Download ppt10a

Slides by
John
Loucks
St. Edward’s
University
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 1
Chapter 10, Part A
Statistical Inference 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
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 2
Estimating the Difference Between
Two Population Means



Let 1 equal the mean of population 1 and 2 equal
the mean of population 2.
The difference between the two population means is
1 - 2.
To estimate 1 - 2, 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.

© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 3
Sampling Distribution of x1  x2

Expected Value
E ( x1  x2 )  1   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
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 4
Interval Estimation of 1 - 2:
s 1 and s 2 Known

Interval Estimate
x1  x2  z / 2
s 12 s 22

n1 n2
where:
1 -  is the confidence coefficient
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 5
Interval Estimation of 1 - 2:
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.
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 6
Interval Estimation of 1 - 2:
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.
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 7
Interval Estimation of 1 - 2:
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.
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 8
Estimating the Difference Between
Two Population Means
Population 1
Par, Inc. Golf Balls
1 = mean driving
distance of Par
golf balls
Population 2
Rap, Ltd. Golf Balls
2 = mean driving
distance of Rap
golf balls
m1 – 2 = 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 – 2
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 9
Point Estimate of 1 - 2
Point estimate of 1  2 = x1  x2
= 275  258
= 17 yards
where:
1 = mean distance for the population
of Par, Inc. golf balls
2 = mean distance for the population
of Rap, Ltd. golf balls
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 10
Interval Estimation of 1 - 2:
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.
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 11
Interval Estimation of 1 - 2:
s 1 and s 2 Known

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Excel Formula Worksheet
A
Par
255
270
294
245
300
262
281
257
268
295
249
291
289
282
B
C
D
E
Rap
Par, Inc.
Rap, Ltd.
266
Sample Size =COUNT(A2:A121)
=COUNT(B2:B81)
238
Sample Mean =AVERAGE(A2:A121)
=AVERAGE(B2:B81)
243
277 Popul. Std. Dev. 15
20
275 Standard Error =SQRT(D5^2/D2+E5^2/E2)
244
239
Confid. Coeff. 0.95
242 Level of Signif. =1-D8
280
z Value =NORM.S.INV(1-D9/2)
261 Margin of Error =D10*D6
276
241
Pt. Est. of Diff. =D3-E3
273
Lower Limit =D13-D11
248
Upper Limit =D13+D11
Note: Rows 16-121 are not shown.
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 12
Interval Estimation of 1 - 2:
s 1 and s 2 Known

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Excel Value Worksheet
A
Par
255
270
294
245
300
262
281
257
268
295
249
291
289
282
C
B
Rap
Sample Size 120
266
Sample Mean 275
238
243
277 Popul. Std. Dev. 15
275 Standard Error 2.622
244
Confid. Coeff. 0.95
239
242 Level of Signif. 0.05
z Value 1.96
280
261 Margin of Error 5.14
276
Pt. Est. of Diff. 17
241
Lower Limit 11.86
273
Upper Limit 22.14
248
E
Rap, Ltd.
D
Par, Inc.
80
258
20
Note: Rows 16-121 are not shown.
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 13
Hypothesis Tests About  1   2:
s 1 and s 2 Known
 Hypotheses
H0 : 1  2  D0 H0 : 1  2  D0 H0 : 1  2  D0
H a : 1  2  D0 H a : 1  2  D0 H a : 1  2  D0
Left-tailed
Right-tailed
Two-tailed
 Test Statistic
z
( x1  x2 )  D0
s 12
n1

s 22
n2
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 14
Hypothesis Tests About  1   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?
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 15
Hypothesis Tests About  1   2:
s 1 and s 2 Known
 p –Value and Critical Value Approaches
1. Develop the hypotheses.
H0: 1 - 2 < 0
Ha: 1 - 2 > 0
where:
1 = mean distance for the population
of Par, Inc. golf balls
2 = mean distance for the population
of Rap, Ltd. golf balls
2. Specify the level of significance.
Righttailed
test
 = .01
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 16
Hypothesis Tests About  1   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

17
 6.49
2.62
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 17
Hypothesis Tests About  1   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.
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 18
Hypothesis Tests About  1   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.
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 19
Excel’s “z-Test: Two Sample for Means” Tool
Step 1 Click the Data tab on the Ribbon
Step 2 In the Analysis group, click Data Analysis
Step 3 Choose z-Test: Two Sample for Means
from the list of Analysis Tools
Step 4 When the z-Test: Two Sample for Means
dialog box appears:
(see details on next slide)
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 20
Excel’s “z-Test: Two Sample for Means” Tool

Excel Dialog Box
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 21
Excel’s “z-Test: Two Sample for Means” Tool

1
2
3
4
5
6
7
8
9
10
11
12
13
Excel Value Worksheet
A
Par
255
270
294
245
300
262
281
257
268
295
249
291
Note:
B C
D
Rap
266
z-Test: Two Sample for Means
238
243
277
Mean
275
Known Variance
244
Observations
239
Hypothesized Mean Difference
242
z
280
P(Z<=z) one-tail
261
z Critical one-tail
276
P(Z<=z) two-tail
241
z Critical two-tail
Rows 14-121 are not shown.
E
F
Par, Inc.
Rap, Ltd.
235
218
225
400
120
80
0
6.483545607
4.50145E-11
2.326341928
9.00291E-11
2.575834515
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 22
Interval Estimation of 1 - 2:
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.
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 23
Interval Estimation of 1 - 2:
s 1 and s 2 Unknown

Interval Estimate
x1  x2  t / 2
s12 s22

n1 n2
Where the degrees of freedom for t/2 are:
2
s s 
  
n1 n2 

df 
2
2
2
2
1  s1 
1  s2 
  
 
n1  1  n1  n2  1  n2 
2
1
2
2
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 24
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.
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 25
Difference Between Two Population Means:
s 1 and s 2 Unknown

Example: Specific Motors
Sample #1
M Cars
24 cars
Sample Size
Sample Mean
29.8 mpg
Sample Std. Dev. 2.56 mpg
Sample #2
J Cars
28 cars
27.3 mpg
1.81 mpg
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 26
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.
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 27
Point Estimate of  1   2
Point estimate of 1  2 = x1  x2
= 29.8 - 27.3
= 2.5 mpg
where:
1 = mean miles-per-gallon for the
population of M cars
2 = mean miles-per-gallon for the
population of J cars
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 28
Interval Estimation of  1   2:
s 1 and s 2 Unknown
The degrees of freedom for t/2 are:
2
 (2.56) (1.81) 



24
28


df 
 24.07  24
2
2
1  (2.56) 2 
1  (1.81) 2 

 


24  1  24  28  1  28 
2
2
With /2 = .05 and df = 24, t/2 = 1.711
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 29
Interval Estimation of  1   2:
s 1 and s 2 Unknown
x1  x2  t / 2
s12 s22
(2.56)2 (1.81)2
  29.8  27.3  1.711

n1 n2
24
28
2.5 + 1.069 or
1.431 to 3.569 mpg
We are 90% confident that the difference between
the miles-per-gallon performances of M cars and J cars
is 1.431 to 3.569 mpg.
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 30
Interval Estimation of  1   2:
s 1 and s 2 Unknown

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Excel Formula Worksheet
A
M
26.1
32.5
31.8
27.6
28.5
33.6
31.7
25.2
26.0
32.0
31.7
30.4
27.6
32.3
30.6
29.5
B
C
D
E
J
Par, Inc.
Rap, Ltd.
25.6
Sample Size =COUNT(A2:A25)
=COUNT(B2:B29)
28.1
Sample Mean =AVERAGE(A2:A25)
=AVERAGE(B2:B29)
27.9 Sample Std. Dev. =STDEV(A2:A25)
=STDEV(B2:B29)
25.3
30.1 Est. of Variance =D4^2/D2+E4^2/E2
27.5
Standard Error =SQRT(D6)
26.0
28.8
Confid. Coeff. 0.90
30.6
Level of Signif. =1-D9
24.4 Degr. of Freedom =D6^2/((1/(D2-1))*(D4^2/D2)^2+(1/(E2-1))*(E4^2/E2)^2))
27.3
t Value =T.INV.2T(D10,D11)
27.5
Margin of Error =D12*D7
26.3
Note:
25.5 Point Est. of Diff.=D3-E3
Rows 18-29
are not shown.
26.3
Lower Limit =D15-D13
24.3
Upper Limit =D15+D13
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 31
Interval Estimation of  1   2:
s 1 and s 2 Unknown

Excel Formula Worksheet
A
B
C
D
E
1
M
J
Par, Inc.
Rap, Ltd.
2 26.1 25.6
Sample Size 24
28
3 32.5 28.1
Sample Mean 29.8
27.3
4 31.8 27.9 Sample Std. Dev. 2.56
1.81
5 27.6 25.3
6 28.5 30.1
Est. of Variance 0.39007
7 33.6 27.5
Standard Error 0.62456
8 31.7 26.0
9 25.2 28.8
Confid. Coeff. 0.90
10 26.0 30.6
Level of Signif. 0.10
11 32.0 24.4 Degr. of Freedom 24.07
12 31.7 27.3
t Value 1.711
13 30.4 27.5
Margin of Error 1.069
14 27.6 26.3
Note:
15 32.3 25.5 Point Est. of Diff. 2.5
Rows 18-29
16 30.6 26.3
Lower Limit 1.431
are not shown.
17 29.5 24.3
Upper Limit 3.569
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
Slide 32
or duplicated, or posted to a publicly accessible website, in whole or in part.
Hypothesis Tests About  1   2:
s 1 and s 2 Unknown

Hypotheses
H0 : 1  2  D0 H0 : 1  2  D0 H0 : 1  2  D0
H a : 1  2  D0 H a : 1  2  D0 H a : 1  2  D0
Left-tailed

Right-tailed
Two-tailed
Test Statistic
t
( x1  x2 )  D0
2
1
2
2
s
s

n1 n2
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 33
Hypothesis Tests About  1   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-per-gallon performance of J
cars?
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 34
Hypothesis Tests About  1   2:
s 1 and s 2 Unknown
 p –Value and Critical Value Approaches
1. Develop the hypotheses.
H0: 1 - 2 < 0
Ha: 1 - 2 > 0
Righttailed
test
where:
1 = mean mpg for the population of M cars
2 = mean mpg for the population of J cars
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 35
Hypothesis Tests About  1   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

(29.8  27.3)  0
(2.56)2 (1.81)2

24
28
 4.003
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 36
Hypothesis Tests About  1   2:
s 1 and s 2 Unknown
 p –Value Approach
4. Compute the p –value.
The degrees of freedom for t are:
2
 (2.56) (1.81) 



24
28 

df 
 40.566  41
2
2
1  (2.56) 2 
1  (1.81) 2 

 


24  1  24  28  1  28 
2
2
Because t = 4.003 > t.005 = 1.683, the p–value < .005.
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 37
Hypothesis Tests About  1   2:
s 1 and s 2 Unknown
 p –Value Approach
5. Determine whether to reject H0.
Because p–value <  = .05, we reject H0.
We are at least 95% confident that the miles-pergallon (mpg) performance of M cars is greater than
the miles-per-gallon performance of J cars.
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 38
Hypothesis Tests About  1   2:
s 1 and s 2 Unknown
 Critical Value Approach
4. Determine the critical value and rejection rule.
For  = .05 and df = 41, t.05 = 1.683
Reject H0 if t > 1.683
5. Determine whether to reject H0.
Because 4.003 > 1.683, we reject H0.
We are at least 95% confident that the miles-pergallon (mpg) performance of M cars is greater than
the miles-per-gallon performance of J cars.
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 39
Excel’s “z-Test: Two-Sample
Assuming Unequal Variances” Tool
Step 1 Click the Data tab on the Ribbon
Step 2 In the Analysis group, click Data Analysis
Step 3 Choose t-Test: Two-Sample Assuming
Unequal Variances from the list of
Analysis Tools
Step 4 When the t-Test: Two-Sample Assuming
Unequal Variances dialog box appears:
(see details on next slide)
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 40
Excel’s “z-Test: Two-Sample
Assuming Unequal Variances” Tool

Excel Dialog Box
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 41
Excel’s “z-Test: Two-Sample
Assuming Unequal Variances” Tool

Excel Value Worksheet
A
1
2
3
4
5
6
7
8
9
10
11
12
13
B
Mcar Jcar
26.1
32.5
31.8
27.6
28.5
33.6
31.7
25.2
26.0
32.0
31.7
30.4
Note:
C
D
E
F
z-Test: Two-Sample Assuming Unequal Variances
25.6
28.1
27.9
Mean
25.3
Variance
30.1
Observations
27.5
Hypothesized Mean Difference
26.0
df
28.8
t Stat
30.6
P(T<=t) one-tail
24.4
t Critical one-tail
27.3
P(T<=t) two-tail
27.5
t Critical two-tail
Rows 14-121 are not shown.
Mcar
29.79583
6.555199
24
0
41
3.99082
1.33000E-04
1.682879
0.000266
2.019542
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Jcar
27.30357
3.272209
28
Slide 42
End of Chapter 10, Part A
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 43