comparing sample means
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Comparing Sample Means. Confidence Intervals. Difference Between Two Means. The point estimate for the difference is. Population means, independent samples. *. x 1 – x 2. σ 1 and σ 2 known. σ 1 and σ 2 unknown, n 1 and n 2 30. σ 1 and σ 2 unknown, n 1 or n 2 < 30. - PowerPoint PPT PresentationTRANSCRIPT
Comparing Sample Means
Confidence Intervals
AP Statistics Chap 13-2
Difference Between Two Means
Population means, independent samples
σ1 and σ2 known
σ1 and σ2 unknown, n1 and n2 30
σ1 and σ2 unknown, n1 or n2 < 30
The point estimate for the difference is
x1 – x2*
AP Statistics Chap 13-3
Independent Samples
Population means, independent samples
σ1 and σ2 known
σ1 and σ2 unknown, n1 and n2 30
σ1 and σ2 unknown, n1 or n2 < 30
Different data sources Unrelated Independent
Sample selected from one population has no effect on the sample selected from the other population
Use the difference between 2 sample means
Use z test or t test
*
AP Statistics Chap 13-4
Population means, independent samples
σ1 and σ2 known
σ1 and σ2 unknown, n1 and n2 30
σ1 and σ2 unknown, n1 or n2 < 30
σ1 and σ2 known
Assumptions: Samples are randomly and independently drawn
population distributions are normal or both sample sizes are 30
Population standard deviations are known
*
AP Statistics Chap 13-5
Population means, independent samples
σ1 and σ2 known
σ1 and σ2 unknown, n1 and n2 30
σ1 and σ2 unknown, n1 or n2 < 30
…and the standard error of x1 – x2 is
When σ1 and σ2 are known and both populations are normal or both sample sizes are at least 30, the test statistic is a z-value…
2
22
1
21
xx nσ
nσσ
21
σ1 and σ2 known
AP Statistics Chap 13-6
Population means, independent samples
σ1 and σ2 known
σ1 and σ2 unknown, n1 and n2 30
σ1 and σ2 unknown, n1 or n2 < 30
2
22
1
21
/221nσ
nσzxx
The confidence interval for μ1 – μ2 is:
σ1 and σ2 known
AP Statistics Chap 13-7
Population means, independent samples
σ1 and σ2 known
σ1 and σ2 unknown, n1 and n2 30
σ1 and σ2 unknown, n1 or n2 < 30
σ1 and σ2 unknown, large samples
Assumptions: Samples are randomly and independently drawn
both sample sizes are 30
Population standard deviations are unknown
AP Statistics Chap 13-8
Population means, independent samples
σ1 and σ2 known
σ1 and σ2 unknown, n1 and n2 30
σ1 and σ2 unknown, n1 or n2 < 30
σ1 and σ2 unknown, large samples
Forming interval estimates:
use sample standard deviation s to estimate σ
the test statistic is a z value
AP Statistics Chap 13-9
Population means, independent samples
σ1 and σ2 known
σ1 and σ2 unknown, n1 and n2 30
σ1 and σ2 unknown, n1 or n2 < 30
2
22
1
21
/221ns
nszxx
The confidence interval for μ1 – μ2 is:
σ1 and σ2 unknown, large samples
*
AP Statistics Chap 13-10
Population means, independent samples
σ1 and σ2 known
σ1 and σ2 unknown, n1 and n2 30
σ1 and σ2 unknown, n1 or n2 < 30
σ1 and σ2 unknown, small samples
The confidence interval for μ1 – μ2 is:
2 2
1 21 2 /2
1 2
s sx xn n
t
AP Statistics Chap 13-11
ExampleYou’re a financial analyst for a brokerage firm. Is there a difference in dividend yield between stocks listed on the NYSE & NASDAQ? You collect the following data: NYSE NASDAQNumber 21 25Sample mean 3.27 2.53Sample std dev 1.30 1.16
Form a 95% confidence intervalfor the true mean.
Example
2 2
1 21 2 /2
1 2
s sx xn n
t
2 21.3 1.163.27 2.53 2.07921 25
.02,1.5
Example: TI-84
Comparing Sample Means
Hypothesis Tests
AP Statistics Chap 13-15
Hypothesis Tests forTwo Population Proportions
Lower tail test:
H0: μ1 μ2
HA: μ1 < μ2
i.e.,
H0: μ1 – μ2 0HA: μ1 – μ2 < 0
Upper tail test:
H0: μ1 ≤ μ2
HA: μ1 > μ2
i.e.,
H0: μ1 – μ2 ≤ 0HA: μ1 – μ2 > 0
Two-tailed test:
H0: μ1 = μ2
HA: μ1 ≠ μ2
i.e.,
H0: μ1 – μ2 = 0HA: μ1 – μ2 ≠ 0
Two Population Means, Independent Samples
AP Statistics Chap 13-16
Hypothesis tests for μ1 – μ2
Population means, independent samples
σ1 and σ2 known
σ1 and σ2 unknown, n1 and n2 30
σ1 and σ2 unknown, n1 or n2 < 30
Use a z test statistic
Use s to estimate unknown σ , approximate with a z test statistic
Use s to estimate unknown σ , use a t test statistic and pooled standard deviation
AP Statistics Chap 13-17
Population means, independent samples
σ1 and σ2 known
σ1 and σ2 unknown, n1 and n2 30
σ1 and σ2 unknown, n1 or n2 < 30
2
22
1
21
2121
nσ
nσ
μμxxz
The test statistic for μ1 – μ2 is:
σ1 and σ2 known
AP Statistics Chap 13-18
Population means, independent samples
σ1 and σ2 known
σ1 and σ2 unknown, n1 and n2 30
σ1 and σ2 unknown, n1 or n2 < 30
σ1 and σ2 unknown, large samples
The test statistic for μ1 – μ2 is:
2
22
1
21
2121
ns
ns
μμxxz
AP Statistics Chap 13-19
Population means, independent samples
σ1 and σ2 known
σ1 and σ2 unknown, n1 and n2 30
σ1 and σ2 unknown, n1 or n2 < 30
σ1 and σ2 unknown, small samples
1 2 1 2
2 21 2
1 2
x x μ μt
s sn n
The test statistic for μ1 – μ2 is:
AP Statistics Chap 13-20
Two Population Means, Independent Samples
Lower tail test:
H0: μ1 – μ2 0HA: μ1 – μ2 < 0
Upper tail test:
H0: μ1 – μ2 ≤ 0HA: μ1 – μ2 > 0
Two-tailed test:
H0: μ1 – μ2 = 0HA: μ1 – μ2 ≠ 0
/2 /2
-z -z/2z z/2
Reject H0 if z < -z Reject H0 if z > z Reject H0 if z < -z/2
or z > z/2
Hypothesis tests for μ1 – μ2
AP Statistics Chap 13-21
ExampleYou’re a financial analyst for a brokerage firm. Is there a difference in dividend yield between stocks listed on the NYSE & NASDAQ? You collect the following data: NYSE NASDAQNumber 21 25Sample mean 3.27 2.53Sample std dev 1.30 1.16
Test the hypothesis that there is a difference in dividend yield.
Example 1 2 1 2
2 21 2
1 2
x x μ μt
s sn n
2 2
3.27 2.53 0t 2.019
1.3 1.1621 25
P-value = .0285(2) = .057
Example: TI-84
Comparing Sample Means
Matched Pair Comparisons
AP Statistics Chap 13-25
Paired Samples Tests Means of 2 Related Populations
Paired or matched samples Repeated measures (before/after) Use difference between paired values:
Eliminates Variation Among Subjects Assumptions:
Both Populations Are Normally Distributed Or, if Not Normal, use large samples
Paired samples
d = x1 - x2
AP Statistics Chap 13-26
Paired DifferencesThe ith paired difference is di , where
Paired samples
di = x1i - x2i
The point estimate for the population mean paired difference is d :
n2
ii 1
d
(d d)s
n
n
dd
n
1ii
The sample standard deviation is
n is the number of pairs in the paired sample
AP Statistics Chap 13-27
Paired Differences
The confidence interval for d isPaired samples
n2
ii 1
d
(d d)s
n
nstd d
/2
Where t/2 has
n - 1 d.f. and sd is:
(continued)
n is the number of pairs in the paired sample
AP Statistics Chap 13-28
The test statistic for d isPaired samples
n2
ii 1
d
(d d)s
n
ns
μdtd
d
Where t/2 has n - 1 d.f.
and sd is:
n is the number of pairs in the paired sample
Hypothesis Testing for Paired Samples
AP Statistics Chap 13-29
Lower tail test:
H0: μd 0HA: μd < 0
Upper tail test:
H0: μd ≤ 0HA: μd > 0
Two-tailed test:
H0: μd = 0HA: μd ≠ 0
Paired Samples
Hypothesis Testing for Paired Samples
/2 /2
-t -t/2t t/2
Reject H0 if t < -t Reject H0 if t > t Reject H0 if t < -t/2 or t > t/2 Where t has n - 1 d.f.
(continued)
AP Statistics Chap 13-30
Assume you send your salespeople to a “customer service” training workshop. Is the training effective? You collect the following data:
Paired Samples Example
Number of Complaints: (2) - (1)Salesperson Before (1) After (2) Difference, di
C.B. 6 4 - 2 T.F. 20 6 -14 M.H. 3 2 - 1 R.K. 0 0 0 M.O. 4 0 - 4 -21
d = di
n
2i
d
(d d)s
n5.67
= -4.2
AP Statistics Chap 13-31
Has the training made a difference in the number of complaints (at the 0.01 level)?
- 4.2d =
1.6655.67/04.2
n/sμdt
d
d
H0: μd = 0HA: μd 0
Test Statistic:
Critical Value = ± 4.604 d.f. = n - 1 = 4
Reject
/2 - 4.604 4.604
Decision: Do not reject H0
(t stat is not in the reject region)
Conclusion: There is not a significant change in the number of complaints.
Paired Samples: Solution
Reject
/2
- 1.66 = .01