1. Horng-Chyi HorngStatistics II_Five19 Inference for the Difference in Means -Two Normal Distributions and Variance Unknown &5-3 (&9-3) n Why?

2. Horng-Chyi HorngStatistics II_Five20

3. Horng-Chyi HorngStatistics II_Five21 Replacing  by S p, we have

4. Horng-Chyi HorngStatistics II_Five22

5. Horng-Chyi HorngStatistics II_Five23 Example 9-5 Two catalysts are being analyzed to determine how they affect the mean yield of a chemical process. Specially, catalyst 1 is currently in use, but catalyst 2 is acceptable. Since catalyst 2 is cheaper, it should be adopted, providing it does not change the process yield. A test is run in the pilot plant and results in the data shown in Table 9-2. Is there any difference between the mean yields? Use  = 0.05 and assume equal variances.

6. Horng-Chyi HorngStatistics II_Five24 Table 9-2 Catalyst Yield Data, Example 9-5

7. Horng-Chyi HorngStatistics II_Five25

8. Horng-Chyi HorngStatistics II_Five26

9. Horng-Chyi HorngStatistics II_Five27 is distributed approximately as t with degrees of freedom given by

10. Horng-Chyi HorngStatistics II_Five28

11. Horng-Chyi HorngStatistics II_Five29

12. Horng-Chyi HorngStatistics II_Five30

13. Horng-Chyi HorngStatistics II_Five31 C.I. on the Difference in Means

14. Horng-Chyi HorngStatistics II_Five32 C.I. on the Difference in Means

15. Horng-Chyi HorngStatistics II_Five33

16. Horng-Chyi HorngStatistics II_Five34

17. Horng-Chyi HorngStatistics II_Five35 Paired t-Test n When the observations on the two populations of interest are collected in pairs. Let (X 11, X 21 ), (X 12, X 22 ), …, (X 1n, X 2n ) be a set of n paired observations, in which X 1j ~(  1,  1 2 ) and X 2j ~(  2,  2 2 ) and D j = X 1j – X 2j, j = 1, 2, …, n. Then, to test H 0 :  1 =  2 is the same as performing a one-sample t-test H 0 :  D = 0 since Let (X 11, X 21 ), (X 12, X 22 ), …, (X 1n, X 2n ) be a set of n paired observations, in which X 1j ~(  1,  1 2 ) and X 2j ~(  2,  2 2 ) and D j = X 1j – X 2j, j = 1, 2, …, n. Then, to test H 0 :  1 =  2 is the same as performing a one-sample t-test H 0 :  D = 0 since  D = E(X 1 -X 2 ) = E(X 1 )-E(X 2 ) =  1 -  2  D = E(X 1 -X 2 ) = E(X 1 )-E(X 2 ) =  1 -  2 &5-4 (&9-4)

18. Horng-Chyi HorngStatistics II_Five36

19. Horng-Chyi HorngStatistics II_Five37 Example 9-9 An article in the Journal of Strain Analysis compares several methods for predicting the shear strength for steel plate girders. Data for two of these methods are shown in Table below. We wish to determine whether there is any difference between the two methods.

20. Horng-Chyi HorngStatistics II_Five38

21. Horng-Chyi HorngStatistics II_Five39 Paired Vs. Unpaired If the experimental units are relatively homogeneous (small  ) and the correlation within pairs is small, the gain in precision attributable to pairing will be offset by the loss of degrees of freedom, so an independent-sample experiment should be used. If the experimental units are relatively heterogeneous (large  ) and there is large positive correlation within pairs, the paired experiment should be used. Typically, this case occurs when the experimental units are the same for both treatments; as in Example 9-9, the same girders were used to test the two methods.

22. Horng-Chyi HorngStatistics II_Five40 A Confidence Interval for  D

23. Horng-Chyi HorngStatistics II_Five41 Table 9-5 Time in Seconds to Parallel Park Two Automobiles

24. Horng-Chyi HorngStatistics II_Five42