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Chapter 17
Two-Sample Problems
BPS - 3rd Ed.
Chapter 17
1
Two-Sample Problems
 The
goal of inference is to compare the
responses to two treatments or to
compare the characteristics of two
populations.
 We have a separate sample from each
treatment or each population.
– Each sample is separate. The units are not
matched, and the samples can be of
differing sizes.
BPS - 3rd Ed.
Chapter 17
2
Case Study
Exercise and Pulse Rates
A study if performed to compare the mean resting
pulse rate of adult subjects who regularly exercise
to the mean resting pulse rate of those who do not
regularly exercise.
n
mean
std. dev.
Exercisers
29
66
8.6
Nonexercisers
31
75
9.0
This is an example of when to use the two-sample t procedures.
BPS - 3rd Ed.
Chapter 17
3
Conditions for Comparing Two Means
 We
have two independent SRSs, from two
distinct populations
– that is, one sample has no influence on the other-matching violates independence
– we measure the same variable for both samples.
 Both
populations are Normally distributed
– the means and standard deviations of the
populations are unknown
– in practice, it is enough that the distributions have
similar shapes and that the data have no strong
outliers.
BPS - 3rd Ed.
Chapter 17
4
Two-Sample t Procedures
 In
order to perform inference on the difference
of two means (m1 – m2), we’ll need the standard
deviation of the observed difference x1  x2 :
2
σ1
n1
BPS - 3rd Ed.

2
σ2
n2
Chapter 17
5
Two-Sample t Procedures
We don’t know the population
standard deviations s1 and s2.
 Problem:
 Solution:
Estimate them with s1 and s2. The
result is called the standard error, or estimated
standard deviation, of the difference in the
sample means.
SE 
BPS - 3rd Ed.
2
s1
n1

Chapter 17
2
s2
n2
6
Two-Sample t Confidence Interval


Draw an SRS of size n1 form a Normal population with
unknown mean m1, and draw an independent SRS of
size n2 form another Normal population with unknown
mean m2.
A confidence interval for m1 – m2 is:
x1  x2   t 
2
s1
n1

2
s2
n2
– here t* is the critical value for confidence level C for the
t(k) density curve. The degrees of freedom k are equal
to the smaller of n1 – 1 and n2 – 1.
BPS - 3rd Ed.
Chapter 17
7
Case Study
Exercise and Pulse Rates
Find a 95% confidence interval for the difference in
population means (nonexercisers minus exercisers).
2
2
2 (8.6)2
s
s

(9.0)
1
x1  x2  t
 2  75  66  2.048

n1 n2
31
29
 9  4.65
 4.35 to 13.65
“We are 95% confident that the difference in mean
resting pulse rates (nonexercisers minus exercisers) is
between 4.35 and 13.65 beats per minute.”
BPS - 3rd Ed.
Chapter 17
8
Two-Sample t Significance Tests


Draw an SRS of size n1 form a Normal population with
unknown mean m1, and draw an independent SRS of
size n2 form another Normal population with unknown
mean m2.
To test the hypothesis H0: m1 = m2, the test statistic is:
t

( x1  x 2 )  ( μ1  μ 2 )
2
s1
n1


2
s2

n2
x1  x 2
2
s1
n1

2
s2
n2
Use P-values for the t(k) density curve. The degrees of
freedom k are equal to the smaller of n1 – 1 and n2 – 1.
BPS - 3rd Ed.
Chapter 17
9
P-value for Testing Two Means

Ha: m1 > m2


Ha: m1 < m2


P-value is the probability of getting a value as large or
larger than the observed test statistic (t) value.
P-value is the probability of getting a value as small or
smaller than the observed test statistic (t) value.
Ha: m1 m2

P-value is two times the probability of getting a value as
large or larger than the absolute value of the observed test
statistic (t) value.
BPS - 3rd Ed.
Chapter 17
10
Case Study
Exercise and Pulse Rates
Is the mean resting pulse rate of adult subjects who
regularly exercise different from the mean resting pulse
rate of those who do not regularly exercise?

Null: The mean resting pulse rate of adult subjects who
regularly exercise is the same as the mean resting pulse
rate of those who do not regularly exercise? [H0: m1 = m2]

Alt: The mean resting pulse rate of adult subjects who
regularly exercise is different from the mean resting pulse
rate of those who do not regularly exercise? [Ha : m1 ≠ m2]
Degrees of freedom k = 28 (smaller of 31 – 1 and 29 – 1).
BPS - 3rd Ed.
Chapter 17
11
Case Study
1.
Hypotheses:
2.
Test Statistic:
t 
H0: m1 = m2
x1  x 2
2
s1
n1

2
s2
Ha: m1 ≠ m2
75  66

(9.0)
n2
31
2

 3.961
(8.6)
2
29
3.
P-value:
P-value = 2P(T > 3.961) = 0.000207 (using a computer)
P-value is smaller than 2(0.0005) = 0.0010 since t = 3.961 is
greater than t* = 3.674 (upper tail area = 0.0005) (Table C)
4.
Conclusion:
Since the P-value is smaller than a = 0.001, there is very strong
evidence that the mean resting pulse rates are different for the two
populations (nonexercisers and exercisers).
BPS - 3rd Ed.
Chapter 17
12
Robustness of t Procedures




The two-sample t procedures are more robust than
the one-sample t methods, particularly when the
distributions are not symmetric.
When the two populations have similar distribution
shapes, the probability values from the t table are
quite accurate, even when the sample sizes are as
small as n1 = n2 = 5.
When the two populations have different distribution
shapes, larger samples are needed.
In planning a two-sample study, it is best to choose
equal sample sizes. In this case, the probability
values are most accurate.
BPS - 3rd Ed.
Chapter 17
13
Using the t Procedures




Except in the case of small samples, the assumption that
each sample is an independent SRS from the population of
interest is more important than the assumption that the two
population distributions are Normal.
Small sample sizes (n1 + n2 < 15): Use t procedures if
each data set appears close to Normal (symmetric, single
peak, no outliers). If a data set is skewed or if outliers are
present, do not use t.
Medium sample sizes (n1 + n2 ≥ 15): The t procedures
can be used except in the presence of outliers or strong
skewness in a data set.
Large samples: The t procedures can be used even for
clearly skewed distributions when the sample sizes are
large, roughly n1 + n2 ≥ 40.
BPS - 3rd Ed.
Chapter 17
14
Details of t Degrees of Freedom



Using degrees of freedom k as the smallest of
n1 – 1 and n2 – 1 is only a rough approximation
to the actual degrees of freedom for the twosample t procedures.
A better approximation that is used by software
uses a function of the sample sizes and sample
standard deviations to compute degrees of
freedom df.
Use of df in the two-sample t procedures gives
more accurate results than when using k.
BPS - 3rd Ed.
Chapter 17
15
Details of t Degrees of Freedom
BPS - 3rd Ed.
Chapter 17
16
Case Study
Exercise and Pulse Rates
Compute the degrees of freedom df used by software
to analyze these data using two-sample t procedures.
df 
2


 n n 
 1 2
2
2
2
2
1  s1 
1  s2 
   
n 1 n  n 1 n 
1  1
2  2 
s2 s2
1  2

2
  9.0   8.6  
 31  29 


2
2
2
2
1   9.0  
1   8.6  

 

30  31  28  29 




2
2
 57.97
This is the degrees of freedom value used by software
when computing critical values and P-values.
BPS - 3rd Ed.
Chapter 17
17
Comparing Two
Population Standard Deviations


Suppose we have independent SRSs from two Normal
populations with unknown means and standard deviations:
a sample of size n1 from N(m1, s1) and a sample of size n2
from N(m2, s2).
To test the hypothesis of equal spread,
H0: s1 = s2
Ha: s1 ≠ s2 ,
use the ratio of sample variances, called the F statistic:
F = s12 / s22

Since s1 and s2 each have associated degrees of freedom
(n1 – 1 and n2 – 1, respectively), F has two types of
degrees of freedom:
– Numerator degrees of freedom: j = n1 – 1
– Denominator degrees of freedom: k = n2 – 1
BPS - 3rd Ed.
Chapter 17
18
Characteristics of F Distributions






The F distributions are right-skewed.
F(j, k) is the notation for the F distribution with j
numerator degrees of freedom and k denominator
degrees of freedom.
The area under the density curve equals 1
Since the F statistic takes on only positive values, the
F distribution has no probability below 0.
The peak of the F density curve is near 1.
If s1 = s2, we would expect s1 and s2 to be close in size,
yielding an F statistic near 1
– values of F far from 1 provide evidence against the null
hypothesis of equal standard deviations.
BPS - 3rd Ed.
Chapter 17
19
Characteristics of F Distributions
Density curve for the F(9, 10) distribution.
BPS - 3rd Ed.
Chapter 17
20
Using Table D
F Distribution Critical Values

Table D gives the upper tail area p critical points of the F
distribution for upper tail areas p = 0.100, 0.050, 0.025,
0.010, and 0.001. For example, the critical points for the
F(15, 20) distribution are:
p
F*

0.100
1.84
0.050
2.20
0.025
2.57
0.010
3.09
0.001
4.56
Since the F distributions are not symmetric about 0 (as are
the standard Normal and t distributions), and Table D only
gives upper tail areas, we shall eliminate the need for lowertail critical values by always computing F as the larger s2
divided by the smaller s2.
BPS - 3rd Ed.
Chapter 17
21
F Test Statistics
If you do not use statistical software, arrange the
two-sided F test as follows:
BPS - 3rd Ed.
Chapter 17
22
Case Study
Reading Ability
Two methods of teaching children to read are being investigated to
see if one provides more consistent improvement in reading ability
across students. Method 1 is implemented in one randomly
chosen class of 16 students, while another class of 16 students is
randomly chosen to represent Method 2. The standard deviations
of the improvement in reading ability scores (as measured by a
standardized test) are s1 = 3.905 and s2 = 3.186. Test to see if the
standard deviations differ.
Degrees of freedom: j = 15 and k = 15.
p
0.100
0.050
0.025
0.010
0.001
F*
1.97
2.40
2.86
3.52
5.54
BPS - 3rd Ed.
Chapter 17
23
Case Study
1.
Hypotheses:
2.
Test Statistic:
H 0 : s1 = s2
F
2
s1
2
s2

H a : s1 ≠ s2
(3.905)
2
(3.186)
2

15.249
 1.502
10.151
3.
P-value:
P-value = 2P(F > 1.502) = 0.4398 (using a computer)
P-value is larger than 2(0.100) = 0.200 since F = 1.502 is
smaller than F* = 1.97 (upper tail area = 0.100) (Table D)
4.
Conclusion:
Since the P-value is larger than a = 0.200, there is not strong
evidence that the standard deviations differ for the two teaching
methods; thus, neither gives more consistent improvement.
BPS - 3rd Ed.
Chapter 17
24