Comparing 2 Population Means

1. Comparing 2 Population Means • Goal is to compare the mean response (or other quantity) of two different populations. • We sample from two groups of individuals. • We have one measurement or observation per individual. • These two groups are assumed to be independent. • A common use is comparing the mean response of experimental and control groups in a randomized, controlled experiment.

2. Formulating hypotheses • Label the means of populations 1 and 2 as m1 and m2. • Null hypothesis says mean population responses are the same • H0: m1 - m2= 0. • Possible alternative hypotheses: • Ha: m1 - m2> 0 (population 1 mean bigger) • Ha: m1 - m2< 0 (population 2 mean bigger) • Ha: m1 - m2 0 (means are different)

3. What test statistic do we use? • To estimate m1 – m2, we use , where is the sample mean from population 1 and is the sample mean from population 2.

4. Two-sample t procedures • We rarely know s1 and s2 (the standard deviations of the two populations). • As in the case of the one-sample test for m, if we don’t know s, we estimate it using s (sample standard deviation). • Since we made 2 estimates, our test statistic doesn’t have an exact t distribution (except in the special case that s1 = s2).

5. Two-Sample t Statistic With H0: m1m2 and n1 and n2 being the sample sizes from populations 1 and 2, respectively.

6. Approximating the t distribution for a 2-sample test statistic • We can approximate the distribution of a 2-sample test statistic using a t distribution – the trick is figuring out the degrees of freedom (d.f.). • We can approximate the d.f. with the smaller of (n1 1) and (n21). • Software allows for a slightly better approximation. • These methods yield similar results, with the first being more conservative.

7. Ex: Accountants vs. Planners • Random samples of accountants and financial planners were taken • The groups were asked about their starting salaries. • Is there a significant difference in mean starting salaries? Use alpha=10%. • Data in P:\Data\MATH\Swall\plan&cpa.mtw

8. Robustness of 2-sample procedures • In general, 2-sample t procedures are more robust than 1-sample t methods. • When the 2 population distributions have similar shapes and the sample sizes are equal, the t procedures can be used for small sample sizes. • Larger sample sizes are needed when population distributions have different shapes. • Procedures are more robust when sample sizes equal.

9. Two-Sample t CIs • The confidence interval for m1 – m2 given byhas confidence level C. Here, t* is the value of the t(k) distribution with area C between –t* and t*. The degrees of freedom (k) are estimated by software or chosen to be the smaller of n1–1 and n2–1.

10. Example • Pat wants to compare the cost of one- and two-bedroom apartments locally. She collects data for a random sample of 10 advertisements of each type. • Here are the rents for the two-bedroom apartments (in $/month):595, 500, 580, 650, 675, 675, 750, 500, 495, 670 • Here are the rents for one-bedroom apartments:500, 650, 600, 505, 450, 550, 515, 495, 650, 395 • Find a 95% CI for the additional cost of a second bedroom.

11. Example • A study of iron deficiency among infants compared samples of infants following different feeding regimens. One group contained breast-fed infant, while children in another group were fed a standard baby formula without any iron supplements. Here are summary results on blood hemoglobin levels at 12 months of age:Is there significant evidence that the mean hemoglobin level is higher among breast-fed babies?