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Lesson 11 – R
Review of Chapter 11
Comparison of Two Populations
Similarities in Hypothesis Test
Processes
Parameter
Two Means
(Independent)
Two Means
(Dependent)
Two
Proportions
Two
Variances
Two
Std Devs
H0:
μ1 = μ2
μ1 = μ2
p1 = p2
σ12 = σ22
σ1 = σ2
(2-tailed) H1:
μ1 ≠ μ2
μ1 ≠ μ2
p1 ≠ p2
σ12 ≠ σ22
σ1 ≠ σ2
(L-tailed) H1:
μ1 < μ2
μ1 < μ2
p1 < p2
σ12 < σ22
σ1 < σ2
(R-tailed) H1:
μ1 > μ2
μ1 > μ2
p1 > p2
σ12 > σ22
σ1 > σ2
Test statistic
Difference
Difference
Difference
Ratio
Ratio
Critical value
Student t
Student t
Normal
F
F
Putting it All Together
Provided np(1-p) ≥ 10 and n ≤ 0.05N
for each sample,
use normal distribution with
p1 – p2
z0 = --------------------------------1
1
p (1- p)
--- + --n1
n2
Provided each sample size ≥ 30 or
differences come from population that
is normally distributed, use tdistribution with n-1 degrees of
freedom and
(x1 – x2) – (μ1 – μ2 )
t0 = ------------------------------s12
s22
----- + ----n1
n2
where
x1 + x 2
p = -----------n1 + n2
proportion, p
What parameters
are addressed in
the claim?
μ
Independent
Dependent or
Independent
Sampling
Dependent
s or s2
Provided data are
normally distributed,
use F-distribution with
s12
F0 = -----------s22
Provided each sample size ≥ 30
or differences come from
population that is normally
distributed, use t-distribution
with n-1 degrees of freedom and
d – μd
t0 = -----------sd / n
Multiple Choice Evaluations
Which of the following is a characteristic of dependent
(or matched-pairs) samples?
1) The observations from the sample are matched
to the hypothesis being tested
2) The observations from sample 1 and sample 2
are paired with each other
3) The observations from sample 1 are independent
of the observations from sample 2
4) The mean of sample 2 depends on the mean of
sample 1
Section 1
Multiple Choice Evaluations
When we have dependent (or matched-pairs) data, then
we should
1) Test the equality of their squares using a chisquare distribution
2) Test the equality of their means using
proportions
3) Test the equality of their variances using a
normal distribution
4) Test their differences using a student tdistribution
Section 1
Multiple Choice Evaluations
A researcher collected data from two sets of patients,
both chosen at random from a large population of
patients. If there is no interaction between the two
groups, then this is an example of
1)
2)
3)
4)
Independent samples
Dependent samples
Stratified sampling
Descriptive statistics
Section 2
Multiple Choice Evaluations
The standard deviation for the difference of two means
from independent samples involves
1)
2)
3)
4)
The standard deviation of each sample
The mean of each sample
The difference between the two means
All of the above
Section 2
Multiple Choice Evaluations
To compare two population proportions, we
1) Multiply the two proportions and take the square
root
2) Subtract one proportion from the other and
divide by the appropriate standard deviation
3) Subtract one proportion from the other and use
the chi-square distribution
4) Divide one proportion by the other and use the
normal distribution
Section 3
Multiple Choice Evaluations
If a researcher requires 100 subjects in sample 1 and
100 subjects in sample 2 to achieve a particular margin
of error, then the number of subjects required to halve
the margin of error is
1) 200 subjects in each of sample 1 and sample 2
2) 200 subjects in sample 1 and 100 subjects in
sample 2
3) 50 subjects in each of sample 1 and sample 2
4) 400 subjects in each of sample 1 and sample 2
Section 3
Multiple Choice Evaluations
To test for the equality of two population standard
deviations, we use for the test statistic
1) The difference between the two standard
deviations
2) The ratio of the two standard deviations
3) The product of the two standard deviations
4) The ratio of the squares of the two standard
deviations
Section 4
Multiple Choice Evaluations
To test for the equality of two population standard
deviations, we use
1)
2)
3)
4)
The normal distribution
The beta distribution
The F distribution
The chi-square distribution
Section 4
Summary and Homework
• Summary
– We can test whether sample data from two different
samples supports a hypothesis claim about a
population mean, proportion, or standard deviation
– For two population means, there are two cases
• Dependent (or matched-pair) samples
• Independent samples
– All of these tests follow very similar processes as
hypothesis tests on one sample
• Homework
– pg 625 – 628; 1, 4, 6, 7, 10, 11, 14, 15, 21
Homework Answers
• 4) Dependent
• 6) a) 2.06
b) 0.49
• 10) a) Reject H0
b) [-12.44, 0.04]
c) Reject H0
• 14) a) H0: p1 = p2
H1: p1 < p2
b) z0 = -0.62
c) zc = -1.645
d) p = 0.2611 FTR H0