1. 6.3 Two-Sample Inference for Means November 17, 2003

2. Paired Differences Matched Pairs Design experimental plan where the experimental units are divided into halves and two treatments are randomly assigned to the halves Attempting to determine if there is a significant difference between the mean responses of the treatments

3. Procedure Obtain the difference between responses for each experimental unit Analyze the differences using a one- sample approach If a large sample is obtained, use critical values from the Standard Normal distribution (z) Otherwise, use critical values from the corresponding t distribution

4. Example 9 pg 370 Students worked with a company on the monitoring of the operation of an end-cut router in the manufacture of a wood product. They measured the critical dimensions of a number of pieces of a type as they came off the router. Both a leading- edge and a trailing-edge measurement were made on each piece. Both were to have a target value of.172 in.

5. PieceLeading Edge Trailing Edge 1.168.169 2.170.168 3.165.168 4.165.168 5.170.168

6. PieceLeading Edge Trailing Edge Difference 1.168.169-.001 2.170.168.002 3.165.168-.003 4.165.168-.003 5.170.168.002

7. Confidence Interval

8. Hypothesis Test Is there a significant difference between the measurements at α =.01? Ho: µ = 0 Ha: µ ≠ 0

9. Independent Samples The goal of this type of inference is to compare the mean response of two variables (or treatments) when the data are not paired or matched A key assumption that will be made is that the separate samples used to collect information concerning the two variables are independent One sample does not influence the other sample in any way Furthermore, one must assume that both sampled populations are normally distributed

10. Large Sample Comparison The quantity of interest is a linear combination of population means, namely µ 1 - µ 2 The above quantity will be estimated by As a result, various quantities of the sampling distribution of the difference, under the assumption of equality between the means, need to be developed Ho: µ 1 - µ 2 = # Ha: µ 1 - µ 2 ≠ #

11. Test Statistic

12. Confidence Interval

13. Example A company research effort involved finding a workable geometry for molded pieces of a solid. A comparison was made between the weight of molded pieces and the weight of irregularly shaped pieces that could be poured into the same container. A series of 30 attempts to pack both the molded and the irregular pieces of the solid were compared. Is there enough evidence to suggest that the irregular pieces produced higher weights?

14. The Data 1 = molded n 1 = 30 s 1 = 9.31 = 164.65 2=irregular n 2 = 30 s 2 = 8.51 = 179.65

15. Small Samples If at least one sample size is small, then use critical values from a t distribution for constructing confidence intervals and performing hypothesis tests with degrees of freedom obtained by Satterthwaite’s Approximation

16. Satterthwaite’s Approximation

17. Test Statistic

18. Confidence Interval

19. Example The data shown gives spring lifetimes under two different levels of stress (900 and 950 N/mm 2 ). Do the data give evidence of a significant difference at α =.05? 950 Level900 Level 225 171 198 187 189 135 162 135 117 162 216 162 153 216 225 216 306 225 243 189

20. Minitab Output Descriptive Statistics: 950, 900 Variable N Mean Median TrMean StDev SE Mean 950 10 168.3 166.5 167.6 33.1 10.5 900 10 215.1 216.0 211.5 42.9 13.6

21. Assignment Page 385: #3, #4