1. Special Case: Paired Sample T-Test Examples Paired-sample? A.CarRadialBelted 1 ** **Radial, Belted tires 2 ** ** placed on each car. 3 ** ** 4 ** ** B.Person Pre Post 1 ** **Pre- and post-test 2 ** **administered to each 3 ** **person. 4 ** ** C.Student Test1 Test2 1 ** **5 scores from test 1, 2 ** **5 scores from test 2. 3 ** ** 4 ** **

2. Example* Nine steel plate girders were subjected to two methods for predicting sheer strength. Partial data are as follows: GirderKarlsruheLehighdifference, d 1 1.186 1.061 2 1.151 0.992 9 1.559 1.052 Conduct a paired-sample t-test at the 0.05 significance level to determine if there is a difference between the two methods. * adapted from Montgomery & Runger, Applied Statistics and Probability for Engineers.

3. Example (cont.) Hypotheses: H 0 : μ D = 0 H 1 : μ D ≠ 0 t __________ = ______ Calculate difference scores (d), mean and standard deviation, and t calc … d = 0.2736 s d = 0.1356 t calc = ______________________________

4. What does this mean? Draw the picture: Decision: Conclusion:

5. Goodness-of-Fit Tests Procedures for confirming or refuting hypotheses about the distributions of random variables. Hypotheses: H 0 : The population follows a particular distribution. H 1 : The population does not follow the distribution. Examples: H 0 : The data come from a normal distribution. H 1 : The data do not come from a normal distribution.

6. Goodness of Fit Tests (cont.) Test statistic is χ 2 –Draw the picture –Determine the critical value χ 2 with parameters α, ν = k – 1 Calculate χ 2 from the sample Compare χ 2 calc to χ 2 crit Make a decision about H 0 State your conclusion

7. Tests of Independence Hypotheses H 0 : independence H 1 : not independent Example Choice of pension plan. 1. Develop a Contingency Table Worker Type Pension Plan Total #1#2#3 Salaried16014040340 Hourly4060 160 Total200 100500

8. Example 2. Calculate expected probabilities P(#1 ∩ S) = _______________E(#1 ∩ S) = _____________ P(#1 ∩ H) = _______________E(#1 ∩ H) = _____________ (etc.) Worker Type Pension Plan Total #1#2#3 Salaried16014040340 Hourly4060 160 Total200 100500 #1#2#3 S (exp.) H (exp.)

9. Hypotheses 3.Define Hypotheses H 0 : the categories (worker & plan) are independent H 1 : the categories are not independent 4. Calculate the sample-based statistic = ________________________________________ = ______

10. The Test 5. Compare to the critical statistic, χ 2 α, r where r = (a – 1)(b – 1) for our example, say α = 0.01 χ 2 _____ = ___________ Decision: Conclusion:

11. Homework for Wednesday, Nov. 10 pp. 319-323: 25, 27 Pp. 345-346: 12, 13

12. Homework