1. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 9 Inferences Based on Two Samples

2. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. 9.1 z Tests and Confidence Intervals for a Difference Between Two Population Means

3. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. The Difference Between Two Population Means Assumptions: 1. X 1,…,X m is a random sample from a population with 2. Y 1,…,Y n is a random sample from a population with 3. The X and Y samples are independent of one another New Notation m: sample size 1 n: sample size 2

4. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Expected Value and Standard Deviation of The expected value is The standard deviation is So is an estimator of Think of this as the parameter.

5. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Null hypothesis: Test statistic value: Test Procedures for Normal Populations With Known Variances same

6. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Alt. Hypothesis = P(Type II Error) Similar to p. 330 formulas

7. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Large-Sample Tests The assumptions of normal population distributions and known values of are unnecessary. The Central Limit Theorem guarantees that has approximately a normal distribution. Rule of thumb: Both m, n>40

8. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Large-Sample Tests Use of the test statistic value along with previously stated rejection regions based on z critical values give large-sample tests whose significance levels are approximately m, n >40 Usually zero

9. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Confidence Interval for with a confidence level of Provided m and n are large, a CI for is confidence bounds can be found by replacing

10. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. 9.2 The Two-Sample t Test and Confidence Interval

11. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Assumptions Both populations are normal, so that X 1,…,X m is a random sample from a normal distribution and so is Y 1,…,Y n. The plausibility of these assumptions can be judged by constructing a normal probability plot of the x i ’s and another of the y i ’s. Normality assumption important for (small-sample) t-tests!

12. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. t Distribution When the population distributions are both normal, the standardized variable has approximately a t distribution…

13. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. df v can be estimated from the data by t Distribution (round down to the nearest integer) Yuck! Don’t do by hand if you can help it.

14. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Two-Sample CI for with a confidence level of The two-sample CI for is

15. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Null hypothesis: Test statistic value: Two-Sample t Test Usually zero

16. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Alternative Hypothesis Rejection Region for Approx. Level Test or The Two-Sample t Test

17. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Pooled t Procedures Assume two populations are normal and have equal variances. If denotes the common variance, it can be estimated by combining information from the two samples. Standardizing using the pooled estimator gives a t variable based on m + n – 2 df. Important: pooled t assumes equal variances

18. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Pooled sample variance Usage in formulas:

19. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. 9.3 Analysis of Paired Data

20. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Paired Data (Assumptions) The data consists of n independently selected pairs (X 1,Y 1 ),…, (X n,Y n ), with Let D 1 = X 1 – Y 1, …, D n = X n – Y n. The D i ’s are assumed to be normally distributed with mean value and variance Important: A natural pairing must exist! Bottom line: Two-sample problem becomes a one-sample problem!

21. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Null hypothesis: Test statistic value: The Paired t Test are the sample mean and standard deviation of the d i ’s. Usually zero

22. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Alternative Hypothesis Rejection Region for Level Test or The Paired t Test Nothing new here!

23. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Confidence Interval for The paired t CI for is confidence bounds can be found by replacing Nothing new here! For large samples, you could use Z test and CI

24. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Paired Data and Two-Sample t Independence between X and Y Positive dependence Remember: Smaller variance means better estimates

25. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Pros and Cons of Pairing 1.For great heterogeneity and large correlation within experimental units, the loss in degrees of freedom will be compensated for by an increased precision associated with pairing (use pairing). 2.If the units are relatively homogeneous and the correlation within pairs is not large, the gain in precision due to pairing will be outweighed by the decrease in degrees of freedom (use independent samples). Usually, we’re in case 1; use pairing if possible.

26. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. 9.4 Inferences Concerning a Difference Between Population Proportions

27. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Difference Between Population Proportions Let X ~Bin(m,p 1 ) and Y ~Bin(n,p 2 ) with X and Y independent variables. Then (q i = 1 – p i )

28. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Large-Samples Null hypothesis: Test statistic value:

29. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

30. Confidence Interval for p 1 – p 2 Note: Standard error here is slightly different than for test!