Lesson 11 - 1

1. Lesson 11 - 1 Inference about Two Means -Dependent Samples

2. Objectives • Distinguish between independent and dependent sampling • Test claims made regarding matched pairs data • Construct and interpret confidence intervals about the population mean difference of matched pairs

3. Vocabulary • Robust – minor deviations from normality will not affect results • Independent – when the individuals selected for one sample do not dictate which individuals are in the second sample • Dependent – when the individuals selected for one sample determine which individuals are in the second sample; often referred to as matched pairs samples

4. Now What • Chapter 10 covered a variety of models dealing with one population • The mean parameter for one population • The proportion parameter for one population • The standard deviation parameter for one population • However, many real-world applications need techniques to compare two populations • Our Chapter 10 techniques do not do these

5. Two Population Examples • We want to test whether a certain treatment helps or not … the measurements are the “before” measurement and the “after” measurement • We want to test the effectiveness of Drug A versus Drug B … we give 40 patients Drug A and 40 patients Drug B … the measurements are the Drug A and Drug B responses • Two precision manufacturers are bidding for our contract … they each have some precision (standard deviation) … are their precisions significantly different

6. Types of Two Samples An independentsampleis when individuals selected for one sample have no relationship to the individuals selected for the other • Examples • 50 samples from one store compared to 50 samples from another • 200 patients divided at random into two groups of 100 each A dependentsampleis one when each individual in the first sample is directly matched to one individual in the second • Examples • Before and after measurements (a specific person’s before and the same person’s after) • Experiments on identical twins (twins matched with each other

7. Match Pair Designs • Remember back to Chapter 1 discussions on design of experiments: the dependent samples were often called matched-pairs • Matched-pairs is an appropriate term because each observation in sample 1 is matched to exactly one in sample 2 • The person before  the person after • One twin  the other twin • An experiment done on a person’s left eye  the same experiment done on that person’s right eye

8. Terms • d-bar or d – the mean of the differences of the two samples • sd is the standard deviation of the differenced data x1 – x2 = d 30 – 25 = 5 23 – 27 = - 4

9. Requirements Testing a claim regarding the difference of two means using matched pairs • Sample is obtained using simple random sampling • Sample data are matched pairs • Differences are normally distributed with no outliers or the sample size, n, is large (n ≥ 30)

10. Classical and P-Value Approach – Matched Pairs -tα -tα/2 tα tα/2 d Test Statistic: t0 = --------- sd/√n P-Value is thearea highlighted -|t0| |t0| t0 t0 Critical Region Remember to add the areas in the two-tailed!

11. Confidence Interval – Matched Pairs Lower Bound: d – tα/2 · sd/√n Upper Bound: d + tα/2 · sd/√n tα/2 is determined using n - 1 degrees of freedom d is the mean of the differenced data sd is the standard deviation of the differenced data Note: The interval is exact when population is normally distributed and approximately correct for nonnormal populations, provided that n is large.

12. Two-sample, dependent, T-Test on TI • If you have raw data: • enter data in L1 and L2 • define L3 = L1 – L2 (or vice versa – depends on alternative Hypothesis) • L1 – L2 STOL3 • Press STAT, TESTS, select T-Test • raw data: List set to L3 and freq to 1 • summary data: enter as before

13. Example Problem Carowinds quality control manager feels that people are waiting in line for the new roller coaster too long. To determine is a new loading and unloading procedure is effective in reducing wait time, she measures the amount of time people are waiting in line for 7 days and obtains the following data. A normality plot and a box plot indicate that the differences are apx normal with no outliers. Test the claim that the new procedure reduces wait time at the α=0.05 level of significance.

14. Example Problem Cont. seem to be met from problem info • Requirements: • HypothesisH0: H1: • Test Statistic: • Critical Value: • Conclusion: Mean wait time the same (d-bar = 0, new-old) Mean wait time reduced (d-bar < 0, new-old) d-bar - 0 t0 = ---------------------- sd / n = -1.220, p = 0.1286 tc(9-1,0.05) = -1.860, α = 0.05 Fail to Reject H0 : not enough evidence to show that new procedure reduces wait times

15. Summary and Homework • Summary • Two sets of data are dependent, or matched-pairs, when each observation in one is matched directly with one observation in the other • In this case, the differences of observation values should be used • The hypothesis test and confidence interval for the difference is a “mean with unknown standard deviation” problem, one which we already know how to solve • Homework • pg 582-587; 1, 2, 4-8, 12, 15, 18, 19

16. HW Answers 6) independent 8) dependent 12a) your task 12b) d-bar = -1.075 sd = 3.833 12c) Fail to reject H0 12d) [-5.82, 3.67] 18) example problem in class