1. Two Samples z-test We know how to deal with one sample z-test. Now we look at two independent samples.

2. Introduction Suppose we want to compare two samples: look at the difference between their averages. For example: in box A, the average is 110, and the SD of the box is 60. In box B, the average is 90, and the SD is 40. Now we draw one sample at random independently from each of the boxes. From box A, 400 draws are made. From box B, 100 draws are made.

3. Introduction

5. Example 1 The National Assessment of Education Progress (NAEP) monitors trends in school performance. Each year, NAEP administers tests on several subjects to a nationwide sample of 17-year-olds who are in school. The reading test was given in 1990 and again in 2004. The average score went down from 290 to 285. The difference is 5 points. Is the difference real or due to chance? In fact, the NAEP design was quite complicated, but we could consider a simplified version: the test was administered to a nationwide simple random sample of 1,000 students. In 1990, the SD of the scores was 40. In 2004, the SD was 37.

6. Example 1

8. Example 2 In 1999, NAEP found that 13% of the 17-year-old students had taken calculus, compared to 17% in 2004. Suppose they took a simple random sample of size 1,000 in each time. Is the difference real, or due to chance?

9. Example 2

11. Example 3 The model can also be applied to (randomized control) experiments: There are 200 subjects in a small clinical trial on vitamin C. Half the subjects are assigned at random to treatment (2,000 mg of vitamin C daily) and half to control (2,000 mg of placebo). Over the period of the experiment, the treatment group averaged 2.3 colds, and the SD was 3.1. The controls did a little worse: they averaged 2.6 colds and the SD was 2.9. Q: Is the difference in averages statistically significant?

12. Example 3

14. Remark In example 3, there are 2 problems in dealing with the SE: 1. The draws are made without replacement. (Inflates the SE.) 2. The two averages are dependent. (Cuts the SE back down.) But the computation can still be applied, even though there are problems. This is because these two problems cancel out each other. Therefore, the two-sample z-test here still applies.