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1 Slide© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
John LoucksSt. Edward’sUniversity
...........
SLIDES . BY
2 Slide© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Chapter 10, Part A 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: Matched Samples
Inferences About the Difference Between Two Population Means: s 1 and s 2 Unknown
3 Slide© 2014 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 Known
Interval Estimation of m 1 – m 2 Hypothesis Tests About m 1 – m 2
4 Slide© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Estimating the Difference BetweenTwo 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 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
5 Slide© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
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 m m
Standard Deviation (Standard Error)
s s sx x n n1 2
12
1
22
2
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
6 Slide© 2014 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 m1 - m2: s 1 and s 2 Known
2 21 2
1 2 / 21 2
x x zn ns s
where: 1 - is the confidence coefficient
7 Slide© 2014 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 m1 - m2: 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.
8 Slide© 2014 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 m1 - m2: 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.
9 Slide© 2014 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 m1 - m2: 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.
10 Slide© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Estimating the Difference BetweenTwo Population Means
m1 – m2 = difference between the mean distances
x1 - x2 = Point Estimate of m1 – m2
Population 1Par, Inc. Golf Ballsm1 = mean driving
distance of Pargolf balls
Population 2Rap, Ltd. Golf Ballsm2 = mean driving
distance of Rapgolf balls
Simple random sample of n2 Rap golf ballsx2 = sample mean distance for the Rap golf balls
Simple random sample of n1 Par golf ballsx1 = sample mean distance for the Par golf balls
11 Slide© 2014 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 m1 - m2
Point estimate of m1 m2 =x x1 2
where:m1 = mean distance for the population of Par, Inc. golf ballsm2 = mean distance for the population of Rap, Ltd. golf balls
= 275 258= 17 yards
12 Slide© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
x x zn n1 2 2
12
1
22
2
2 217 1 96 15
1202080
s s
/ . ( ) ( )
Interval Estimation of m1 - m2:s 1 and s 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
13 Slide© 2014 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 Known
Hypotheses
1 2 02 21 2
1 2
( )x x Dz
n ns s
m m 1 2 0: aH Dm m 0 1 2 0: H Dm m 0 1 2 0: H D
m m 1 2 0: aH Dm m 0 1 2 0: H Dm m 1 2 0: aH D
Left-tailed Right-tailed Two-tailed Test Statistic
14 Slide© 2014 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 = .01, that themean driving distance of Par, Inc. golf balls is greater than the mean driving distance of Rap, Ltd. golf balls?
15 Slide© 2014 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: m1 - m2 < 0
Ha: m1 - m2 > 0where: m1 = mean distance for the population of Par, Inc. golf balls m2 = 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. = .01
16 Slide© 2014 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
s s
1 2 02 21 2
1 2
( )x x Dz
n n
2 2(235 218) 0 17 6.492.62(15) (20)
120 80
z
17 Slide© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
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 < = .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.
18 Slide© 2014 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 Known
5. Determine whether to reject H0.Because z = 6.49 > 2.33, we reject H0.
Critical Value Approach
For = .01, z.01 = 2.334. 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.
19 Slide© 2014 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 m 1 – m 2 Hypothesis Tests About m 1 – m 2
20 Slide© 2014 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 m1 - m2:s 1 and s 2 Unknown
When s 1 and s 2 are unknown, we will:
• replace z/2 with t/2.
• use the sample standard deviations s1 and s2
as estimates of s 1 and s 2 , and
21 Slide© 2014 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 21 2
1 2 / 21 2
s sx x tn n
Where the degrees of freedom for t/2 are:
Interval Estimation of m1 - m2: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
22 Slide© 2014 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.
23 Slide© 2014 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
24 Slide© 2014 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
25 Slide© 2014 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 m1 m2 =x x1 2
Point Estimate of m 1 m 2
where:m1 = mean miles-per-gallon for the population of M carsm2 = mean miles-per-gallon for the population of J cars
= 29.8 - 27.3= 2.5 mpg
26 Slide© 2014 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 t/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 /2 = .05 and df = 24, t/2 = 1.711
27 Slide© 2014 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 1.71124 28
s sx x tn 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
28 Slide© 2014 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 02 21 2
1 2
( )x x Dts sn n
m m 1 2 0: aH Dm m 0 1 2 0: H Dm m 0 1 2 0: H D
m m 1 2 0: aH Dm m 0 1 2 0: H Dm m 1 2 0: aH D
Left-tailed Right-tailed Two-tailed Test Statistic
29 Slide© 2014 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?
30 Slide© 2014 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: m1 - m2 < 0
Ha: m1 - m2 > 0where: m1 = mean mpg for the population of M cars m2 = 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
31 Slide© 2014 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.
= .05 p –Value and Critical Value Approaches
Hypothesis Tests About m 1 m 2:s 1 and s 2 Unknown
1 2 02 2 2 21 2
1 2
( ) (29.8 27.3) 0 4.003(2.56) (1.81)
24 28
x x Dts sn n
32 Slide© 2014 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 Approach4. Compute the p –value.The degrees of freedom for t 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
33 Slide© 2014 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 < = .05, we reject H0.
Hypothesis Tests About m 1 m 2:s 1 and s 2 Unknown
34 Slide© 2014 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 = .05 and df = 41, t.05 = 1.683Reject 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?.
35 Slide© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
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
36 Slide© 2014 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: 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.
37 Slide© 2014 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: 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.
38 Slide© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
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
39 Slide© 2014 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: md = 0
Ha: md Let md = 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
40 Slide© 2014 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. = .05
Inferences About the Difference BetweenTwo Population Means: Matched Samples
p –Value and Critical Value Approaches
3. Compute the value of the test statistic.
d dni ( ... ) .7 6 5
102 7
s d dndi
( ) . .2
176 1
92 9
2.7 0 2.942.9 10d
d
dts n
m
41 Slide© 2014 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 < = .05, we reject H0.
Inferences About the Difference BetweenTwo Population Means: Matched Samples
p –Value Approach
42 Slide© 2014 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 = .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?
43 Slide© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
End of Chapter 10Part A