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

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 unbiased estimator of

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

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

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.

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

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.

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)

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

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.

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

19. 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

20. 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.

21. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Alternative Hypothesis Rejection Region for Level Test or The Paired t Test

22. 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

23. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Paired Data and Two-Sample t Independence between X and Y Positive dependence

24. 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).

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

26. 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 )

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

28. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Alternative Hypothesis Rejection Region or Large-Samples Valid provided

29. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Alt. Hypothesis General Expressions for

30. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Alt. Hypothesis General Expressions for where

31. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Sample Size For the case m = n, the level test has type II error probability at the alternative values p 1, p 2 with p 1 – p 2 = d when

32. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Confidence Interval for p 1 – p 2

33. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. 9.5 Inferences Concerning Two Population Variances

34. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. The F Distribution The F probability distribution has parameters v 1 (number of numerator df) and v 2 (number of denominator df). If X 1 and X 2 are independent chi-squared rv’s with v 1 and v 2 df, then

35. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. The F Distribution Density Curve Property f F density curve Shaded area =

36. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Inferential Methods Let X 1,…,X m and Y 1,…,Y n be random (independent) samples from normal distributions with variances respectively. Let the two sample variances, then

37. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. F Test for Equality of Variances Null hypothesis: Test statistic value:

38. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Alternative Hypothesis Rejection Region or F Test for Equality of Variances

39. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. P-Values for F Tests The P-value for an upper-tailed F test is the area under the F curve with appropriate numerator and denominator df to the right of the calculated f.