Lesson 11 - 1

Inference about Two Means -

Dependent Samples

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

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

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

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

Types of Two Samples

An independent sample is 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 dependent sample is 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

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

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 = 523 – 27 = - 4

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)

Classical and P-Value Approach – Matched Pairs

dTest Statistic: t0 = --------- sd/√n

tα-tα/2 tα/2-tα

Critical Region

P-Value is thearea highlighted

|t0|-|t0|t0 t0

Reject null hypothesis, if

P-value < α

Left-Tailed Two-Tailed Right-Tailed

t0 < - tα

t0 < - tα/2

or

t0 > tα/2

t0 > tα

Remember to add the areas in the two-tailed!

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.

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 STOL3

• Press STAT, TESTS, select T-Test– raw data: List set to L3 and freq to 1– summary data: enter as before

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.

Day Mon Tue Wed Thu Fri Sat Sat Sun Sun

Old 11.6 25.9 20.0 38.2 57.3 32.1 81.8 57.1 62.8

New 10.7 28.3 19.2 35.9 59.2 31.8 75.3 54.9 62.0

Example Problem Cont.

• Requirements:

• HypothesisH0: H1:

• Test Statistic:

• Critical Value:

• Conclusion:

Mean wait time reduced (d-bar < 0, new-old)Mean wait time the same (d-bar = 0, new-old)

seem to be met from problem info

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

= -1.220, p = 0.1286 d-bar - 0t0 = ---------------------- sd / n

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

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