Transcript 11.1

Chapter 11
Section 1
Inference about Two Means:
Dependent Samples
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 1 of 26
Chapter 11 – Section 1
● Learning objectives
1

Distinguish between independent and dependent
sampling
2
 Test hypotheses made regarding matched-pairs data
3
 Construct and interpret confidence intervals about the
population mean difference of matched-pairs data
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 2 of 26
Chapter 11 – Section 1
● Learning objectives
1

Distinguish between independent and dependent
sampling
2
 Test hypotheses made regarding matched-pairs data
3
 Construct and interpret confidence intervals about the
population mean difference of matched-pairs data
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 3 of 26
Chapter 11 – Section 1
● 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, there are many real-world applications
that need techniques to compare two
populations
 Our Chapter 10 techniques do not do these
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 4 of 26
Chapter 11 – Section 1
● Examples of situations with two populations
 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
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 5 of 26
Chapter 11 – Section 1
● In certain cases, the two samples are very
closely tied to each other
● 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
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 6 of 26
Chapter 11 – Section 1
● On the other extreme, the two samples can be
completely independent of each other
● An independent sample is when individuals
selected for one sample have no relationship to
the individuals selected for the other
● Examples
 Fifty samples from one factory compared to fifty
samples from another
 Two hundred patients divided at random into two
groups of one hundred
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 7 of 26
Chapter 11 – Section 1
● The dependent samples are 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
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 8 of 26
Chapter 11 – Section 1
● Learning objectives
1

Distinguish between independent and dependent
sampling
2
 Test hypotheses made regarding matched-pairs data
3
 Construct and interpret confidence intervals about the
population mean difference of matched-pairs data
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 9 of 26
Chapter 11 – Section 1
● The method to analyze matched-pairs is to
combine the pair into one measurement
 “Before” and “After” measurements – subtract the
before from the after to get a single “change”
measurement
 “Twin 1” and “Twin 2” measurements – subtract the 1
from the 2 to get a single “difference between twins”
measurement
 “Left eye” and “Right eye” measurements – subtract
the left from the right to get a single “difference
between eyes” measurement
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 10 of 26
Chapter 11 – Section 1
● Specifically, for the before and after example,
 d1 = person 1’s after – person 1’s before
 d2 = person 2’s after – person 1’s before
 d3 = person 3’s after – person 1’s before
● This creates a new random variable d
● We would like to reformulate our problem into a
problem involving d (just one variable)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 11 of 26
Chapter 11 – Section 1
● How do our hypotheses translate?
 The two means are equal … the mean difference is
zero … μd = 0
 The two means are unequal … the mean difference is
non-zero … μd ≠ 0
● Thus our hypothesis test is
 H0: μd = 0
 H1: μd ≠ 0
 The standard deviation σd is unknown
● We know how to do this!
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 12 of 26
Chapter 11 – Section 1
● To solve
 H0: μd = 0
 H1: μd ≠ 0
 The standard deviation σd is unknown
● This is exactly the test of one mean with the
standard deviation being unknown
● This is exactly the subject covered in section
10.2
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 13 of 26
Chapter 11 – Section 1
● In order for this test statistic to be used, the data
must meet certain conditions
 The sample is obtained using simple random
sampling
 The sample data are matched pairs
 The differences are normally distributed with no
outliers, or the sample size is (n at least 30)
● These are the usual conditions we need to make
our Student’s t calculations
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 14 of 26
Chapter 11 – Section 1
● An example … whether our treatment helps or
not … helps meaning a higher measurement
● The “Before” and “After” results
Before
After
Difference
7.2
6.6
6.5
5.5
8.6
7.7
6.2
5.9
1.4
1.1
– 0.3
0.4
5.9
7.7
1.8
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 15 of 26
Chapter 11 – Section 1
● Hypotheses
 H0: μd = 0 … no difference
 H1: μd > 0 … helps
 (We’re only interested in if our treatment makes things
better or not)
 α = 0.01
● Calculations
 n=5
 d = .88
 sd = .83
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 16 of 26
Chapter 11 – Section 1
● Calculations
 n=5
 d = 0.88
 sd = 0.83
● The test statistic is
d  d
0.88  0
t0 

 2.36
s/ n
0.83 / 5
● This has a Student’s t-distribution with 4 degrees
of freedom
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 17 of 26
Chapter 11 – Section 1
● Use the Student’s t-distribution with 4 degrees of
freedom
● The right-tailed α = 0.01 critical value is 3.75
● 2.36 is less than 3.75 (the classical method)
● Thus we do not reject the null hypothesis
● There is insufficient evidence to conclude that
our method significantly improves the situation
● We could also have used the P-Value method
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 18 of 26
Chapter 11 – Section 1
● Matched-pairs tests have the same various
versions of hypothesis tests
 Two-tailed tests
 Left-tailed tests (the alternatively hypothesis that the
first mean is less than the second)
 Right-tailed tests (the alternatively hypothesis that the
first mean is greater than the second)
 Different values of α
● Each can be solved using the Student’s t
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 19 of 26
Chapter 11 – Section 1
● Each of the types of tests can be solved using
either the classical or the P-value approach
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 20 of 26
Chapter 11 – Section 1
● A summary of the method
 For each matched pair, subtract the first observation
from the second
 This results in one data item per subject with the data
items independent of each other
 Test that the mean of these differences is equal to 0
● Conclusions
 Do not reject that μd = 0
 Reject that μd = 0 ... Reject that the two populations
have the same mean
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 21 of 26
Chapter 11 – Section 1
● Learning objectives
1

Distinguish between independent and dependent
sampling
2
 Test hypotheses made regarding matched-pairs data
3
 Construct and interpret confidence intervals about the
population mean difference of matched-pairs data
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 22 of 26
Chapter 11 – Section 1
● We’ve turned the matched-pairs problem in one
for a single variable’s mean / unknown standard
deviation
 We just did hypothesis tests
 We can use the techniques in Section 9.2 (again,
single variable’s mean / unknown standard deviation)
to construct confidence intervals
● The idea – the processes (but maybe not the
specific calculations) are very similar for all the
different models
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 23 of 26
Chapter 11 – Section 1
● Confidence intervals are of the form
Point estimate ± margin of error
● This is precisely an application of our results for
a population mean / unknown standard deviation
 The point estimate
d
and the margin of error
t / 2
sd
n
for a two-tailed test
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 24 of 26
Chapter 11 – Section 1
● Thus a (1 – α) • 100% confidence interval for the
difference of two means, in the matched-pair
case, is
sd
d  t / 2
n
where tα/2 is the critical value of the Student’s
t-distribution with n – 1 degrees of freedom
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 25 of 26
Summary: Chapter 11 – Section 1
● Two sets of data are dependent, or matchedpairs, 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
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 26 of 26