Comparing 2 population parameters. Chapter 13. Introduction: Two Sample problems.
13.1 We can examine two-sample data graphically by comparing dotplots or stemplots (smaller samples) or histograms or boxplots (larger samples). This chapter details
Note! The difference between independent samples (this chapter) and matched pairs or paired samples (past chapters)
*Note: If each population is normally distributed, then μ1 – μ2 will be too
There are four unknown parameters (the two means and the two SD’s). We want to compare the two population means, either by giving a confidence interval for their difference (μ1 – μ2) or by testing the hypothesis of no difference, H0: μ1 = μ2
Group 1: 7, -4, 18, 17, -3, -5, 1, 10, 11, -1
Group 2: -1, 12, -1, -3, 3, -5, 5, 2, -11, -1, -3
HA: μ1 ≠ μ2 HA: μ1 -μ2 ≠ 0
Independence: “Because of the randomization, we are willing to regard the calcium and placebo groups as two independent samples. We are not sampling without replacement from a population of interest in this case.
We use the T test b/c we don’t know population sigma
P value: There are 9 DF (the smaller of the 2 groups) which is approximately .07
The experiment provides some evidence that calcium reduces blood pressure, but the evidence falls short of the traditional 5% and 1% levels. We would fail to reject the null at either of these significance levels.
90% Confidence interval: (t* is 1.833)
(-.753, 11.299). We are 90% confident that the true mean advantage of calcium over a placebo lies in this interval. Since 0 is in the interval, we cannot reject the null.
*Remember, these are small samples! Bigger samples (and bigger df) give smaller t* values making them easier to ‘beat’ and attain significance!
We compare the populations by doing inference about the difference ρ1 - ρ2
HA: ρ1≠ ρ2 HA: ρ -ρ2 ≠ 0