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IS 310
Business
Statistics
CSU
Long Beach
IS 310 – Business Statistics
Slide 1
Inferences on Two Populations
In the past, we dealt with one population mean and one
population proportion. However, there are
situations where two populations are involved
dealing with two means.
Examples are the following:
O We want to compare the mean salaries of male and female
graduates (two populations and two means).
O We want to compare the mean miles per gallon(MPG) of two
comparable automobile makes (two populations and two
means)
IS 310 – Business Statistics
Slide 2
Statistical Inferences About Means
and Proportions with Two Populations

Inferences About the Difference Between
Two Population Means: s 1 and s 2 Known


Inferences About the Difference Between
Two Population Means: s 1 and s 2 Unknown
IS 310 – Business Statistics
Slide 3
Inferences About the Difference Between
Two Population Means: s 1 and s 2 Known

Interval Estimation of m 1 – m 2

Hypothesis Tests About m 1 – m 2
IS 310 – Business Statistics
Slide 4
Estimating the Difference Between
Two Population Means



Let m1 equal the mean of population 1 and m2 equal
the mean of population 2.
The difference between the two population means is
m1 - m2.
To estimate m1 - m2, we will select a simple random
sample of size n1 from population 1 and a simple
random sample of size n2 from population 2.
Let x1 equal the mean of sample 1 and x2 equal the
mean of sample 2.
 The point estimator of the difference between the
means of the populations 1 and 2 is x1  x2.

IS 310 – Business Statistics
Slide 5
Sampling Distribution of x1  x2

Expected Value
E ( x1  x2 )  m1  m 2

Standard Deviation (Standard Error)
s x1  x2 
s12
n1

s 22
n2
where: s1 = standard deviation of population 1
s2 = standard deviation of population 2
n1 = sample size from population 1
n2 = sample size from population 2
IS 310 – Business Statistics
Slide 6
Interval Estimation of m1 - m2:
s 1 and s 2 Known

Interval Estimate
x1  x2  z / 2
s 12 s 22

n1 n2
where:
1 -  is the confidence coefficient
IS 310 – Business Statistics
Slide 7
Interval Estimation of m1 - m2:
s 1 and s 2 Known

Example: Par, Inc.
Par, Inc. is a manufacturer
of golf equipment and has
developed a new golf ball
that has been designed to
provide “extra distance.”
In a test of driving distance using a mechanical
driving device, a sample of Par golf balls was
compared with a sample of golf balls made by Rap,
Ltd., a competitor. The sample statistics appear on the
next slide.
IS 310 – Business Statistics
Slide 8
Interval Estimation of m1 - m2:
s 1 and s 2 Known

Example: Par, Inc.
Sample Size
Sample Mean
Sample #1
Par, Inc.
120 balls
275 yards
Sample #2
Rap, Ltd.
80 balls
258 yards
Based on data from previous driving distance
tests, the two population standard deviations are
known with s 1 = 15 yards and s 2 = 20 yards.
IS 310 – Business Statistics
Slide 9
Interval Estimation of m1 - m2:
s 1 and s 2 Known

Example: Par, Inc.
Let us develop a 95% confidence interval estimate
of the difference between the mean driving distances of
the two brands of golf ball.
IS 310 – Business Statistics
Slide 10
Estimating the Difference Between
Two Population Means
Population 1
Par, Inc. Golf Balls
m1 = mean driving
distance of Par
golf balls
Population 2
Rap, Ltd. Golf Balls
m2 = mean driving
distance of Rap
golf balls
m1 – m2 = difference between
the mean distances
Simple random sample
of n1 Par golf balls
Simple random sample
of n2 Rap golf balls
x1 = sample mean distance
for the Par golf balls
x2 = sample mean distance
for the Rap golf balls
x1 - x2 = Point Estimate of m1 – m2
IS 310 – Business Statistics
Slide 11
Point Estimate of m1 - m2
Point estimate of m1  m2 = x1  x2
= 275  258
= 17 yards
where:
m1 = mean distance for the population
of Par, Inc. golf balls
m2 = mean distance for the population
of Rap, Ltd. golf balls
IS 310 – Business Statistics
Slide 12
Interval Estimation of m1 - m2:
s 1 and s 2 Known
x1  x2  z / 2
s12
s 22
(15) 2 ( 20) 2

 17  1. 96

n1 n2
120
80
17 + 5.14 or 11.86 yards to 22.14 yards
We are 95% confident that the difference between
the mean driving distances of Par, Inc. balls and Rap,
Ltd. balls is 11.86 to 22.14 yards.
IS 310 – Business Statistics
Slide 13
Hypothesis Tests About m 1  m 2:
s 1 and s 2 Known
 Hypotheses
H0 : m1  m2  D0 H0 : m1  m2  D0 H0 : m1  m2  D0
H a : m1  m2  D0 H a : m1  m2  D0 H a : m1  m2  D0
Left-tailed
Right-tailed
Two-tailed
 Test Statistic
z
( x1  x2 )  D0
s 12
n1
IS 310 – Business Statistics

s 22
n2
Slide 14
Hypothesis Tests About m 1  m 2:
s 1 and s 2 Known

Example: Par, Inc.
Can we conclude, using
 = .01, that the mean driving
distance of Par, Inc. golf balls
is greater than the mean driving
distance of Rap, Ltd. golf balls?
IS 310 – Business Statistics
Slide 15
Hypothesis Tests About m 1  m 2:
s 1 and s 2 Known
 p –Value and Critical Value Approaches
1. Develop the hypotheses.
H0: m1 - m2 < 0
Ha: m1 - m2 > 0
where:
m1 = mean distance for the population
of Par, Inc. golf balls
m2 = mean distance for the population
of Rap, Ltd. golf balls
2. Specify the level of significance.
IS 310 – Business Statistics
 = .01
Slide 16
Hypothesis Tests About m 1  m 2:
s 1 and s 2 Known
 p –Value and Critical Value Approaches
3. Compute the value of the test statistic.
z
( x1  x2 )  D0
s 12
n1
z

s 22
n2
(235  218)  0
(15)2 (20)2

120
80
IS 310 – Business Statistics

17
 6.49
2.62
Slide 17
Hypothesis Tests About m 1  m 2:
s 1 and s 2 Known
 p –Value Approach
4. Compute the p–value.
For z = 6.49, the p –value < .0001.
5. Determine whether to reject H0.
Because p–value <  = .01, we reject H0.
At the .01 level of significance, the sample evidence
indicates the mean driving distance of Par, Inc. golf
balls is greater than the mean driving distance of Rap,
Ltd. golf balls.
IS 310 – Business Statistics
Slide 18
Hypothesis Tests About m 1  m 2:
s 1 and s 2 Known
 Critical Value Approach
4. Determine the critical value and rejection rule.
For  = .01, z.01 = 2.33
Reject H0 if z > 2.33
5. Determine whether to reject H0.
Because z = 6.49 > 2.33, we reject H0.
The sample evidence indicates the mean driving
distance of Par, Inc. golf balls is greater than the mean
driving distance of Rap, Ltd. golf balls.
IS 310 – Business Statistics
Slide 19
Sample Problem
Problem # 7 (10-Page 401; 11-Page 414)
a. H : µ = µ
H : µ > µ
0
1 2
a
1
2
b. Point reduction in the mean duration of games during 2003 = 172 – 166
= 6 minutes
_ _
2
2
c. Test-statistic, z = [( x - x ) – 0] /√ [ (σ / n ) + (σ / n )]
1 2
1
1
2
2
=(172 – 166)/√[ (144/60 + 144/50)]
= 6/2.3 = 2.61
Critical z at
= 1.645
Reject H
0.05
0
Statistical test supports that the mean duration of games in 2003 is less
than that in 2002.
p-value = 1 – 0.9955 = 0.0045
IS 310 – Business Statistics
Slide 20
Inferences About the Difference Between
Two Population Means: s 1 and s 2 Unknown

Interval Estimation of m 1 – m 2

Hypothesis Tests About m 1 – m 2
IS 310 – Business Statistics
Slide 21
Interval Estimation of m1 - m2:
s 1 and s 2 Unknown
When s 1 and s 2 are unknown, we will:
• use the sample standard deviations s1 and s2
as estimates of s 1 and s 2 , and
• replace z/2 with t/2.
IS 310 – Business Statistics
Slide 22
Interval Estimation of µ - µ
1 2

(Unknown
1







and
)
2
Interval estimate
_ _
2
2
(x - x ) ± t
√ (s /n + s /n )
1 2
/2 1 1 2 2
Degree of freedom = n + n - 2
1
2
IS 310 – Business Statistics
Slide 23
Difference Between Two Population Means:
s 1 and s 2 Unknown

Example: Specific Motors
Specific Motors of Detroit
has developed a new automobile
known as the M car. 24 M cars
and 28 J cars (from Japan) were road
tested to compare miles-per-gallon (mpg) performance.
The sample statistics are shown on the next slide.
IS 310 – Business Statistics
Slide 24
Difference Between Two Population Means:
s 1 and s 2 Unknown

Example: Specific Motors
Sample #1
M Cars
24 cars
29.8 mpg
2.56 mpg
Sample #2
J Cars
28 cars
27.3 mpg
1.81 mpg
IS 310 – Business Statistics
Sample Size
Sample Mean
Sample Std. Dev.
Slide 25
Difference Between Two Population Means:
s 1 and s 2 Unknown

Example: Specific Motors
Let us develop a 90% confidence
interval estimate of the difference
between the mpg performances of
the two models of automobile.
IS 310 – Business Statistics
Slide 26
Point Estimate of m 1  m 2
Point estimate of m1  m2 = x1  x2
= 29.8 - 27.3
= 2.5 mpg
where:
m1 = mean miles-per-gallon for the
population of M cars
m2 = mean miles-per-gallon for the
population of J cars
IS 310 – Business Statistics
Slide 27
Interval Estimate of µ - µ
1
2

Interval estimate

2
2
√ (2.56) /24 + (1.81) /28)

29.8 – 27.3 ± t

0.1/2
2.5 ± 1.676 (0.62)
2.5 ± 1.04
1.46 and 3.54
We are 90% confident that the difference between the
average miles per gallon between the J cars and M
cars is between 1.46 and 3.54.




IS 310 – Business Statistics
Slide 28
Hypothesis Tests About m 1  m 2:
s 1 and s 2 Unknown

Hypotheses
H0 : m1  m2  D0 H0 : m1  m2  D0 H0 : m1  m2  D0
H a : m1  m2  D0 H a : m1  m2  D0 H a : m1  m2  D0
Left-tailed

Right-tailed
Two-tailed
Test Statistic
t
( x1  x2 )  D0
2
1
2
2
s
s

n1 n2
IS 310 – Business Statistics
Slide 29
Hypothesis Tests About m 1  m 2:
s 1 and s 2 Unknown

Example: Specific Motors
Can we conclude, using a
.05 level of significance, that the
miles-per-gallon (mpg) performance
of M cars is greater than the miles-pergallon performance of J cars?
IS 310 – Business Statistics
Slide 30
Hypothesis Tests About m 1  m 2:
s 1 and s 2 Unknown
 p –Value and Critical Value Approaches
2. Specify the level of significance.
 = .05
3. Compute the value of the test statistic.
t
( x1  x2 )  D0
s12 s22

n1 n2
IS 310 – Business Statistics

(29.8  27.3)  0
(2.56)2 (1.81)2

24
28
 4.003
Slide 31
Hypothesis Tests of µ - µ
1
2


H : µ =µ
0
1
2
H :µ >µ
a
1
2

Where µ average miles per gallon of M cars
1
µ average miles per gallon of J cars
2

At

Since t-statistic (4.003) is larger than critical t (1.676), we reject
the null hypothesis. This means that the average MPG of M cars
is not equal to that of J cars



= 0.05 with 50 degree of freedom, critical t = 1.676
IS 310 – Business Statistics
Slide 32
End of Chapter 10
Part A
IS 310 – Business Statistics
Slide 33