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Statistics. Chapter 9: Inferences Based on Two Samples: Confidence Intervals and Tests of Hypotheses. Where We’ve Been. Made inferences based on confidence intervals and tests of hypotheses Studied confidence intervals and hypothesis tests for µ , p and 2 - PowerPoint PPT PresentationTRANSCRIPT
Chapter 9: Inferences Based on Two Samples: Confidence Intervals and Tests of Hypotheses
Statistics
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
2
Where We’ve Been
Made inferences based on confidence intervals and tests of hypotheses
Studied confidence intervals and hypothesis tests for µ, p and 2
Selected the necessary sample size for a given margin of error
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
3
Where We’re Going
Learn to use confidence intervals and hypothesis tests to compare two populations
Learn how to use these tools to compare two population means, proportions and variances
Select the necessary sample size for a given margin of error when comparing parameters from two populations
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
4
9.1: Identifying the Target Parameter
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
5
9.2: Comparing Two Population Means: Independent Sampling
Point Estimators → µ 1 - 2 → µ1 - µ2To construct a confidence interval or conduct a hypothesis test, we need the standard deviation:
Singe sample nsx ̂
2
22
1
21
21ˆ
ns
ns
xx Two samples
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
6
9.2: Comparing Two Population Means: Independent Sampling
The Sampling Distribution for (1 - 2)1. The mean of the sampling distribution is (µ1-µ2).2. If the two samples are independent, the standard
deviation of the sampling distribution (the standard error) is
3. The sampling distribution for (1 - 2) is approximately normal for large samples.
2
22
1
21
21ˆ
ns
ns
xx
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
7
9.2: Comparing Two Population Means: Independent Sampling
The Sampling Distribution for (1 - 2)
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
8
9.2: Comparing Two Population Means: Independent Sampling
Large Sample Confidence Interval for (µ1 - µ2 )
2
22
1
21
2/21
2
22
1
21
2/21)(2/21
)(
)()(21
ns
nszxx
nnzxxzxx xx
9.2: Comparing Two Population Means: Independent Sampling
Private Colleges n: 71 Mean: 78.17 Standard Deviation: 9.55 Variance: 91.17
Public Universities n: 32 Mean: 84 Standard Deviation: 9.88 Variance: 97.64
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
9
Two samples concerning retention rates for first-year students at private and public institutions were obtained from the Department of Education’s data
base to see if there was a significant difference in the two types of colleges.
What does a 95% confidence interval tell us about retention rates?
Source: National Center for Education Statistics
9.2: Comparing Two Population Means: Independent Sampling
Private Colleges n: 71 Mean: 78.17 Standard Deviation: 9.55 Variance: 91.17
Public Universities n: 32 Mean: 84 Standard Deviation: 9.88 Variance: 97.64
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
10
08.483.532
64.9771
1.9196.18417.78
)()(2
22
1
21
2/21)(2/21 21
ns
nszxxzxx xx
9.2: Comparing Two Population Means: Independent Sampling
Private Colleges n: 71 Mean: 78.17 Standard Deviation: 9.55 Variance: 91.17
Public Universities n: 32 Mean: 84 Standard Deviation: 9.88 Variance: 97.64
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
11
08.483.532
64.9771
1.9196.18417.78
)()(2
22
1
21
2/21)(2/21 21
ns
nszxxzxx xx
Since 0 is not in the confidence interval, the difference in the sample means appears to indicate a real difference in retention.
9.2: Comparing Two Population Means: Independent Sampling
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
12
H0: (µ1 - µ2) = D0
Ha: (µ1 - µ2) > D0 (< D0)
Rejection region:z < -z (> z)
Test Statistic:
where
H0: (µ1 - µ2) = D0
Ha: (µ1 - µ2) ≠ D0
Rejection region:|z| > z/2
Two-Tailed TestOne-Tailed Test
)(
021
21
)(
xx
Dxxz
2
22
1
21
2
22
1
21
)( 21 ns
ns
nnxx
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
13
9.2: Comparing Two Population Means: Independent Sampling
Conditions Required for Valid Large-Sample Inferences about (µ1 - µ2)
1. The two samples are randomly and independently selected from the target populations.2. The sample sizes are both ≥ 30.
9.2: Comparing Two Population Means: Independent Sampling
Private Colleges n: 71 Mean: 78.17 Standard Deviation: 9.55 Variance: 91.17
Public Universities n: 32 Mean: 84 Standard Deviation: 9.88 Variance: 97.64
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
14
Let’s go back to the retention data and test the hypothesis that there is no significant difference in retention at privates and publics.
08.232
64.9771
17.91
2
22
1
21
)( 21 n
sns
xx
9.2: Comparing Two Population Means: Independent Sampling
799.208.283.50)(
05.0)(:0)(:
)(
21
21
210
21
xx
a
xxz
HH
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
15
Test statistic:
Reject the null hypothesis:
96.1799.2|| 2/
zz
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
16
9.2: Comparing Two Population Means: Independent Sampling
For small samples, the t-distribution can be used with a pooled sample estimator of 2, sp2
2)1()1(
21
222
2112
nn
snsnsp
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
17
9.2: Comparing Two Population Means: Independent Sampling
Small Sample Confidence Interval for (µ1 - µ2 )
21
22/21)(2/21
11)()(21 nn
stxxtxx pxx
The value of t is based on (n1 + n2 -2) degrees of freedom.
9.2: Comparing Two Population Means: Independent Sampling
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
18
H0: (µ1 - µ2) = D0
Ha: (µ1 - µ2) > D0 (< D0)
Rejection region:t < -t (> t)
Test Statistic:
H0: (µ1 - µ2) = D0*
Ha: (µ1 - µ2) ≠ D0
Rejection region:|t| > t/2
Two-Tailed TestOne-Tailed Test
21
2
021
11
)(
nns
Dxxt
p
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
19
9.2: Comparing Two Population Means: Independent Sampling
Conditions Required for Valid Small-Sample Inferences about (µ1 - µ2)
1. The two samples are randomly and independently selected from the target populations.
2. Both sampled populations have distributions that are approximately normal.
3. The population variances are equal.
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
20
Does class time affect performance? The test performance of students in two
sections of international trade, meeting at different times, were compared.
8:00 a.m. ClassMean: 78
Standard Deviation: 14Variance: 196
n: 21
9:30 a.m. ClassMean: 82
Standard Deviation: 17Variance: 289
n: 21
With = .05, test H0 : µ1 = µ2
9.2: Comparing Two Population Means: Independent Sampling
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
21
8:00 a.m. ClassMean: 78
Variance: 196n: 21
9:30 a.m. ClassMean: 82
Variance: 289n: 21
832.
211
2115.242
0)8278(
11
0)( :StatisticTest
5.24222121
289)121(196)121(2
)1()1(
05.0:0:
21
2
21
21
222
2112
21
210
nns
xxt
nnsnsns
HH
p
p
a
9.2: Comparing Two Population Means: Independent Sampling
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
22
8:00 a.m. ClassMean: 78
Variance: 196n: 21
9:30 a.m. ClassMean: 82
Variance: 289n: 21
832.
211
2115.242
0)8278(
11
0)( :StatisticTest
5.24222121
289)121(196)121(2
)1()1(
05.0:0:
21
2
21
21
222
2112
21
210
nns
xxt
nnsnsns
HH
p
p
a
With df = 18 + 24 – 2 = 40, t/2 = t.025 = 2.021. Since out test statistic t = -.812. |t| < t.025.
Do not reject the null hypothesis
9.2: Comparing Two Population Means: Independent Sampling
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
23
8:00 a.m. ClassMean: 72
Variance: 154n: 13
9:30 a.m. ClassMean: 86
Variance: 163n: 21
0 1 2
1 2
1 2
2 21 2
1 2
Several students in the 8:00 a.m. section sleep through the next exam. We can still compare the results, with some modifications.
: 0: 0.05
( ) 0Test Statistic:
a
HH
x xts sn n
(72 86) 0 3.16154 16313 21
9.2: Comparing Two Population Means: Independent Sampling
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
24
8:00 a.m. ClassMean: 72
Variance: 154n: 13
9:30 a.m. ClassMean: 86
Variance: 163n: 21
056.22615.26
12121/163
11313/154
21/16313/154
1/
1/
// Freedom of Degrees
16.3
26,025.
22
2
2
22
22
1
21
21
22
221
21
dft
nns
nns
nsnsv
t
9.2: Comparing Two Population Means: Independent Sampling
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
25
8:00 a.m. ClassMean: 72
Variance: 154n: 13
9:30 a.m. ClassMean: 86
Variance: 163n: 21
056.22615.26
12121/163
11313/154
21/16313/154
1/
1/
// Freedom of Degrees
16.3
26,025.
22
2
2
22
22
1
21
21
22
221
21
dft
nns
nns
nsnsv
t
Since |t| > t.025,df=26 ,, reject the null hypothesis.
9.2: Comparing Two Population Means: Independent Sampling
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
26
9.3: Comparing Two Population Means: Paired Difference Experiments
observed pairs ofnumber n wheresdifference ofdeviation standard sample s where
differencemean sample where
freedom, of degrees )1(with :Sample Small
:Sample Large
for Interval Confidence Difference Paired
2/
2/2/
21d
d
d
d
dd
dd
d
dd
d
dd
x
nnstx
nszx
nzx
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
27
9.3: Comparing Two Population Means: Paired Difference Experiments
Suppose ten pairs of puppies were housetrained using two different methods: one puppy from each pair was paper-trained, with the paper gradually moved outside, and the other was taken out every three hours and twenty minutes after each meal. The number of days until the puppies were considered housetrained (three days straight without an accident) were compared. Nine of the ten paper-trained dogs took longer than the other paired dog to complete training, with the average difference equal to 4 days, with a standard deviation of 3 days.
What is a 90% confidence interval on the difference in successful training?
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
28
9.3: Comparing Two Population Means: Paired Difference Experiments
74.14103833.14
confident 90%
34
1,2/21
d
dndfd
d
d
nstx
sx
d
Since 0 is not in the interval, one program does seem to work more effectively.
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
29
9.3: Comparing Two Population Means: Paired Difference Experiments
dd
dα, n
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Test Tailed-Two
ttzz
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McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
30
9.3: Comparing Two Population Means: Paired Difference Experiments
Suppose 150 items were priced at two online stores, “cport” and “warriorwoman.” Mean difference: $1.75 Standard Deviation: $10.35 Test at the 95% level that the difference in
the two stores is zero.
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
31
9.3: Comparing Two Population Means: Paired Difference Experiments
Suppose 150 items were priced at two online stores, “cport” and “warriorwoman.” Mean difference:
$1.75 Standard Deviation:
$10.35 = .05
07.2150/35.10075.1
/
0:0:
0
0
z
z
nDxz
HH
dd
d
da
d
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
32
9.3: Comparing Two Population Means: Paired Difference Experiments
Suppose 150 items were priced at two online stores, “cport” and “warriorwoman.” Mean difference:
$1.75 Standard Deviation:
$10.35 = .05
07.2150/35.10075.1
/
0:0:
0
0
z
z
nDxz
HH
dd
d
da
d
The critical value of z.05 is
1.96, so we would reject
this null hypothesis.
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
33
9.4: Comparing Two Population Proportions: Independent Sampling
.ˆ and ˆ examiningby and about inferences makecan We
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McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
34
9.4: Comparing Two Population Proportions: Independent Sampling
normal.ely approximat ison distributi sampling thelarge, are samples theIf .3
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McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
35
9.4: Comparing Two Population Proportions: Independent Sampling
2
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112/21
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1
112/21)ˆˆ(2/21
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21
nqp
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pp
pp
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
36
9.4: Comparing Two Population Proportions: Independent Sampling
A group of men and women were asked their opinions on the following important issue: Are the Three Stooges funny?
The results are as follow:Men Women
Yes 290 200
No 50 50
n 340 250
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
37
9.4: Comparing Two Population Proportions: Independent Sampling
Men Womenp .85 .80
q .15 .20
n 340 250
Calculate a 95% confidence interval on the difference in the opinions of men and women.
1 1 2 2/2
1 2
ˆ ˆ( )
.85 .15 .80 .20(.85 .80) 1.96340 250
.05 .062
M Wp q p qp p zn n
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
38
9.4: Comparing Two Population Proportions: Independent Sampling
Men Womenp .85 .80
q .15 .20
n 340 250
Calculate a 95% confidence interval on the difference in the opinions of men and women.
1 1 2 2/2
1 2
ˆ ˆ( )
.85 .15 .80 .20(.85 .80) 1.96340 250
.05 .062
M Wp q p qp p zn n
Since 0 is in the confidence interval, we cannot rule out
the possibility that both genders find the Stooges
equally funny. Nyuk nyuk nyuk.
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
39
9.4: Comparing Two Population Proportions: Independent Sampling
21about Hypothesis ofTest Sample-Large pp
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McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
40
9.4: Comparing Two Population Proportions: Independent Sampling Randy Stinchfield of the University of Minnesota studied the gambling
activities of public school students in 1992 and 1998 (Journal of Gambling Studies, Winter 2001). His results are reported below:
Do these results represent a statistically significant difference at the = .01 level?
1992 1998Survey n 21,484 23,199
Number who gambled 4.684 5,313
Proportion who gambled .218 .229
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
41
9.4: Comparing Two Population Proportions: Independent Sampling
1992 1998Survey n 21,484 23,199
Number who gambled
4.684 5,313
Proportion who gambled
.218 .229
0 1 2 0
1 2 0
/2
Two-Tailed Test: ( ): ( )
Rejection region: | | 2.576
a
H p p DH p p D
z z
1 2
1 2
1 2
1 2
ˆ ˆ( )1 2
1 2
ˆ ˆ( )
4,684 5,313ˆ .22421, 484 23,199
1 1ˆ ˆ
1 1(.224)(.776) .0039521, 484 23,199
ˆ ˆ( ) (.218 .229)Test Statistic: .00395
2.786
p p
p p
x xpn n
pqn n
p pz
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
42
9.4: Comparing Two Population Proportions: Independent Sampling
1992 1998Survey n 21,484 23,199
Number who gambled
4.684 5,313
Proportion who gambled
.218 .229
0 1 2 0
1 2 0
/2
Two-Tailed Test: ( ): ( )
Rejection region: | | 2.576
a
H p p DH p p D
z z
1 2
1 2
1 2
1 2
ˆ ˆ( )1 2
1 2
ˆ ˆ( )
4,684 5,313ˆ .22421, 484 23,199
1 1ˆ ˆ
1 1(.224)(.776) .0039521, 484 23,199
ˆ ˆ( ) (.218 .229)Test Statistic: .00395
2.786
p p
p p
x xpn n
pqn n
p pz
Since the computed value of z, -2.786, is of greater magnitude than the critical value, 2.576, we can reject the null hypothesis at the = .01 level.
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
43
9.4: Comparing Two Population Proportions: Independent Sampling
For valid inferences The two samples must be independent The two sample sizes must be large:
15ˆ and 15ˆ15ˆ and 15ˆ
2222
1111
qnpnqnpn
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
44
9.5: Determining the Sample Size
With a given level of confidence, and a specified sampling error, it is possible to calculate the required sample size Typically, n1 = n2
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
45
9.5: Determining the Sample Size
Sample size needed to estimate (µ1 - µ1) Given (1 - ) and the sampling error (SE)
required
Estimates of 1 and 2 will be needed
2
22
21
22/
21 SE)()(
znn
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
46
9.5: Determining the Sample Size
Suppose you need to estimate the difference between two population means to within 2.2 at the = 5% level. You have good reason to believe the two variances are equal to each other, and equal 15.
How large must n1 and n2 be?
248.232.2
)1515(96.1SE
)()(
21
2
2
21
2
22
21
22/
21
nn
nn
znn
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
47
9.5: Determining the Sample Size
SE = 2.2 = 5% level. 1
2 = 22 =15.
n1 and n2 = ?
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
48
9.5: Determining the Sample Size
Sample size needed to estimate (p1 - p2) Given (1 - ) and the sampling error (SE)
required
Estimates of p1 and p2 will be needed The most conservative values are p1 = p2 =.5
22211
22/
21 SE)()( qpqpznn
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
49
9.5: Determining the Sample Size
Suppose you need to calculate a 90% confidence interval of width .05, with no information about possible values of p1 and p2.
What size do n1 and n2 need to be?
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McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
50
9.5: Determining the Sample Size Suppose you
need to calculate a 90% confidence interval of width .05, with no information about possible values of p1 and p2.
What size do n1 and n2 need to be?
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
51
9.6: Comparing Two Population Variances: Independent Sampling
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
52
9.6: Comparing Two Population Variances: Independent Sampling
Since variances do not follow a normal distribution (due to the squaring involved) we use a different technique for inferences about variances.
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
53
9.6: Comparing Two Population Variances: Independent Sampling
2 22 2 2 2 1 1
1 1 2 2 2 22 2
22 2 1
1 2 22
1
2
If ( ) and ( ) , then .
If , the random variable .
The random variable is a ratio with 1 numeratorand 1 denominator degrees of freed
sE s E s Es
s Fs
F nn
om.
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
54
9.6: Comparing Two Population Variances: Independent Sampling
If the ratio of variances gets too far from 1 in value, in either direction the likelihood that both samples share a population variance drops. For convenience the larger sample variance is usually put in the numerator.
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
55
9.6: Comparing Two Population Variances: Independent Sampling
2 21 2
2 2 21 1 12 2 22 , /2 2 2 , /2
, /2
, /2
A (1 ) 100% Confidence Interval for /
1 1
where leaves ( /2)% of the distribution in a lower tail
and leaves ( /2)% o
U L
L
U
s ss F s F
F
F
1
2
f the distribution in an upper tail.
Degrees of freedom are 1 in the numerator and 1 in the denominator.
nn
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
56
9.6: Comparing Two Population Variances: Independent Sampling
Home Runs, 2003
American League
National League
µ 178.5 169.25
2 1415.8 951.4
n 14 16
Has the designated hitter produced a significant difference in homeruns in the two major leagues?
To use the two-sample t-test, we should check the variance ratio to see if the assumption of equal variances is acceptable.
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
57
9.6: Comparing Two Population Variances: Independent Sampling
2 2 21 1 12 2 22 , /2 2 2 , /2
212
,.025 2 ,.025
2122
1 1
1415.8 1 1415.8 1951.4 951.4
1415.8 1 1415.8951.4 2.92 951.
U L
U L
s ss F s F
F F
2122
14 .32
.5096 4.6504
Home Runs, 2003
American League
National League
µ 178.5 169.25
2 1415.8 951.4
n 14 16
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
58
9.6: Comparing Two Population Variances: Independent Sampling
2 2 21 1 12 2 22 , /2 2 2 , /2
212
,.025 2 ,.025
2122
1 1
1415.8 1 1415.8 1951.4 951.4
1415.8 1 1415.8951.4 2.92 951.
U L
U L
s ss F s F
F F
2122
14 .32
.5096 4.6504
Home Runs, 2003
American League
National League
µ 178.5 169.25
2 1415.8 951.4
n 14 16
Since 1 is in the confidence interval, the assumption of equal variances needed for the t-test is not rejected.
-Test for Equal Population VariancesF
2 20 1 2
2 21 2
22 221 22
1
22 211 22
2
One-Tailed Test
:
: (or )
Test Statistic: if :
or if :
Rejection Region:
a
a
a
H
H
sF Hs
sF HsF F
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
59
9.6: Comparing Two Population Variances: Independent Sampling
2 20 1 2
2 21 2
/2
Two-Tailed Test
:
:Test Statistic:
Larger sample varianceSmaller sample variance
Rejection Region:
a
H
H
F
F F
9.6: Comparing Two Population Variances: Independent Sampling
In 2006, ten fast-growing economies had an average GDP growth rate of 8.69%, with a standard deviation of 1.7. Ten slow-growing economies had an average GDP growth rate of 2.29%, with a standard deviation of .58.* Do slow- and fast-growing economies have similar variability (=5%)?
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
60
* Source: euroekonom.com
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
61
9.6: Comparing Two Population Variances: Independent Sampling
2 20 1 2
2 21 2
.025
Two-Tailed Test
:
:Test Statistic:
Larger sample variance 2.89 8.5Smaller sample variance .34
Rejection Region: 4.03
a
H
H
F
F F
µfast-growing = 8.69%µslow-growing = 2.29%2
fast-growing = 2.892
slow-growing = .34
McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples
62
9.6: Comparing Two Population Variances: Independent Sampling
2 20 1 2
2 21 2
.025
Two-Tailed Test
:
:Test Statistic:
Larger sample variance 2.89 8.5Smaller sample variance .34
Rejection Region: 4.03
a
H
H
F
F F
µfast-growing = 8.69%µslow-growing = 2.29%2
fast-growing = 2.892
slow-growing = .34
Reject the null hypothesis at the =.05 level.