Math 445—Two-Sample and Paired-Data Inference Examples

Example 1

A researcher is interested in a new diet, which he thinks, on average, will help people lose weight. For a sample of 28 women he records their weights both before and after a diet period of 3 months. The weight losses (before weight – after weight) for the sample are shown in the histogram below. Furthermore, the numerical summaries from the sample of weight losses is also shown.

Variable N Mean StDev Minimum Q1 Median Q3 Maximum

Weight Losses (in lbs) 28 5.39 7.84 -12.00 -1.25 6.00 10.50 19.00

Carry out the appropriate significance test: a) state the hypotheses, b) check any conditions of the test, c) calculate the test statistic, d) calculate the P-value, e) define the P-value in the words of the problem, and f) provide a conclusion—you can use a 0.05 significance level. If you find statistical significance, then further investigate the practical significance (say, with a confidence interval—note it would need to be a one-sided confidence bound in this case).

Example 2

To what extent do syntax textbooks, which analyze the structure of sentences, illustrate gender bias? A study was done to address this question. In the study, sentences were sampled from 10 syntax textbooks. One part of the study examined the use of the words “girl,” “boy,” “woman,” and “man.” In the study, the first two words were labeled juvenile and the second two words adult. For a sample from one of the textbooks, of the 60 female references (i.e., “girl” or “woman”) 48 were juvenile (i.e., “girl”), whereas of the 132 males references 52 were juvenile.

Do these data show evidence that for this textbook the proportion of female references that are juvenile is different from the proportion of male references that are juvenile? Carry out the appropriate significance test. Furthermore, create a 95% confidence interval for the difference in proportions of juvenile references for the textbook (be sure to use the adjustment to make the interval “score-like.”)

Example 3

The Survey of Study Habits and Attitudes (SSHA) is a psychological test designed to measure the motivation, study habits, and attitudes toward learning of college students. These factors, along with ability, are important in explaining success in school. A selective private college gives the SSHA to a random sample of 15 male, first-year students, and also to a random sample of 15 female, first-year students. The graphs of the scores are shown below (as are the numerical summaries).

Variable N Mean StDev Minimum Q1 Median Q3 Maximum

SSHA Scores (Males) 15 122.20 6.88 111 118 122 127 134

SSHA Scores (Females) 15 125.93 5.79 116 122 126 130 136

A researcher wonders if there is any difference, on average, between male and female SSHA scores for first-year students at this college. Carry out the appropriate significance test (including all details).