Transcript Chapter 10 Comparisons Involving Means Part A

Chapter 10
Comparisons Involving Means
Part A
• Estimation of the Difference between the Means
of Two Populations: Independent Samples
• Hypothesis Tests about the Difference between
the Means of Two Populations: Independent
Samples
Estimation of the Difference Between the
Means of Two Populations: Independent
Samples
• Point Estimator of the Difference between the Means of
Two Populations
• Sampling Distribution of x1  x2
• Interval Estimate of Large-Sample Case
• Interval Estimate of Small-Sample Case
Point Estimator of the Difference Between
the Means of Two Populations



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 .

Sampling Distribution of x1  x2

Expected Value
E ( x1  x2 )  1   2

Standard Deviation
 x1  x2 
12
n1

 22
n2
where: 1 = standard deviation of population 1
2 = standard deviation of population 2
n1 = sample size from population 1
n2 = sample size from population 2
Interval Estimate of 1 - 2:
Large-Sample Case (n1 > 30 and n2 > 30)
• Interval Estimate with 1 and 2 Known
x1  x2  z / 2  x1  x2
where:
1 -  is the confidence coefficient
Interval Estimate of 1 - 2:
Large-Sample Case (n1 > 30 and n2 > 30)

Interval Estimate with 1 and 2 Unknown
x1  x2  z / 2 sx1  x2
where:
sx1  x2
s12 s22


n1 n2
Interval Estimate of 1 - 2:
Large-Sample Case (n1 > 30 and n2 > 30)
• 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.
Interval Estimate of 1 - 2:
Large-Sample Case (n1 > 30 and n2 > 30)
• Example: Par, Inc.
Sample Size
Sample Mean
Sample Std. Dev.
Sample #1
Par, Inc.
120 balls
235 yards
15 yards
Sample #2
Rap, Ltd.
80 balls
218 yards
20 yards
Point Estimator of the Difference Between
the Means of Two Populations
Population 1
Par, Inc. Golf Balls
Population 2
Rap, Ltd. Golf Balls
1 = mean driving
2 = mean driving
distance of Par
golf balls
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 sample of Par golf ball
x2 = sample mean distance
for sample of Rap golf ball
x1 - x2 = Point Estimate of m1 –
2
Point Estimate of the Difference
Between Two Population Means
Point estimate of 1  2 = x1  x2
= 235  218
= 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
95% Confidence Interval Estimate of the
Difference Between Two Population Means:
Large-Sample Case, 1 and 2 Unknown
Substituting the sample standard
deviations for the population standard
deviation:
x1  x2  z / 2
12
 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.
Using Excel to Develop an
Interval Estimate of 1 – 2: Large-Sample Case

1
2
3
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8
9
10
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12
13
14
15
Formula Worksheet
A
Par
195
230
254
205
260
222
241
217
228
255
209
251
229
220
B
C
Rap
226 Sample Size
198
Mean
203
Stand. Dev.
237
235 Confid. Coeff.
204 Lev. of Signif.
199
z Value
202
240
Std. Error
221 Marg. of Error
206
201 Pt. Est. of Diff.
233
Lower Limit
194
Upper Limit
D
Par, Inc.
120
=AVERAGE(A2:A121)
=STDEV(A2:A121)
0.95
=1-D6
=NORMSINV(1-D7/2)
=SQRT(D4^2*/D2+E4^2/E2)
=D8*D10
=D3-E3
=D13-D11
=D13+D11
Note: Rows 16-121 are not shown.
E
Rap, Ltd.
80
=AVERAGE(A2:A81)
=STDEV(A2:A81)
Using Excel to Develop an
Interval Estimate of 1 – 2: Large-Sample Case

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Value Worksheet
A
Par
195
230
254
205
260
222
241
217
228
255
209
251
229
220
B
C
Rap
226 Sample Size
198
Mean
203
Stand. Dev.
237
235 Confid. Coeff.
204 Lev. of Signif.
199
z Value
202
240
Std. Error
221 Marg. of Error
206
201 Pt. Est. of Diff.
233
Lower Limit
194
Upper Limit
D
Par, Inc.
120
235
15
0.95
0.05
1.960
2.622
5.139
17
11.86
22.14
Note: Rows 16-121 are not shown.
E
Rap, Ltd.
80
218
20
Interval Estimate of 1 - 2:
Small-Sample Case (n1 < 30 and/or n2 < 30)
 Interval Estimate with  12   22   2
x1  x2  z / 2  x1  x2
where:
 x1  x2
1 1
  (  )
n1 n2
2
Interval Estimate of 1 - 2:
Small-Sample Case (n1 < 30 and/or n2 < 30)
• Interval Estimate with  2 Unknown
x1  x2  t / 2 sx1  x2
where:
sx1  x2
1 1
 s (  )
n1 n2
2
2
2
(
n

1
)
s

(
n

1
)
s
1
2
2
s2  1
n1  n2  2
Difference Between Two Population Means:
Small Sample Case

Example: Specific Motors
Specific Motors of Detroit
has developed a new automobile
known as the M car. 12 M cars
and 8 J cars (from Japan) were road
tested to compare miles-per-gallon (mpg)
performance. The sample statistics are shown on the
next slide.
Difference Between Two Population Means:
Small Sample Case

Example: Specific Motors
Sample #1
M Cars
12 cars
29.8 mpg
2.56 mpg
Sample #2
J Cars
8 cars
27.3 mpg
1.81 mpg
Sample Size
Sample Mean
Sample Std. Dev.
Point Estimate of the Difference
Between Two Population Means
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
95% Confidence Interval Estimate of the
Difference Between Two Population Means:
Small-Sample Case
We will make the following
assumptions:
•
The miles per gallon rating is normally
distributed for both the M car and the J car.
•
The variance in the miles per gallon rating
is the same for both the M car and the J car.
95% Confidence Interval Estimate of the
Difference Between Two Population Means:
Small-Sample Case
 We will use a weighted average of the two sample
variances as the pooled estimator of  2.
2
2
2
2
(
n

1
)
s

(
n

1
)
s
11
(
2
.
56
)

7
(
1
.
81
)
1
2
2
s2  1

 5. 28
n1  n2  2
12  8  2
95% Confidence Interval Estimate of the
Difference Between Two Population Means:
Small-Sample Case

Using the t distribution with n1 + n2 - 2 = 18 degrees
of freedom, the appropriate t value is t.025 = 2.101.
x1  x2  t.025
1 1
1 1
s (  )  2. 5  2.101 5. 28(  )
n1 n2
12 8
2
2.5 + 2.2 or .3 to 4.7 miles per
gallon
We are 95% confident that the difference between
the mean mpg ratings of the two car types is .3 to 4.7
mpg (with the M car having the higher mpg).
Using Excel to Develop an
Interval Estimate of 1 – 2: Small-Sample

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Formula Worksheet
A
M Car
25.1
32.2
31.7
27.6
28.5
33.6
30.8
26.2
29.0
31.0
31.7
30.0
B
C
D
J Car
M Car
25.6 Sample Size 12
28.1
Mean =AVERAGE(A2:A13)
27.9
Stand. Dev. =STDEV(A2:A13)
25.3
30.1 Confid. Coeff. 0.95
27.5 Lev. of Signif. =1-D6
25.1
Deg. Freed. =D2+E2-2
28.8
z Value =TINV(D7,D8)
Pool.Est.Var. =((D2-1)*D4^2+(E2-1)*E4^2)/D8
Std. Error =SQRT(D11*(1/D2+1/E2))
Marg. of Error =D9*D12
Pt. Est. of Diff. =D3-E3
Lower Limit =D15-D13
Upper Limit =D15+D13
E
J Car
8
=AVERAGE(B2:B9)
=STDEV(B2:B9)
Using Excel to Develop an
Interval Estimate of 1 – 2: Small-Sample

Value Worksheet
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
A
M Car
25.1
32.2
31.7
27.6
28.5
33.6
30.8
26.2
29.0
31.0
31.7
30.0
B
C
J Car
25.6 Sample Size 12
28.1
Mean 29.8
27.9
Stand. Dev. 2.56
25.3
30.1 Confid. Coeff. 0.95
27.5 Lev. of Signif. 0.05
25.1
Deg. Freed. 18
28.8
z Value 2.101
Pool.Est.Var. 5.2765
Std. Error 1.0485
Marg. of Error 2.2027
Pt. Est. of Diff. 2.4833
Lower Limit 0.2806
Upper Limit 4.6861
D
M Car
E
J Car
8
27.3
1.81
Hypothesis Tests About the Difference
between the Means of Two Populations:
Independent Samples
• Hypotheses
H 0 : 1   2  0
H a : 1   2  0
H 0 : 1   2  0 H 0 : 1   2  0
H a : 1   2  0 H a : 1   2  0
• Test Statistic
Large-Sample
z
( x1  x2 )  ( 1   2 )
12
n1   22
n2
Small-Sample
t
( x1  x2 )  ( 1   2 )
s2 (1 n1  1 n2 )
Hypothesis Tests About the Difference
Between the Means of Two Populations:
Independent Samples, Large-Sample Case

Example: Par, Inc.
Recall that Par, Inc. has
developed a new golf ball that
was designed to provide “extra
distance.” 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.
Hypothesis Tests About the Difference
Between the Means of Two Populations:
Independent Samples, Large-Sample Case

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?
Sample #1
Sample Size
Sample Mean
Sample Std. Dev.
Par, Inc.
120 balls
235 yards
15 yards
Sample #2
Rap, Ltd.
80 balls
218 yards
20 yards
Hypothesis Tests About the Difference
Between the Means of Two Populations:
Independent Samples, Large-Sample Case
 Using the Test Statistic
1. Determine 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
Hypothesis Tests About the Difference
Between the Means of Two Populations:
Independent Samples, Large-Sample Case
 Using the Test Statistic
2. Specify the level of significance.
3. Select the test statistic.
z
 = .01
( x1  x 2 )  (  1   2 )
 12
n1
4. State the rejection rule.

 22
n2
Reject H0 if z > 2.33
Hypothesis Tests About the Difference
Between the Means of Two Populations:
Independent Samples, Large-Sample Case
 Using the Test Statistic
5. Compute the value of the test statistic.
z
( x1  x 2 )  (  1   2 )
 12
n1
z

 22
n2
(235  218)  0
(15)2 (20)2

120
80

17
 6.49
2.62
Hypothesis Tests About the Difference
Between the Means of Two Populations:
Independent Samples, Large-Sample Case
 Using the Test Statistic
6. Determine whether to reject H0.
z = 6.49 > z.01 = 2.33, so 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.
Using Excel to Conduct a
Hypothesis Test about 1 – 2: Large Sample Case

Excel’s “z-Test: Two Sample for Means” Tool
Step 1 Select the Tools menu
Step 2 Choose the Data Analysis option
Step 3 Choose z-Test: Two Sample for Means
from the list of Analysis Tools
… continued
Using Excel to Conduct a
Hypothesis Test about 1 – 2: Large Sample Case

Excel’s “z-Test: Two Sample for Means” Tool
Step 4 When the z-Test: Two Sample for Means
dialog box appears:
Enter A1:A121 in the Variable 1 Range box
Enter B1:B81 in the Variable 2 Range box
Type 0 in the Hypothesized Mean
Difference box
Type 225 in the Variable 1 Variance
(known) box
Type 400 in the Variable 2 Variance
(known) box
… continued
Using Excel to Conduct a
Hypothesis Test about 1 – 2: Large Sample Case

Excel’s “z-Test: Two Sample for Means” Tool
Step 4 (continued)
Select Labels
Type .01 in the Alpha box
Select Output Range
Enter D4 in the Output Range box
(Any upper left-hand corner cell indicating
where the output is to begin may be entered)
Click OK
Using Excel to Conduct a
Hypothesis Test about 1 – 2: Large Sample Case
Using Excel to Conduct a
Hypothesis Test about 1 – 2: Large Sample Case

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Value Worksheet
A
Par
195
230
254
205
260
222
241
217
228
255
209
251
229
220
B C
D
Rap
226
Sample Variance
198
203
z-Test: Two Sample for Means
237
235
204
Mean
199
Known Variance
202
Observations
240
Hypothesized Mean Difference
221
z
206
P(Z<=z) one-tail
201
z Critical one-tail
233
P(Z<=z) two-tail
194
z Critical two-tail
Note: Rows 16-121 are not shown.
E
Par, Inc.
225
F
Rap, Ltd.
400
Par, Inc.
Rap, Ltd.
235
218
225
400
120
80
0
6.483545607
4.50145E-11
2.326341928
9.00291E-11
2.575834515
Using Excel to Conduct a
Hypothesis Test about 1 – 2: Large Sample Case
 Using the p Value
4. Compute the value of the test statistic.
The Excel worksheet states z = 6.48
5. Compute the p–value.
The Excel worksheet states p-value = 4.501E-11
6. Determine whether to reject H0.
Because p–value <  = .01, we reject H0.