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