SLIDES . BY
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Transcript of SLIDES . BY
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SLIDES . BY
John LoucksSt. Edward’sUniversity
...........
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Chapter 10, Part A Comparisons Involving Means, Experimental
Design, and Analysis of Variance
Inferences About the Difference Between Two Population Means: s 1 and s 2 Known
Inferences About the Difference Between Two Population Means: Matched Samples
Inferences About the Difference Between Two Population Means: s 1 and s 2 Unknown
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Inferences About the Difference BetweenTwo Population Means: s 1 and s 2 Known
Interval Estimation of m 1 – m 2
Hypothesis Tests About m 1 – m 2
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Estimating the Difference BetweenTwo 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 equal the mean of sample 1 and
equal the mean of sample 2.
x1 x2
The point estimator of the difference between the
means of the populations 1 and 2 is .x x1 2
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Expected Value
Sampling Distribution of x x1 2
E x x( )1 2 1 2
Standard Deviation (Standard Error)
x x n n1 2
12
1
22
2
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
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6 6 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Interval Estimate
Interval Estimation of 1 - 2: s 1 and s 2 Known
2 21 2
1 2 / 21 2
x x zn n
where: 1 - is the confidence coefficient
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Interval Estimation of 1 - 2: s 1 and s 2 Known
In a test of driving distance using a mechanicaldriving device, a sample of Par golf balls wascompared with a sample of golf balls made by Rap,Ltd., a competitor. The sample statistics appear onthe next slide.
Par, Inc. is a manufacturer of golf equipment andhas developed a new golf ball that has been designed to provide “extra distance.”
Example: Par, Inc.
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8 8 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Example: Par, Inc.
Interval Estimation of 1 - 2: s 1 and s 2 Known
Sample SizeSample Mean
Sample #1Par, Inc.
Sample #2Rap, Ltd.
120 balls 80 balls275 yards 258 yards
Based on data from previous driving distancetests, the two population standard deviations areknown with s 1 = 15 yards and s 2 = 20 yards.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Interval Estimation of 1 - 2: s 1 and s 2 Known
Example: Par, Inc.
Let us develop a 95% confidence interval estimateof the difference between the mean driving distances ofthe two brands of golf ball.
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Estimating the Difference BetweenTwo Population Means
m1 – m2 = difference between the mean distances
x1 - x2 = Point Estimate of m1 – m2
Population 1Par, Inc. Golf Balls
m1 = mean driving distance of Par
golf balls
Population 2Rap, Ltd. Golf Balls
m2 = mean driving distance of Rap
golf balls
Simple random sample of n2 Rap golf balls
x2 = sample mean distance for the Rap golf balls
Simple random sample of n1 Par golf balls
x1 = sample mean distance for the Par golf balls
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Point Estimate of 1 - 2
Point estimate of 1 - 2 =x x1 2
where:1 = mean distance for the population of Par, Inc. golf balls2 = mean distance for the population of Rap, Ltd. golf balls
= 275 - 258
= 17 yards
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x x zn n1 2 2
12
1
22
2
2 2
17 1 9615120
2080
/ .( ) ( )
Interval Estimation of 1 - 2: 1 and 2 Known
We are 95% confident that the difference betweenthe mean driving distances of Par, Inc. balls and Rap,Ltd. balls is 11.86 to 22.14 yards.
17 + 5.14 or 11.86 yards to 22.14 yards
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Hypothesis Tests About m 1 - m 2:s 1 and s 2 Known
Hypotheses
1 2 0
2 21 2
1 2
( )x x Dz
n n
1 2 0: aH D 0 1 2 0: H D 0 1 2 0: H D
1 2 0: aH D 0 1 2 0: H D 1 2 0: aH D
Left-tailed Right-tailed Two-tailed
Test Statistic
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14 14 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Example: Par, Inc.
Hypothesis Tests About m 1 - m 2:s 1 and s 2 Known
Can we conclude, using a = .01, that themean driving distance of Par, Inc. golf balls is greater than the mean driving distance of Rap, Ltd. golf balls?
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or duplicated, or posted to a publicly accessible website, in whole or in part.
H0: 1 - 2 < 0
Ha: 1 - 2 > 0where: 1 = mean distance for the population of Par, Inc. golf balls 2 = mean distance for the population of Rap, Ltd. golf balls
1. Develop the hypotheses.
p –Value and Critical Value Approaches
Hypothesis Tests About m 1 - m 2:s 1 and s 2 Known
2. Specify the level of significance. a = .01
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16 16 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
3. Compute the value of the test statistic.
Hypothesis Tests About m 1 - m 2:s 1 and s 2 Known
p –Value and Critical Value Approaches
1 2 0
2 21 2
1 2
( )x x Dz
n n
2 2
(235 218) 0 17 6.49
2.62(15) (20)120 80
z
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p –Value Approach
4. Compute the p–value.
For z = 6.49, the p –value < .0001.
Hypothesis Tests About m 1 - m 2:s 1 and s 2 Known
5. Determine whether to reject H0.
Because p–value < a = .01, we reject H0.
At the .01 level of significance, the sample evidenceindicates the mean driving distance of Par, Inc. golfballs is greater than the mean driving distance of Rap,Ltd. golf balls.
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Hypothesis Tests About m 1 - m 2:s 1 and s 2 Known
5. Determine whether to reject H0.
Because z = 6.49 > 2.33, we reject H0.
Critical Value Approach
For a = .01, z.01 = 2.33
4. Determine the critical value and rejection rule.
Reject H0 if z > 2.33
The sample evidence indicates the mean drivingdistance of Par, Inc. golf balls is greater than the meandriving distance of Rap, Ltd. golf balls.
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19 19 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Inferences About the Difference BetweenTwo Population Means: s 1 and s 2
Unknown Interval Estimation of m1 – m2
Hypothesis Tests About m1 – m2
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20 20 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Interval Estimation of 1 - 2:s 1 and s 2 Unknown
When s 1 and s 2 are unknown, we will:
• replace za/2 with ta/2.
• use the sample standard deviations s1 and s2
as estimates of s 1 and s 2 , and
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2 21 2
1 2 / 21 2
s sx x t
n n
Where the degrees of freedom for ta/2 are:
Interval Estimation of 1 - 2:s 1 and s 2 Unknown
Interval Estimate
22 21 2
1 22 22 2
1 2
1 1 2 2
1 11 1
s sn n
dfs s
n n n n
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22 22 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Example: Specific Motors
Difference Between Two Population Means:
s 1 and s 2 Unknown
Specific Motors of Detroit has developed a newAutomobile known as the M car. 24 M cars and 28 Jcars (from Japan) were road tested to compare miles-per-gallon (mpg) performance. The sample statisticsare shown on the next slide.
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23 23 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Difference Between Two Population Means:
s 1 and s 2 Unknown Example: Specific Motors
Sample SizeSample MeanSample Std. Dev.
Sample #1M Cars
Sample #2J Cars
24 cars 28 cars29.8 mpg 27.3 mpg2.56 mpg 1.81 mpg
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24 24 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Difference Between Two Population Means:
s 1 and s 2 Unknown
Let us develop a 90% confidence interval estimate of the difference between the mpg performances of the two models of automobile.
Example: Specific Motors
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25 25 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Point estimate of 1 - 2 =x x1 2
Point Estimate of m 1 - m 2
where:1 = mean miles-per-gallon for the population of M cars2 = mean miles-per-gallon for the population of J cars
= 29.8 - 27.3
= 2.5 mpg
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26 26 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Interval Estimation of m 1 - m 2:s 1 and s 2 Unknown
The degrees of freedom for ta/2 are:22 2
2 22 2
(2.56) (1.81)24 28
24.07 241 (2.56) 1 (1.81)
24 1 24 28 1 28
df
With a/2 = .05 and df = 24, ta/2 = 1.711
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27 27 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Interval Estimation of m 1 - m 2:s 1 and s 2 Unknown
2 2 2 21 2
1 2 / 21 2
(2.56) (1.81) 29.8 27.3 1.711
24 28
s sx x t
n n
We are 90% confident that the difference betweenthe miles-per-gallon performances of M cars and J carsis 1.431 to 3.569 mpg.
2.5 + 1.069 or 1.431 to 3.569 mpg
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28 28 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Hypothesis Tests About m 1 - m 2:s 1 and s 2 Unknown
Hypotheses
1 2 0
2 21 2
1 2
( )x x Dt
s sn n
1 2 0: aH D 0 1 2 0: H D 0 1 2 0: H D
1 2 0: aH D 0 1 2 0: H D 1 2 0: aH D
Left-tailed Right-tailed Two-tailed Test Statistic
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29 29 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Example: Specific Motors
Hypothesis Tests About m 1 - m 2:s 1 and s 2 Unknown
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?
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30 30 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
H0: 1 - 2 < 0
Ha: 1 - 2 > 0where: 1 = mean mpg for the population of M cars 2 = mean mpg for the population of J cars
1. Develop the hypotheses.
p –Value and Critical Value Approaches
Hypothesis Tests About m 1 - m 2:s 1 and s 2 Unknown
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31 31 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
2. Specify the level of significance.
3. Compute the value of the test statistic.
a = .05
p –Value and Critical Value Approaches
Hypothesis Tests About m 1 - m 2:s 1 and s 2 Unknown
1 2 0
2 2 2 21 2
1 2
( ) (29.8 27.3) 0 4.003
(2.56) (1.81)24 28
x x Dt
s sn n
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32 32 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Hypothesis Tests About m 1 - m 2:s 1 and s 2 Unknown
p –Value Approach
4. Compute the p –value.
The degrees of freedom for ta are:
Because t = 4.003 > t.005 = 1.683, the p–value < .005.
22 2
2 22 2
(2.56) (1.81)24 28
40.566 411 (2.56) 1 (1.81)
24 1 24 28 1 28
df
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33 33 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
5. Determine whether to reject H0.
We are at least 95% confident that the miles-per-gallon (mpg) performance of M cars is greater than the miles-per-gallon performance of J cars.
p –Value Approach
Because p–value < a = .05, we reject H0.
Hypothesis Tests About m 1 - m 2:s 1 and s 2 Unknown
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34 34 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
4. Determine the critical value and rejection rule.
Critical Value Approach
Hypothesis Tests About m 1 - m 2:s 1 and s 2 Unknown
For a = .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-per-gallon (mpg) performance of M cars is greater than the miles-per-gallon performance of J cars.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
With a matched-sample design each sampled item provides a pair of data values.
This design often leads to a smaller sampling error
than the independent-sample design because variation between sampled items is eliminated as a source of sampling error.
Inferences About the Difference BetweenTwo Population Means: Matched Samples
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Example: Express Deliveries
Inferences About the Difference BetweenTwo Population Means: Matched Samples
A Chicago-based firm has documents that mustbe quickly distributed to district offices throughout the U.S. The firm must decide between two deliveryservices, UPX (United Parcel Express) and INTEX(International Express), to transport its documents.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Example: Express Deliveries
Inferences About the Difference BetweenTwo Population Means: Matched Samples
In testing the delivery times of the two services,the firm sent two reports to a random sample of its district offices with one report carried by UPX and the other report carried by INTEX. Do the data onthe next slide indicate a difference in mean deliverytimes for the two services? Use a .05 level ofsignificance.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
3230191615181410 716
25241515131515 8 911
UPX INTEX DifferenceDistrict OfficeSeattleLos AngelesBostonClevelandNew YorkHoustonAtlantaSt. LouisMilwaukeeDenver
Delivery Time (Hours)
7 6 4 1 2 3 -1 2 -2 5
Inferences About the Difference BetweenTwo Population Means: Matched Samples
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or duplicated, or posted to a publicly accessible website, in whole or in part.
H0: d = 0
Ha: d Let d = the mean of the difference values for the two delivery services for the population of district offices
1. Develop the hypotheses.
Inferences About the Difference BetweenTwo Population Means: Matched Samples
p –Value and Critical Value Approaches
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40 40 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
2. Specify the level of significance. a = .05
Inferences About the Difference BetweenTwo Population Means: Matched Samples
p –Value and Critical Value Approaches
3. Compute the value of the test statistic.
ddni ( ... )
.7 6 5
102 7
sd dndi
( ) ..
2
176 1
92 9
2.7 0 2.94
2.9 10d
d
dt
s n
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41 41 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
5. Determine whether to reject H0.
We are at least 95% confident that there is a difference in mean delivery times for the two services.
4. Compute the p –value.
For t = 2.94 and df = 9, the p–value is between.02 and .01. (This is a two-tailed test, so we double the upper-tail areas of .01 and .005.)
Because p–value < a = .05, we reject H0.
Inferences About the Difference BetweenTwo Population Means: Matched Samples
p –Value Approach
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42 42 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
4. Determine the critical value and rejection rule.
Inferences About the Difference BetweenTwo Population Means: Matched Samples
Critical Value Approach
For a = .05 and df = 9, t.025 = 2.262.
Reject H0 if t > 2.262
5. Determine whether to reject H0.
Because t = 2.94 > 2.262, we reject H0.
We are at least 95% confident that there is a difference in mean delivery times for the two services.
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End of Chapter 10Part A