Download d + t α/2

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
x1 – x2 = d
30 – 25 = 5
23 – 27 = - 4
• sd is the standard deviation of the
differenced data
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
P-Value is the
area highlighted
-|t0|
t0
|t0|
-tα/2
-tα
t0
tα/2
tα
Critical Region
Remember to add the areas in the two-tailed!
Test Statistic:
d
t0 = --------sd/√n
Reject null hypothesis, if
P-value < α
Left-Tailed
Two-Tailed
Right-Tailed
t0 < - tα
t0 < - tα/2
or
t0 > tα/2
t0 > t α
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 STOL3
• 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.
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
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.
Example Problem Cont.
• Requirements: seem to be met from problem info
• Hypothesis
H0: Mean wait time the same (d-bar = 0, new-old)
H1: Mean wait time reduced (d-bar < 0, new-old)
d-bar - 0
• Test Statistic: t0 = ---------------------- = -1.220,
sd / n
p = 0.1286
• Critical Value: tc(9-1,0.05) = -1.860, α = 0.05
• Conclusion: Fail to Reject H0 : not enough evidence to
show that new procedure reduces wait times
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