Section 11.1 – Inference About Two Population Proportions
A sampling method is independent when the individuals selected for one sample do not dictate
which individuals are to be in the second sample.
A sampling method is dependent (matched pairs) when the individuals selected to be in one sample
are used to determine the individuals in the second sample. These are often referred to as
“matched-pairs” samples (see section 11.2). It is possible for an individual to be matched against
himself.
Notation:
p1 = proportion for population 1
n1 = size of sample 1
x1 = number of successes in sample 1
x
pˆ 1  1
n1
p2 = proportion for population 2
n2 = size of sample 2
x2 = number of successes in sample 2
x
pˆ2  1
n1
Requirements:
1. Samples obtained independently by simple
random sampling or data come from a
randomized experiment
2. n1  pˆ1  qˆ1  10 and n2  pˆ2  qˆ2  10
3. Sample size no more than 5% of the population
size.
Hypotheses:
Ho: p1 = p2
H1: p1 ≠ p2 or p1 > p2
or p1 < p2
TEST STATISTIC and P-VALUE come from the calculator.
STAT  TESTS  2-PropZTest (#6)
We will only consider the TEST STATISTIC and P-VALUE in this chapter.
You will not have to find critical values or draw pictures!
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Example: In October of 1947, the Gallup organization surveyed 1100 adult Americans and asked, “Do
you totally abstain from drinking alcoholic beverages?”. Of the 1100 adults surveyed, 407 said “yes”.
In July 2010, the same question was asked of 1100 adult Americans and 333 said “yes”. Has the
proportion of adult Americans who totally abstain from alcohol changed? Use a 0.05 significance
level.
I.
Ho:
H1:
II.
α =
III.
T.S. = z =
IV.
P-value =
V.
Reject Ho or Do Not Reject Ho
VI.
(conclusion)
Example: In a study of red/green color blindness, 500 men and 2,100 women were randomly
selected and tested. Among the men, 45 demonstrated color blindness. Among the women, 6
demonstrated color blindness. Is there sufficient evidence at α = 0.05 to support the claim that men
have a higher rate of red/green color blindness than women?
I.
Ho:
H1:
II.
α =
III.
T.S. = z =
IV.
P-value =
V.
Reject Ho or Do Not Reject Ho
VI.
(conclusion)
2
When you calculate a confidence interval in this chapter (a confidence interval of the difference
between 2 parameters), there are 3 possible outcomes:
 ,  
means that the difference between the two values is negative.
There is a significant difference between the values.
 ,  
means that the difference between the two values is positive.
There is a significant difference between the values.
 ,  
means that there is a possibility that the difference is 0.
There is NOT a significant difference between the values.
•If the confidence interval contains 0, we say there is NO significant difference between
the values.
•If the confidence interval does not contain 0, we say there IS a significant difference
between the values.
Confidence Interval Estimate for p1 – p2 :
STAT  TEST  2-PropZInt (#B)
Example: The body mass index (BMI) is one measure that is used to judge whether an individual is
overweight or not. A BMI between 20 and 25 indicates that one is at a normal weight. In a survey of
750 men and 750 women, the Gallup organization found that 203 men and 270 women were of
normal weight. Construct a 90% confidence interval estimate of the difference in the two
proportions. Based on the result, does there appear to be a significant difference in the proportion of
men and women who are of normal weight?
Example: Among 5000 items of randomly selected baggage handled by American Airlines, 22 were
lost. Among 4000 items of randomly selected baggage handled by Delta Airlines, 15 were lost. Use
the sample data to construct a 95% confidence interval estimate of the difference between the two
rates of lost baggage. Based on the result, does there appear to be a significant difference in the lost
luggage rates?
3
Section 11.2 – Inference about Two Means: Dependent Matched Pairs
Notation:
Requirements:
μd = mean value of the differences d for the
1. Sample obtained by simple random sampling or
population of paired data
data come from a matched-pairs design experiment
2. The sample data are dependent matched pairs.
d = individual difference between two values 3. The differences are normally distributed with no
in a matched pair
outliers or the number of matched pairs is large
(n ≥ 30).
4.
Sample
values are independent
d = mean value of the differences d for the
paired sample data (average of “x – y” values)
sd = standard deviation of the differences d
for the paired sample data
n = number of pairs of data
Hypotheses:
Ho:
H1:
μd = 0
μd ≠ 0 or
μd > 0 or
μd < 0
TEST STATISTIC and P-VALUE come from the calculator.
STAT  TESTS  T-Test (#2)
µ0 will be 0
Select “Data”
Choose L3
Enter “before” data in L1.
Enter “after” data in L2.
Go to the title bar of L3 and make L3 = L1 – L2
Hints:
•If you want L1 to be bigger than L2 (ie L1 > L2), do a right-tailed test.
μd > 0
•If you want L1 to be smaller than L2 (ie L1 < L2) do a left-tailed test.
μd < 0
4
Example: In low-speed crash test of five BMW cars, the repair costs were computed for a factory
authorized repair center and an independent repair center. Is there sufficient evidence at α = 0.01
to support the claim that the independent center has lower repair costs?
Authorized repair center
$797
$571
$904
$1147
$418
Independent repair center
$523
$488
$875
$911
$297
I.
Ho:
H1:
II.
α =
III.
T.S. = t =
IV.
P-value =
V.
Reject Ho or Do Not Reject Ho
VI.
(conclusion)
5
Example: A researcher wanted to compare the pulse rates of identical twins to see whether there
was any difference. Eight sets of twins were selected. At α = 0.05, is there enough evidence to
support the claim that there is a difference in the pulse rates of twins?
Twin A
Twin B
I.
87
83
92
95
78
79
83
83
88
86
90
93
84
80
93
86
Ho:
H1:
II.
α =
III.
T.S. = t =
IV.
P-value =
V.
Reject Ho or Do Not Reject Ho
VI.
(conclusion)
6
Example: To test the belief than sons are taller than their fathers, a student randomly selects 13
fathers who have adult male children. She records the height of both the father and son in inches
and obtains the following data. Are sons taller than their fathers? Use α = 0.10.
Father
Son
70.3
74.1
I.
67.1
69.2
70.9
66.9
66.8
69.2
72.8
68.9
70.4
70.2
71.8
70.4
70.1
69.3
69.9
75.8
70.8
72.3
70.2
69.2
70.4
68.6
72.4
73.9
Ho:
H1:
II.
α =
III.
T.S. = t =
IV.
P-value =
V.
Reject Ho or Do Not Reject Ho
VI.
(conclusion)
7
Confidence Interval Estimate for the population mean difference μd :
STAT  TEST  T-Interval (#8)
Select “Data” and Choose L3
Example: In an experiment conducted online at the University of Mississippi, participants are asked
to react to a stimulus. In one experiment, the participant must press a key upon seeing a blue screen.
Their reaction time is measured (in seconds). The same person is asked to press a key upon seeing a
red screen, again with reaction time measured. The results for 6 randomly sampled participants are
as follows. Construct a 95% confidence interval about the population mean difference in response
time. Is there a significant difference in the response times?
Blue
0.582
0.481
0.841
0.267
0.685
0.450
Red
0.408
0.407
0.512
0.402
0.456
0.533
Example: Listed below are the heights of candidates who won presidential elections and the heights
of the candidates with the next highest number of popular votes. The data are in chronological
order, so the corresponding heights from the two lists are matched. A well-known theory is that
winning candidates tend to be taller than the losing candidates. Find the 99% confidence interval of
the mean difference in the candidates. Is there a significant difference in the heights (ie, does height
appear to be an important factor in winning the presidency)?
Won the
Presidency
Runner-Up
71
74.5
74
73
73
74
68
69.5
69.5 71.5
72
71
75
72
70.5
69
74
70
71
72
70
67
72
71.5
70
68
71
72
70
72
72
72
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Section 11.3 – Inference About Two Means: Independent Samples
Notation:
µ1 = mean for population 1
n1 = size of sample 1
x1 = mean of sample 1
s1 = standard deviation of sample 1
Requirements:
1. Samples obtained by simple random sampling or
data come from a randomized experiment
2. The sample data are independent.
3. The populations are normally distributed or the
sample sizes are large (n ≥ 30).
4. Sample size is no more than 5% of the population
µ2 = mean for population 1
n2 = size of sample 1
x2 = mean of sample 1
s2 = standard deviation of sample 1
Hypotheses:
Ho: μ1 = μ2
H1: μ1 ≠ μ2 or
μ1 > μ2 or
μ1 < μ2
TEST STATISTIC and P-VALUE come from the calculator.
STAT  TEST  2-SampTTest (#4)
For “pooled” select “no” (default is no)
Example: For a sample of 1,657 men (ages 65-74), the mean weight is 164 lbs and the standard
deviation is 27 lbs. For a sample of 804 men (ages 25-34), the mean weight is 176 lbs and the
standard deviation is 35 lbs. Test the claim that the older men have a lower mean weight than the
younger men. Use α = 0.01.
I.
Ho:
H1:
II.
α =
III.
T.S. = t =
IV.
P-value =
V.
Reject Ho or Do Not Reject Ho
VI.
(conclusion)
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Example: A study of zinc-deficient mothers was conducted to determine whether zinc
supplementation during pregnancy results in babies with increased weights at birth. Is there
sufficient evidence at α = 0.05 to support the claim that zinc supplementation does result in increased
birth weight? (weights measured in grams)
Zinc Supplement
Placebo
n1 = 294
n2 = 286
x1 = 3214
x 2 = 3088
s1 = 669
s2 = 728
I.
Ho:
H1:
II.
α =
III.
T.S. = t =
IV.
P-value =
V.
Reject Ho or Do Not Reject Ho
VI.
(conclusion)
Confidence Interval Estimate for μ1 – μ2:
STAT  TEST  2-SampTInterval (#0)
Example: A randomized trial tested the effectiveness of diets on adults. Among 40 subjects using the
Weight Watchers diet, the mean weight loss after one year was 3 lb with a standard deviation of 4.9
lb. Among 40 subjects using the Atkins diet, the mean weight loss after one year was 2.1 lb with a
standard deviation of 4.8 lb. Construct a 95% confidence interval estimate of the difference between
the population means. Does there appear to be a significant difference in the effectiveness of the
two diets?
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