Chapter 7: SAMPLING DISTRIBUTIONS & POINT ESTIMATION OF PARAMETERS. Part 2: Differences in sample means

Transcription

1 Chapter 7: SAMPLING DISTRIBUTIONS & POINT ESTIMATION OF PARAMETERS Part 2: Differences in sample means What if we re interested in estimating the difference in means between two populations? Value of interest: µ 1 µ 2 Estimator: X1 X Pop'n 1 Pop'n Y The above picture shows two populations with different means, µ 1 µ

2 Pop'n 1 Pop'n Y If the populations had the same mean, then the two distributions would be on top of each other (no distinction), and µ 1 µ 2 = 0. If the population means are the same, we would think that X 1 and X 2 should be about the same and the difference X 1 X 2 to be near zero. We want to know the behavior of our estimator X 1 X 2. So far, we ve only discussed the behavior of X. 2

3 The sampling distribution of X1 X 2 : We will assume the sample from each group was taken independent of each other (two independent samples). E( X 1 X 2 ) = E( X 1 ) E( X 2 ) = µ 1 µ 2 where µ 1 is the population mean of pop n 1 where µ 2 is the population mean of pop n 2 V ( X 1 X 2 ) = V ( X 1 ) + V ( X 2 ) {since independent} = σ2 1 n 1 + σ2 2 n 2 where σ 2 1 where σ 2 2 is the population variance of pop n 1 is the population variance of pop n 2 3

4 X 1 X 2 is a random variable with E( X 1 X 2 ) = µ 1 µ 2 and V ( X 1 X 2 ) = σ2 1 n 1 + σ2 2 n 2 So, we have the expected value and the variance of this random variable of interest. But we d like to know the full distribution of the r.v. 4

5 IF both original populations were normal, then X 1 and X 2 are linear combinations of normal random variables, and X 1 X 2 is also a linear combination of normals... so X 1 X 2 N(µ 1 µ 2, σ2 1 n 1 + σ2 2 n 2 ) Again, we have a random variable of interest X 1 X 2 that has a normal distribution with known predictable behavior. What if both original populations were NOT normal? If n 1 and n 2 are both greater than 30, then we can apply the central limit theorem to show that X 1 X 2 is again, normally distributed. 5

6 Approximate Sampling Distribution for X 1 X 2 If we have two independent populations with means µ 1 and µ 2 and variances σ 2 1 and σ2 2, and if X1 and X 2 are sample means of two independent random samples of size n 1 and n 2 from the two populations, then the sampling distribution of Z = ( X 1 X 2 ) (µ 1 µ 2 ) σ 2 1 n 1 + σ2 2 n 2 is approximately standard normal (if the conditions of the central limit theorem apply). If the original populations were normal to begin with, then Z is exactly a standard normal. 6

7 Example: Difference in means A random sample of n 1 =20 observations are taken from a normal population with mean 30. A random sample of n 2 =25 observations are taken from a different normal population with mean 27. Both populations have σ 2 = 8. What is the probability that X1 X 2 exceeds 5? 7

8 Example: Picture tube brightness (problem 7-14 p.248) A consumer electronics company is comparing the brightness of two different types of picture tubes. Type A is the present model, and is thought to have a population mean brightness of 100 and a known standard deviation of 16. Type B has an unknown mean brightness and standard deviation equal to type A. If µ B exceeds µ A, the manufacturer would like to adopt type B for use. A random sample of 25 is taken from each type... 8

9 The observed difference in sample means is x B x A = 6.75 (so, the sample mean brightness for type B was higher than the sample mean for type A, but is it high enough). What decision should they make? 9