11- 1 chapter eleven mcgraw-hill/irwin © 2005 the mcgraw-hill companies, inc., all rights reserved
TRANSCRIPT
11- 1
Chapter
Eleven
McGraw-Hill/Irwin
© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved.
11- 2Chapter Eleven
Two-Sample Tests of Two-Sample Tests of HypothesisHypothesisGOALS
When you have completed this chapter, you will be able to:
TWOConduct a test of hypothesis regarding the difference in two population proportions.
THREE
Conduct a test of hypothesis about the mean difference between paired or dependent observations.
ONE
Conduct a test of hypothesis about the difference between two independent population means.
11- 3Chapter Eleven continued
Two Sample Tests of Two Sample Tests of HypothesisHypothesis
GOALSWhen you have completed this chapter, you will be able to:
FOUR
Understand the difference between dependent and independent samples.
11- 4
Comparing two populations
Does the
distribution of the differences in sample
means have a
mean of 0?
Comparing two populations
If both samples contain at least 30 observations we use the z distribution as the test statistic.
No assumptions about the shape of the populations are required.
The samples are from independent populations.
The formula for computing the value of z is:
2
22
1
21
21
n
s
n
s
XXz
11- 5
EXAMPLE 1
with a standard deviation of $7,000 for a sample of 35 households. At the .01 significance level can we conclude the mean income in Bradford is more?
Two cities, Bradford and Kane are separated only by the Conewango River. There is competition between the two cities. The local
paper recently reported that the mean household income in Bradford is $38,000 with a standard deviation of $6,000 for a sample of 40 households. The same article reported the mean income in Kane is $35,000
11- 6
Example 1 continued
Step 2
State the level of significance. The .01 significance level is
stated in the problem.
Step 3
Find the appropriate test statistic. Because both
samples are more than 30, we can use z as the test statistic.
Step 1 State the null and
alternate hypotheses.H0: µB < µK
H1: µB > µK
Step 4 State the decision rule.The null hypothesis is rejected if z is greater than 2.33 or p < .01.
11- 7
Example 1 continued
98.1
35
)000,7($
40
)000,6($
000,35$000,38$22
z
Step 5: Compute the value of z and make a decision.
The p(z > 1.98) is .0239 for a one-tailed test of significance.
Because the computed Z of 1.98
< critical Z of 2.33, the p-value of .0239 > of .01, the decision is to not reject the null hypothesis. We cannot conclude that the mean household income in Bradford is larger.
11- 8
21
21
nn
XXpc
Two Sample Tests of ProportionsTwo Sample Tests of Proportions investigate whether two samples came from populations with an equal proportion of successes.
The two samples are pooled using the following formula.
where X1 and X2 refer to the number of successes in the respective samples of n1 and n2.
The value of the test statistic is computed from the following formula.
21
21
)1()1(
n
pp
n
pp
ppz
cccc
where X1 and X2 refer to the number of successes in the respective samples of n1 and n2.
11- 9
Example 2
Are unmarried workers more likely to be absent from work than married workers? A sample of 250 married workers showed 22 missed more than 5 days last year, while a sample of 300 unmarried workers
showed 35 missed more than five days. Use a .05 significance level.
11- 10
Example 2 continued
The null and the alternate hypothesesH0: U < M H1: U > M
The null hypothesis is rejected if the computed value of z is greater than 1.65 or the p-value < .05.
The pooled proportion
250300
2235
cp
= .1036
11- 11
Example 2 continued
10.1
250
)1036.1(1036.
300
)1036.1(1036.250
22
300
35
z
The p(z > 1.10) = .136 for a one-tailed test of significance.
Because a calculated z of 1.10 < a critical z of 1.96, p of .136 > of .05, the null hypothesis is not rejected. We cannot conclude that a higher proportion of unmarried workers miss more days in a year than the married workers.
11- 12
Small Sample Tests of Means
The required assumptions1. Both populations must follow
the normal distribution.2. The populations must have
equal standard deviations.3. The samples are from
independent populations.
Small Sample Tests of MeansSmall Sample Tests of MeansThe t distribution is used as the test statistic if one or more of the samples have less than 30 observations.
11- 13
Small sample test of means continued
2
)1()1(
21
222
2112
nn
snsns p
21
2
21
11
nns
XXt
p
Step Two: Determine the value of t from the following formula.
Finding the value of the test statistic requires two steps.
Step One: Pool the sample standard deviations.
11- 14
Example 3
A recent EPA study compared the highway fuel economy of domestic and imported passenger cars. A sample of 15 domestic cars revealed a mean of 33.7 mpg with a standard deviation of 2.4 mpg.
A sample of 12 imported cars revealed a mean of 35.7 mpg with a standard deviation of 3.9. At the .05 significance level can the EPA conclude that the mpg is higher on the imported cars?
11- 15
Example 3 continued
Step 1 State the null and
alternate hypotheses. H0: µD > µI
H1: µD < µI
Step 2
State the level of significance. The .05 significance level is
stated in the problem.
Step 3
Find the appropriate test statistic. Both samples are less than 30, so we use the t distribution.
11- 16
Example 3 continued
918.921215
)9.3)(112()4.2)(115(
2
))(1())(1(
22
21
222
2112
nn
snsns p
Step 4
The decision rule is to reject H0 if t<-1.708 or if p-value < .05. There are n-1 or 25
degrees of freedom.
Step 5 We compute the pooled variance.
11- 17
Example 3 continued
640.1
12
1
15
1312.8
7.357.33
11
21
2
21
nns
XXt
p
We compute the value of t as follows.
11- 18
Since a computed z of –1.64 > critical z of –1.71, the p-value of .0567 > of .05, H0 is not rejected. There is insufficient sample evidence to claim a higher mpg on the imported cars.
P(t < -1.64) = .0567 for a one-tailed t-test.
Example 3 continued
11- 19
Hypothesis Testing Involving Paired Observations
Dependent samples are samples that are paired or related in some fashion.
Independent samples are samples that are not related in any way.
If you wished to buy a car you would look at the same car at two (or more) different dealerships and compare the prices.
If you wished to measure the effectiveness of a new diet you would weigh the dieters at the start and at the finish of the program.
11- 20
Hypothesis Testing Involving Paired Observations
td
s nd
/
dsd
Use the following test when the samples are dependent:
where is the mean of the differences is the standard deviation of the differencesn is the number of pairs (differences)
11- 21
EXAMPLE 4
An independent testing agency is comparing the daily rental cost for renting a compact car from Hertz and Avis. A random sample of eight cities revealed the following information. At the .05 significance level can the testing agency conclude that there is a difference in the rental charged?
City Hertz ($)
Avis ($)
Atlanta 42 40
Chicago 56 52
Cleveland 45 43
Denver 48 48
Honolulu 37 32
Kansas City 45 48
Miami 41 39
Seattle 46 50
11- 22
Example 4 continued
Step 4
H0 is rejected if
t < -2.365 or t > 2.365;
or if p-value < .05.
We use the t distribution with n-1 or 7 degrees of freedom.
Step 2 The stated
significance level is .05.
Step 3 The appropriate
test statistic is the paired t-test.
Step 1Ho: d = 0H1: d = 0
Step 5Perform the
calculations and make a decision.
11- 23
Example 4 continued
City Hertz Avis d d2
Atlanta 42 40 2 4
Chicago 56 52 4 16
Cleveland 45 43 2 4
Denver 48 48 0 0
Honolulu 37 32 5 25
Kansas City 45 48 -3 9
Miami 41 39 2 4
Seattle 46 50 -4 16
11- 24
Example 4 continued
00.18
0.8
n
dd
1623.3
188
878
1
222
n
n
dd
sd
894.081623.3
00.1
ns
dt
d
11- 25
Example 4 continued
P(t>.894) = .20 for a one-tailed t-test at 7 degrees of freedom.
Because 0.894 is less than the critical value, the p-value of .20 > a of .05, do not reject the null hypothesis. There is no difference in the mean amount charged by Hertz and Avis.
11- 26
Advantage of dependent samples:Reduction in variation in the sampling distribution
Disadvantage of dependent samples:Degrees of freedom
are halved
Comparing dependent and independent samples
The same subjects
measured at two different points
in time.
Two types of dependent samplesTwo types of dependent samples
Matched or paired
observations