QMIS 220, by Dr. M. Zainal Chapter 8 Student Lecture Notes 8-1 Business Statistics Dr. Mohammad Zainal Chapter 10 Estimation and Hypothesis Testing for Two Population Parameters Department of Economics ECON 509 Chapter Goals After completing this chapter, you should be able to: Test hypotheses or form interval estimates for two independent population means Standard deviations known Standard deviations unknown two means from paired samples the difference between two population proportions ECON 509, by Dr. M. Zainal Chap 10-2
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QMIS 220, by Dr. M. Zainal
Chapter 8 Student Lecture Notes 8-1
Business Statistics
Dr. Mohammad Zainal
Chapter 10
Estimation and Hypothesis Testing
for Two Population Parameters
Department of Economics
ECON 509
Chapter Goals
After completing this chapter, you should be
able to:
Test hypotheses or form interval estimates for
two independent population means
Standard deviations known
Standard deviations unknown
two means from paired samples
the difference between two population proportions
ECON 509, by Dr. M. Zainal Chap 10-2
QMIS 220, by Dr. M. Zainal
Chapter 8 Student Lecture Notes 8-2
Estimation for Two Populations
Estimating two
population values
Population
means,
independent
samples
Paired
samples
Population
proportions
Group 1 vs. independent
Group 2
Same group before vs. after
treatment
Proportion 1 vs. Proportion 2
Examples:
ECON 509, by Dr. M. Zainal Chap 10-3
Difference Between Two Means
Population means,
independent
samples
σ1 and σ2 known
σ1 and σ2 unknown but assumed equal
σ1 and σ2 unknown, not assumed equal
Goal: Form a confidence
interval for the difference
between two population
means, μ1 – μ2
The point estimate for the
difference is
x1 – x2
*
ECON 509, by Dr. M. Zainal Chap 10-4
QMIS 220, by Dr. M. Zainal
Chapter 8 Student Lecture Notes 8-3
Independent Samples
Population means,
independent
samples
σ1 and σ2 known
Different data sources
Unrelated
Independent
Sample selected from
one population has no
effect on the sample
selected from the other
population
Use the difference between
2 sample means
*
σ1 and σ2 unknown but assumed equal
σ1 and σ2 unknown, not assumed equal
ECON 509, by Dr. M. Zainal Chap 10-5
Population means,
independent
samples
σ1 and σ2 known
σ1 and σ2 known
Assumptions:
Samples are randomly and
independently drawn
population distributions are
normal or both sample sizes
are 30
Population standard
deviations are known
* σ1 and σ2 unknown but assumed equal
σ1 and σ2 unknown, not assumed equal
ECON 509, by Dr. M. Zainal Chap 10-6
QMIS 220, by Dr. M. Zainal
Chapter 8 Student Lecture Notes 8-4
Population means,
independent
samples
σ1 and σ2 known …and the standard error of
x1 – x2 is
When σ1 and σ2 are known and
both populations are normal or
both sample sizes are at least 30,
the test statistic is a z value…
2
2
2
1
2
1
xx n
σ
n
σσ
21
(continued)
σ1 and σ2 known
* σ1 and σ2 unknown but assumed equal
σ1 and σ2 unknown, not assumed equal
ECON 509, by Dr. M. Zainal Chap 10-7
Population means,
independent
samples
σ1 and σ2 known
2
2
2
1
2
1/221
n
σ
n
σxx z
The confidence interval for
μ1 – μ2 is:
σ1 and σ2 known (continued)
* σ1 and σ2 unknown but assumed equal
σ1 and σ2 unknown, not assumed equal
ECON 509, by Dr. M. Zainal Chap 10-8
QMIS 220, by Dr. M. Zainal
Chapter 8 Student Lecture Notes 8-5
Population means,
independent
samples
σ1 and σ2 known
σ1 and σ2 unknown, large samples
Assumptions: Samples are randomly and independently drawn
Population standard
deviations are unknown
The two standard deviations
are equal
* σ1 and σ2 unknown but assumed equal
σ1 and σ2 unknown, not assumed equal
ECON 509, by Dr. M. Zainal Chap 10-9
Population means,
independent
samples
σ1 and σ2 known
σ1 and σ2 unknown, large samples
(continued)
* σ1 and σ2 unknown but assumed equal
Forming interval estimates:
The population standard
deviations are assumed equal,
so use the two sample
standard deviations and pool
them to estimate σ
the test statistic is a t value
with (n1 + n2 – 2) degrees
of freedom σ1 and σ2 unknown, not assumed equal
ECON 509, by Dr. M. Zainal Chap 10-10
QMIS 220, by Dr. M. Zainal
Chapter 8 Student Lecture Notes 8-6
Population means,
independent
samples
σ1 and σ2 known
σ1 and σ2 unknown, large samples
(continued)
* σ1 and σ2 unknown but assumed equal
The pooled standard
deviation is
2nn
s1ns1ns
21
2
22
2
11p
σ1 and σ2 unknown, not assumed equal
ECON 509, by Dr. M. Zainal Chap 10-11
Population means,
independent
samples
σ1 and σ2 known
σ1 and σ2 unknown, large samples
(continued)
* σ1 and σ2 unknown but assumed equal
21
p/221
n
1
n
1stxx
The confidence interval for
μ1 – μ2 is:
Where t/2 has (n1 + n2 – 2) d.f.,
and
2nn
s1ns1ns
21
2
22
2
11p
σ1 and σ2 unknown, not assumed equal
ECON 509, by Dr. M. Zainal Chap 10-12
QMIS 220, by Dr. M. Zainal
Chapter 8 Student Lecture Notes 8-7
Population means,
independent
samples
σ1 and σ2 known
σ1 and σ2 unknown, small samples
Assumptions: populations are normally distributed
there is a reason to believe
that the populations do not
have equal variances
samples are independent
*
σ1 and σ2 unknown but assumed equal
σ1 and σ2 unknown, not assumed equal
ECON 509, by Dr. M. Zainal Chap 10-13
Population means,
independent
samples
σ1 and σ2 known
σ1 and σ2 unknown, small samples
Forming interval
estimates:
The population variances
are not assumed equal, so
we do not pool them
the test statistic is a t value
with degrees of freedom
given by:
(continued)
*
σ1 and σ2 unknown but assumed equal
1n
/ns
1n
/ns
)/ns/n(sdf
2
2
2
2
2
1
2
1
2
1
2
2
2
21
2
1σ1 and σ2 unknown, not assumed equal
ECON 509, by Dr. M. Zainal Chap 8-14
QMIS 220, by Dr. M. Zainal
Chapter 8 Student Lecture Notes 8-8
Population means,
independent
samples
σ1 and σ2 known 2
2
2
1
2
1α/221
n
s
n
stxx
The confidence interval for
μ1 – μ2 is:
σ1 and σ2 unknown, small samples
(continued)
Where t/2 has d.f. given by
*
σ1 and σ2 unknown but assumed equal
1n
/ns
1n
/ns
)/ns/n(sdf
2
2
2
2
2
1
2
1
2
1
2
2
2
21
2
1σ1 and σ2 unknown, not assumed equal
ECON 509, by Dr. M. Zainal Chap 8-15
Hypothesis Tests for the Difference Between Two Means
Testing Hypotheses about μ1 – μ2
Use the same situations discussed already:
Standard deviations known
Standard deviations unknown
Assumed equal
Assumed not equal
ECON 509, by Dr. M. Zainal Chap 10-16
QMIS 220, by Dr. M. Zainal
Chapter 8 Student Lecture Notes 8-9
Hypothesis Tests for Two Population Proportions
Lower tail test:
H0: μ1 μ2
HA: μ1 < μ2
i.e.,
H0: μ1 – μ2 0
HA: μ1 – μ2 < 0
Upper tail test:
H0: μ1 ≤ μ2
HA: μ1 > μ2
i.e.,
H0: μ1 – μ2 ≤ 0
HA: μ1 – μ2 > 0
Two-tailed test:
H0: μ1 = μ2
HA: μ1 ≠ μ2
i.e.,
H0: μ1 – μ2 = 0
HA: μ1 – μ2 ≠ 0
Two Population Means, Independent Samples
ECON 509, by Dr. M. Zainal Chap 10-17
Hypothesis tests for μ1 – μ2
Population means, independent samples
σ1 and σ2 known Use a z test statistic
Use sp to estimate unknown
σ , use a t test statistic with
n1 + n2 – 2 d.f.
Use s1 and s2 to estimate
unknown σ1 and σ2 , use a t
test statistic and calculate the
required degrees of freedom
σ1 and σ2 unknown but assumed equal
σ1 and σ2 unknown, not assumed equal
ECON 509, by Dr. M. Zainal Chap 10-18
QMIS 220, by Dr. M. Zainal
Chapter 8 Student Lecture Notes 8-10
Population means,
independent
samples
σ1 and σ2 known
2
2
2
1
2
1
2121
n
σ
n
σ
μμxxz
The test statistic for
μ1 – μ2 is:
σ1 and σ2 known
* σ1 and σ2 unknown but assumed equal
σ1 and σ2 unknown, not assumed equal
ECON 509, by Dr. M. Zainal Chap 10-19
Population means,
independent
samples
σ1 and σ2 known
σ1 and σ2 unknown, large samples
* σ1 and σ2 unknown but assumed equal
Where t has (n1 + n2 – 2) d.f.,
and
2nn
s1ns1ns
21
2
22
2
11p
21
p
2121
n
1
n
1s
μμxxt
The test statistic for
μ1 – μ2 is:
σ1 and σ2 unknown, not assumed equal
ECON 509, by Dr. M. Zainal Chap 10-20
QMIS 220, by Dr. M. Zainal
Chapter 8 Student Lecture Notes 8-11
Population means,
independent
samples
σ1 and σ2 known
σ1 and σ2 unknown, small samples
The test statistic for
μ1 – μ2 is:
*
σ1 and σ2 unknown but assumed equal
2
2
2
1
2
1
2121
n
s
n
s
μμxxt
Where t has d.f. given by
1n
/ns
1n
/ns
)/ns/n(sdf
2
2
2
2
2
1
2
1
2
1
2
2
2
21
2
1σ1 and σ2 unknown, not assumed equal
ECON 509, by Dr. M. Zainal Chap 8-21
Two Population Means, Independent Samples
Lower tail test:
H0: μ1 – μ2 0
HA: μ1 – μ2 < 0
Upper tail test:
H0: μ1 – μ2 ≤ 0
HA: μ1 – μ2 > 0
Two-tailed test:
H0: μ1 – μ2 = 0
HA: μ1 – μ2 ≠ 0
/2 /2
-z -z/2 z z/2
Reject H0 if z < -z Reject H0 if z > z Reject H0 if z < -z/2
or z > z/2
Hypothesis tests for μ1 – μ2
Example: σ1 and σ2 known:
ECON 509, by Dr. M. Zainal Chap 10-22
QMIS 220, by Dr. M. Zainal
Chapter 8 Student Lecture Notes 8-12
Pooled sp t Test Example σ1 and σ2 unknown, assumed equal
You’re a financial analyst for a brokerage firm. Is there a
difference in dividend yield between stocks listed on the
NYSE & NASDAQ? You collect the following data:
NYSE NASDAQ
Number 21 25
Sample mean 3.27 2.53
Sample std dev 1.30 1.16
Assuming equal variances, is
there a difference in average
yield ( = 0.05)?
ECON 509, by Dr. M. Zainal Chap 10-23
Calculating the Test Statistic
1.2256
22521
1.161251.30121
2nn
s1ns1ns
22
21
2
22
2
11p
2.040
25
1
21
11.2256
02.533.27
n
1
n
1s
μμxxt
21
p
2121
The test statistic is:
Where:
ECON 509, by Dr. M. Zainal Chap 10-24
QMIS 220, by Dr. M. Zainal
Chapter 8 Student Lecture Notes 8-13
Solution
H0: μ1 - μ2 = 0 i.e. (μ1 = μ2)
HA: μ1 - μ2 ≠ 0 i.e. (μ1 ≠ μ2)
= 0.05
df = 21 + 25 - 2 = 44
Critical Values: t = ± 2.0154
Test Statistic: Decision:
Conclusion:
Reject H0 at = 0.05
There is evidence that
the means are different.
t 0 2.0154 -2.0154
.025
Reject H0 Reject H0
.025
2.040
2.040
25
1
21
11.2256
2.533.27t
ECON 509, by Dr. M. Zainal Chap 10-25
Paired Samples
Tests Means of 2 Related Populations
Paired or matched samples
Repeated measures (before/after)
Use difference between paired values:
Eliminates Variation Among Subjects
Assumptions:
Both Populations Are Normally Distributed
Or, if Not Normal, use large samples
Paired
samples
d = x1 - x2
ECON 509, by Dr. M. Zainal Chap 10-26
QMIS 220, by Dr. M. Zainal
Chapter 8 Student Lecture Notes 8-14
Paired Differences
The ith paired difference is di , where
Paired
samples di = x1i - x2i
The point estimate for
the population mean
paired difference is d :
1n
)d(d
s
n
1i
2
i
d
n
d
d
n
1i
i
The sample standard
deviation is
n is the number of pairs in the paired sample
ECON 509, by Dr. M. Zainal Chap 10-27
Paired Differences
The confidence interval for d is Paired
samples
1n
)d(d
s
n
1i
2
i
d
n
std d
Where t has n - 1 d.f. and sd is:
(continued)
n is the number of pairs in the paired sample
ECON 509, by Dr. M. Zainal Chap 10-28
QMIS 220, by Dr. M. Zainal
Chapter 8 Student Lecture Notes 8-15
The test statistic for d is Paired
samples
1n
)d(d
s
n
1i
2
i
d
n
s
μdt
d
d
Where t has n - 1 d.f.
and sd is:
n is the
number
of pairs
in the
paired
sample
Hypothesis Testing for Paired Samples
ECON 509, by Dr. M. Zainal Chap 10-29
Lower tail test:
H0: μd 0
HA: μd < 0
Upper tail test:
H0: μd ≤ 0
HA: μd > 0
Two-tailed test:
H0: μd = 0
HA: μd ≠ 0
Paired Samples
Hypothesis Testing for Paired Samples
/2 /2
-t -t/2 t t/2
Reject H0 if t < -t Reject H0 if t > t Reject H0 if t < -t/2
or t > t/2 Where t has n - 1 d.f.
(continued)
ECON 509, by Dr. M. Zainal Chap 10-30
QMIS 220, by Dr. M. Zainal
Chapter 8 Student Lecture Notes 8-16
Assume you send your salespeople to a “customer
service” training workshop. Is the training effective?
You collect the following data:
Paired Samples Example
Number of Complaints: (2) - (1)
Salesperson Before (1) After (2) Difference, di
C.B. 6 4 - 2
T.F. 20 6 -14
M.H. 3 2 - 1
R.K. 0 0 0
M.O. 4 0 - 4
-21
d = di
n
5.67
1n
)d(ds
2
i
d
= -4.2
ECON 509, by Dr. M. Zainal Chap 10-31
Has the training made a difference in the number of
complaints (at the 0.05 level)?
- 4.2 d =
1.6655.67/
04.2
n/s
μdt
d
d
H0: μd = 0
HA: μd 0
Test Statistic:
Critical Value = ± 2.7765 d.f. = n - 1 = 4
Reject
/2
- 2.7765 2.7765
Decision: Do not reject H0
(t stat is not in the reject region)
Conclusion: There is not a
significant change in the
number of complaints.
Paired Samples: Solution
Reject
/2
- 1.66 = .05
ECON 509, by Dr. M. Zainal Chap 10-32
QMIS 220, by Dr. M. Zainal
Chapter 8 Student Lecture Notes 8-17
Two Population Proportions
Goal: Form a confidence interval for
or test a hypothesis about the
difference between two population
proportions, π1 – π2
The point estimate for
the difference is p1 – p2
Population
proportions
Assumptions:
n1π1 5 , n1(1-π1) 5
n2π2 5 , n2(1-π2) 5
ECON 509, by Dr. M. Zainal Chap 10-33
Confidence Interval for Two Population Proportions
Population
proportions
2
22
1
1121
n
)p(1p
n
)p(1pzpp
The confidence interval for
π1 – π2 is:
ECON 509, by Dr. M. Zainal Chap 10-34
QMIS 220, by Dr. M. Zainal
Chapter 8 Student Lecture Notes 8-18
Hypothesis Tests for Two Population Proportions
Population proportions
Lower tail test:
H0: π1 π2
HA: π1 < π2
i.e.,
H0: π1 – π2 0
HA: π1 – π2 < 0
Upper tail test:
H0: π1 ≤ π2
HA: π1 > π2
i.e.,
H0: π1 – π2 ≤ 0
HA: π1 – π2 > 0
Two-tailed test:
H0: π1 = π2
HA: π1 ≠ π2
i.e.,
H0: π1 – π2 = 0
HA: π1 – π2 ≠ 0
ECON 509, by Dr. M. Zainal Chap 10-35
Two Population Proportions
Population
proportions
21
21
21
2211
nn
xx
nn
pnpnp
The pooled estimate for the
overall proportion is:
where x1 and x2 are the numbers from
samples 1 and 2 with the characteristic of interest
Since we begin by assuming the null
hypothesis is true, we assume π1 = π2
and pool the two p estimates
ECON 509, by Dr. M. Zainal Chap 10-36
QMIS 220, by Dr. M. Zainal
Chapter 8 Student Lecture Notes 8-19
Two Population Proportions
Population
proportions
21
2121
n
1
n
1)p(1p
ππppz
The test statistic for
π1 – π2 is:
(continued)
ECON 509, by Dr. M. Zainal Chap 10-37
Hypothesis Tests for Two Population Proportions
Population proportions
Lower tail test:
H0: π1 – π2 0
HA: π1 – π2 < 0
Upper tail test:
H0: π1 – π2 ≤ 0
HA: π1 – π2 > 0
Two-tailed test:
H0: π1 – π2 = 0
HA: π1 – π2 ≠ 0
/2 /2
-z -z/2 z z/2
Reject H0 if z < -z Reject H0 if z > z Reject H0 if z < -z/2
or z > z/2
ECON 509, by Dr. M. Zainal Chap 10-38
QMIS 220, by Dr. M. Zainal
Chapter 8 Student Lecture Notes 8-20
Example: Two population Proportions
Is there a significant difference between the
proportion of men and the proportion of
women who will vote Yes on Proposition A?
In a random sample, 36 of 72 men and 31 of
50 women indicated they would vote Yes
Test at the .05 level of significance
ECON 509, by Dr. M. Zainal Chap 10-39
The hypothesis test is:
H0: π1 – π2 = 0 (the two proportions are equal)
HA: π1 – π2 ≠ 0 (there is a significant difference between proportions)
The sample proportions are:
Men: p1 = 36/72 = .50
Women: p2 = 31/50 = .62
.549122
67
5072
3136
nn
xxp
21
21
The pooled estimate for the overall proportion is:
Example: Two population Proportions
(continued)
ECON 509, by Dr. M. Zainal Chap 10-40
QMIS 220, by Dr. M. Zainal
Chapter 8 Student Lecture Notes 8-21
The test statistic for π1 – π2 is:
Example: Two population Proportions
(continued)
.025
-1.96 1.96
.025
-1.31
Decision: Do not reject H0
Conclusion: There is not
significant evidence of a
difference in the proportion
who will vote yes between
men and women.
1.31
50
1
72
1.549)(1.549
0.62.50
n
1
n
1)p(1p
ππppz
21
2121
Reject H0 Reject H0
Critical Values = ±1.96 For = .05
ECON 509, by Dr. M. Zainal Chap 10-41
Two Sample Tests in EXCEL
For independent samples:
Independent sample z test with variances known:
Data | data analysis | z-Test: Two Sample for Means
Independent sample t test with variance unknown:
Data | data analysis | t-Test: Two Sample Assuming Equal
Variances
Data | data analysis | t-Test: Two Sample Assuming Unequal
Variances
For paired samples (t test):
Data | data analysis… | t-Test: Paired Two Sample for Means
ECON 509, by Dr. M. Zainal Chap 10-42
QMIS 220, by Dr. M. Zainal
Chapter 8 Student Lecture Notes 8-22
Chapter Summary
Compared two independent samples
Formed confidence intervals for the differences between two
means
Performed z test for the differences in two means
Performed t test for the differences in two means
Compared two related samples (paired samples)
Formed confidence intervals for the paired difference
Performed paired sample t tests for the mean difference
Compared two population proportions
Formed confidence intervals for the difference between two
population proportions
Performed z test for two population proportions
ECON 509, by Dr. M. Zainal Chap 10-43
Problems
ECON 509, by Dr. M. Zainal
A business analyst took a random sample of 32 advertising
managers from across the United States and a similar random
sample is taken of 34 auditing managers. They were asked
what their annual salary is. The analyst wants to test whether
there is a difference in the average wage of an advertising
manager and the auditing manager. The resulting salary data
are listed in the table below with the sample means, population
standard deviations and variances.
164.264
253.16
700.70
32
2
1
1
1
1
x
n
411.166
900.12
187.62
34
2
2
2
2
2
x
n
Advertising Auditing
Chap 10-44
QMIS 220, by Dr. M. Zainal
Chapter 8 Student Lecture Notes 8-23
Chap 10-45 ECON 509, by Dr. M. Zainal
Chap 10-46 ECON 509, by Dr. M. Zainal
QMIS 220, by Dr. M. Zainal
Chapter 8 Student Lecture Notes 8-24
Chap 10-47 ECON 509, by Dr. M. Zainal
Problems
ECON 509, by Dr. M. Zainal
A consumer test group wants to determine the difference in
gasoline mileage of cars using regular gas and cars using
premium gas. Their researchers divided a fleet of 100 cars of
the same make in half and tested each car on one tank of gas.
50 of the cars were filled with regular gas and the rest filled
with premium gas. The sample average for the regular gasoline
group was 21.45 mpg, and the sample average for the
premium gasoline was 24.60. Assume the population standard
deviation of the regular gasoline is 3.46 mpg, and the
population standard deviation of the premium gas is 2.99 mpg.
Construct a 95% confidence interval to estimate the difference
in the mean gas mileage between the cars using regular and
premium gasoline.
Chap 10-48
QMIS 220, by Dr. M. Zainal
Chapter 8 Student Lecture Notes 8-25
Chap 10-49 ECON 509, by Dr. M. Zainal
Problems
ECON 509, by Dr. M. Zainal
At a certain company, new employees are expected to attend a
3 days seminar (Method A) to learn about the company. At the
end of the seminar, they are tested to measure their knowledge
about the company. The management decided to experiment
another training procedure (Method B) which ends with the
same knowledge test but saves a lot of time and money. To test
the effectiveness of the new method (B), managers randomly
selected two groups and the results were recorded in the
following tables. Using = 5%, the managers want to
determine whether there is a significant difference in the mean
score of the two groups assuming the score of the test are
normally distributed and the population variances are
approximately equal.
Chap 10-50
QMIS 220, by Dr. M. Zainal
Chapter 8 Student Lecture Notes 8-26
Training Method A
56 51 45
47 52 43
42 53 52
50 42 48
47 44 44
Training Method B
59
52
53
54
57
56
55
64
53
65
53
57
495.19
73.47
15
2
1
1
1
s
x
n
273.18
5.56
12
2
2
2
2
s
x
n
Chap 10-51 ECON 509, by Dr. M. Zainal
Chap 10-52 ECON 509, by Dr. M. Zainal
QMIS 220, by Dr. M. Zainal
Chapter 8 Student Lecture Notes 8-27
Chap 10-53 ECON 509, by Dr. M. Zainal
Chap 10-54 ECON 509, by Dr. M. Zainal
QMIS 220, by Dr. M. Zainal
Chapter 8 Student Lecture Notes 8-28
Problems
ECON 509, by Dr. M. Zainal
A coffee manufacturer is interested in estimating the difference
in the average daily coffee consumption of a regular coffee
drinker and decaffeinated coffee drinker. Its researcher
randomly selects 13 regular coffee drinkers and asks how
many cups of coffee per day they drink. He randomly locates
15 decaffeinated coffee drinkers and how many cups of coffee
per day they drink. The average for the regular coffee drinkers
is 4.35 cups, with a standard deviation of 1.2 cups. The
average of the decaffeinated-coffee drinkers is 6.84 cups, with
a standard deviation of 1.42 cups. The researcher assumes, for
each population, that the daily consumption is normally
distributed, and their variances are approximately equal. He
wants to construct a 95% confidence interval to estimate the
difference in the averages of the two populations.
Chap 10-55
Chap 10-56 ECON 509, by Dr. M. Zainal
QMIS 220, by Dr. M. Zainal
Chapter 8 Student Lecture Notes 8-29
Chap 10-57 ECON 509, by Dr. M. Zainal
Problems
ECON 509, by Dr. M. Zainal
Suppose a stock market investor is interested in determining
whether there is a significant difference in P/E (price to
earning) ratio for companies from one year to the next. He
randomly samples nine companies and records the P/E ratios
for each of these companies at the end of the year 1 and 2.
Assume there is no prior information to indicate whether P/E
ratios have gone up or down. Also, assume that the P/E ratios
are normally distributed in the population. The data are shown
in the next table. (Use = 1%)
Chap 10-58
QMIS 220, by Dr. M. Zainal
Chapter 8 Student Lecture Notes 8-30
Chap 10-59 ECON 509, by Dr. M. Zainal
Chap 10-60 ECON 509, by Dr. M. Zainal
QMIS 220, by Dr. M. Zainal
Chapter 8 Student Lecture Notes 8-31
ECON 509, by Dr. M. Zainal
10-61
Chap 10-62 ECON 509, by Dr. M. Zainal
QMIS 220, by Dr. M. Zainal
Chapter 8 Student Lecture Notes 8-32
Problems
ECON 509, by Dr. M. Zainal
Time magazine reported the result of a telephone poll of 800
adult Americans. The question posed of the Americans who
were surveyed was: "Should the federal tax on cigarettes be
raised to pay for health care reform?" The results of the survey