “P-values are exact if the population distribution is normal and are approximately correct for large n in other cases.” Later in the section (pp. 515-517) they expound on some of these “other cases”. Describe these cases and indicate how large n should be in each. Are there situations in which you should be wary of the validity of a t procedure even for large n?

    If the data in the sample appears almost normal, it is probably OK to use t procedures (confidence intervals and hypothesis tests). If the data appears not to be normal but is not strongly skewed and has no outliers, a sample size of n > 15 should still yield valid results using a t procedure. If quite skewed, you should still be safe with n > 40. However, if the data has outliers, the t procedures are less accurate than they would otherwise be, though you can rely on them to give conservative estimates even then.

  2. Finish this sentence appropriately:
    There is a strong relationship between the two inference procedures: confidence intervals and hypothesis tests. Specifically, when a sample mean is used to determine a level C confidence interval, the result of a two-sided hypothesis test for null hypothesis H0: m = m0 will generally be significant at the a = 1 - C level whenever ...

    ... m0 is not inside the level C confidence interval.

  3. How do matched pairs inference procedures differ from those we have learned so far?

    Remember that matched pairs experiments fall into two types:

    1. each experimental unit undergoes two treatments.

    2. the experimental units have a natural pairing (wife and husband, identical twins, etc.), and the pairs are divided so that one receives one treatment and one receives the other.

    While we may measure a quantitative variable on each subject (in both treatment groups), generally what we want to know about is the difference in this variables value between pairs (i.e., the difference in a person's score under one treatment vs. the other; the difference in the husband's and wife's scores, etc.). Once we have determined the various values of this random variable that is the difference in scores among pairs, the matched pairs t inference procedures are carried out in exactly the same fashion as we have learned. In the case of a significance test, the null hypothesis is usually H0: m = 0 (i.e., that there is no difference in performance between the two groups).

  4. Example 7.12 demonstrates a sign test-approach for a matched pairs experiment. Notice how the quantitative data (scores and difference in scores) is considered categorically (either a person did better after special training or did not), the count of those doing better after training has a binomial distribution, and the null hypothesis is that the probability of success due to the training is p = 1/2. Under what circumstances would it be better to use such a test than a matched-pairs t test?

    A matched-pairs t test had already been applied to the data of Example 7.12 in Example 7.8. The sign-test analysis was suggested because of the presence of an outlier in this data. Remember that the presence of outliers can throw off the results of any t procedure.