1. Chapter 11 Hypothesis Testing II
Statistics for
Business and Economics 6th Edition
Chapter 11
Hypothesis Testing II

2. Chapter Goals Setelah menyelesaikan bab ini, anda akan mampu :
Melakukan uji hipotesis untuk perbedaan dua rata-rata populasi
Data berpasangan
Populasi Independen, variansi populasi diketahui
Populasi Independen, variansi populasi tak diketahui
Melakukan uji hipotesis dua proporsi (sampel besar )
Use the F table to find critical F values
Complete an F test for the equality of two variances

3. Two Sample Tests Two Sample Tests Population Means, Matched Pairs
Population Means, Independent Samples
Population Proportions
Population Variances
Examples:
Same group before vs. after treatment
Group 1 vs. independent Group 2
Proportion 1 vs. Proportion 2
Variance 1 vs.
Variance 2
(Note similarities to Chapter 9)

4. Matched Pairs Tests Means of 2 Related Populations di = xi - yi
Paired or matched samples
Repeated measures (before/after)
Use difference between paired values:
Compute the average and standard dev of di
Assumptions:
Both Populations Are Normally Distributed
di = xi - yi

5. Test Statistic: Matched Pairs
The test statistic for the mean difference is a t value, with n – 1 degrees of freedom:
Where
D0 = hypothesized mean difference
sd = sample standard dev. of differences
n = the sample size (number of pairs)

6. Decision Rules: Matched Pairs
Paired Samples
Lower-tail test:
H0: μx – μy  0
H1: μx – μy < 0
Upper-tail test:
H0: μx – μy ≤ 0
H1: μx – μy > 0
Two-tail test:
H0: μx – μy = 0
H1: μx – μy ≠ 0 a a
a/2
a/2
-ta ta
-ta/2
ta/2
Reject H0 if t < -tn-1, a
Reject H0 if t > tn-1, a
Reject H0 if t < -tn-1 , a/2
or t > tn-1 , a/2
has n - 1 d.f.
Where

7. Matched Pairs Example  d = = - 4.2
Assume you send your salespeople to a “customer service” training workshop. Has the training made a difference in the number of complaints? You collect the following data:  di
d =
Complaints:
S.person Before (1) After (2) di
-21 n
= - 4.2

8. Critical Value = ± 4.604 d.f. = n - 1 = 4
Matched Pairs: Solution
Has the training made a difference
in the number of complaints?
Reject
Reject
H0: μx – μy = 0
H1: μx – μy  0
/2
/2
 = .01
d =
- 4.2
- 1.66
Critical Value = ± d.f. = n - 1 = 4
Decision: Do not reject H0
(t stat is not in the reject region)
Test Statistic:
Conclusion: There is not a significant change in the number of complaints.

9. Difference Between Two Means
Tujuan: Menguji perbedaan dua rata-rata populasi independent, μx – μy
Sumber data berbeda, pop 1 dan pop 2
Tidak berhubungan atau berpasangan
Independen
Sampel yang diambil dari satu populasi tidak berpengaruh terhadap sampel dari populasi lain

10. Difference Between Two Means
Population means, independent samples
σx2 and σy2 known
Test statistic is a z value
σx2 and σy2 unknown
σx2 and σy2 assumed equal
Test statistic is a a value from the Student’s t distribution
σx2 and σy2 assumed unequal

11. Population means, independent samples
σx2 and σy2 Known
Population means, independent samples
Assumptions:
Samples are randomly and
independently drawn
both population distributions
are normal
Population variances are
known *
σx2 and σy2 known
σx2 and σy2 unknown

12. Population means, independent samples
σx2 and σy2 Known
When σx2 and σy2 are known and both populations are normal, the variance of X – Y is
Population means, independent samples *
σx2 and σy2 known
…and the statistic
has a standard normal distribution
σx2 and σy2 unknown

13. Hypothesis Tests for Two Population Means
Two Population Means, Independent Samples
Lower-tail test:
H0: μx  μy
H1: μx < μy
i.e.,
H0: μx – μy  0
H1: μx – μy < 0
Upper-tail test:
H0: μx ≤ μy
H1: μx > μy
i.e.,
H0: μx – μy ≤ 0
H1: μx – μy > 0
Two-tail test:
H0: μx = μy
H1: μx ≠ μy
i.e.,
H0: μx – μy = 0
H1: μx – μy ≠ 0

14. Two Population Means, Independent Samples, Variances Known
Decision Rules
Two Population Means, Independent Samples, Variances Known
Lower-tail test:
H0: μx – μy  0
H1: μx – μy < 0
Upper-tail test:
H0: μx – μy ≤ 0
H1: μx – μy > 0
Two-tail test:
H0: μx – μy = 0
H1: μx – μy ≠ 0 a a
a/2
a/2
-za za
-za/2
za/2
Reject H0 if z < -za
Reject H0 if z > za
Reject H0 if z < -za/2
or z > za/2

15. σx2 and σy2 Unknown, Assumed Equal
Assumptions:
Samples are randomly and
independently drawn
Populations are normally
distributed
Population variances are
unknown but assumed equal
Population means, independent samples
σx2 and σy2 known
σx2 and σy2 unknown *
σx2 and σy2 assumed equal
σx2 and σy2 assumed unequal

16. σx2 and σy2 Unknown, Assumed Equal
Test Hypotheses:
The population variances
are assumed equal, so use
the two sample standard
deviations and pool them to
estimate σ
use a t value with
(nx + ny – 2) degrees of
freedom
Population means, independent samples
σx2 and σy2 known
σx2 and σy2 unknown *
σx2 and σy2 assumed equal
σx2 and σy2 assumed unequal

17. Test Statistic, σx2 and σy2 Unknown, Equal
The test statistic for
μx – μy is:
σx2 and σy2 unknown *
σx2 and σy2 assumed equal
σx2 and σy2 assumed unequal
Where t has (n1 + n2 – 2) d.f.,
and

18. σx2 and σy2 Unknown, Assumed Unequal
Assumptions:
Samples are randomly and
independently drawn
Populations are normally
distributed
Population variances are
unknown and assumed
unequal
Population means, independent samples
σx2 and σy2 known
σx2 and σy2 unknown
σx2 and σy2 assumed equal *
σx2 and σy2 assumed unequal

19. σx2 and σy2 Unknown, Assumed Unequal
Forming interval estimates:
The population variances are
assumed unequal, so a pooled
variance is not appropriate
use a t value with  degrees
of freedom, where
Population means, independent samples
σx2 and σy2 known
σx2 and σy2 unknown
σx2 and σy2 assumed equal *
σx2 and σy2 assumed unequal

20. Test Statistic, σx2 and σy2 Unknown, Unequal
The test statistic for
μx – μy is:
σx2 and σy2 unknown
σx2 and σy2 assumed equal *
σx2 and σy2 assumed unequal
Where t has  degrees of freedom:

21. Two Population Means, Independent Samples, Variances Unknown
Decision Rules
Two Population Means, Independent Samples, Variances Unknown
Lower-tail test:
H0: μx – μy  0
H1: μx – μy < 0
Upper-tail test:
H0: μx – μy ≤ 0
H1: μx – μy > 0
Two-tail test:
H0: μx – μy = 0
H1: μx – μy ≠ 0 a a
a/2
a/2
-ta ta
-ta/2
ta/2
Reject H0 if t < -tn-1, a
Reject H0 if t > tn-1, a
Reject H0 if t < -tn-1 , a/2
or t > tn-1 , a/2
Where t has n - 1 d.f.

22. Pooled Variance t Test: Example
You are 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 NASDAQ Number
Sample mean
Sample std dev
Assuming both populations are
approximately normal with equal variances, is there a difference in average yield ( = 0.05)?

23. Calculating the Test Statistic
The test statistic is:

24. Solution t Decision: Reject H0 at a = 0.05 Conclusion:
H0: μ1 - μ2 = 0 i.e. (μ1 = μ2)
H1: μ1 - μ2 ≠ 0 i.e. (μ1 ≠ μ2)
 = 0.05
df = = 44
Critical Values: t = ±
Test Statistic:
.025
.025
2.0154 t
2.040
Decision:
Conclusion:
Reject H0 at a = 0.05
There is evidence of a difference in means.

25. Two Population Proportions
Goal: Test hypotheses for the difference between two population proportions, Px – Py
Assumptions:
Both sample sizes are large,
nP(1 – P) > 9
Population proportions

26. Two Population Proportions
The random variable
is approximately normally distributed
Population proportions

27. Test Statistic for Two Population Proportions
The test statistic for
H0: Px – Py = 0
is a z value:
Population proportions
Where
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

28. Decision Rules: Proportions
Population proportions
Lower-tail test:
H0: px – py  0
H1: px – py < 0
Upper-tail test:
H0: px – py ≤ 0
H1: px – py > 0
Two-tail test:
H0: px – py = 0
H1: px – py ≠ 0 a a
a/2
a/2
-za za
-za/2
za/2
Reject H0 if z < -za
Reject H0 if z > za
Reject H0 if z < -za/2
or z > za/2

29. Example: Two Population Proportions
Is there a significant difference between the proportion of men and the proportion of women who will vote Yes on Proposition A?
In a random sample, 36 of 72 men and 31 of 50 women indicated they would vote Yes
Test at the .05 level of significance

30. Example: Two Population Proportions
(continued)
The hypothesis test is:
H0: PM – PW = 0 (the two proportions are equal)
H1: PM – PW ≠ 0 (there is a significant difference between
proportions)
The sample proportions are:
Men: = 36/72 = .50
Women: = 31/50 = .62
The estimate for the common overall proportion is:
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

31. Example: Two Population Proportions
Reject H0
Reject H0
The test statistic for PM – PW = 0 is:
.025
.025
-1.96
1.96
-1.31
Decision: Do not reject H0
Conclusion: There is not significant evidence of a difference between men and women in proportions who will vote yes.
Critical Values = ±1.96
For  = .05

32. Hypothesis Tests for Two Variances
Population
Variances
Goal: Test hypotheses about two
population variances
H0: σx2  σy2
H1: σx2 < σy2
Lower-tail test
F test statistic
H0: σx2 ≤ σy2
H1: σx2 > σy2
Upper-tail test
H0: σx2 = σy2
H1: σx2 ≠ σy2
Two-tail test
The two populations are assumed to be independent and normally distributed

33. Hypothesis Tests for Two Variances
The random variable
Tests for Two
Population
Variances
F test statistic
Has an F distribution with (nx – 1) numerator degrees of freedom and (ny – 1) denominator degrees of freedom
Denote an F value with 1 numerator and 2 denominator degrees of freedom by

34. Test Statistic Tests for Two
Population
Variances
The critical value for a hypothesis test about two population variances is
F test statistic
where F has (nx – 1) numerator degrees of freedom and (ny – 1) denominator degrees of freedom

35. Decision Rules: Two Variances
Use sx2 to denote the larger variance.
H0: σx2 = σy2
H1: σx2 ≠ σy2
H0: σx2 ≤ σy2
H1: σx2 > σy2
/2  F F
Do not
reject H0
Reject H0
Do not
reject H0
Reject H0
rejection region for a two-tail test is:
where sx2 is the larger of the two sample variances

36. Example: F Test
You are a financial analyst for a brokerage firm. You want to compare dividend yields between stocks listed on the NYSE & NASDAQ. You collect the following data: NYSE NASDAQ Number Mean Std dev Is there a difference in the variances between the NYSE & NASDAQ at the  = 0.10 level?

37. F Test: Example Solution
Form the hypothesis test:
H0: σx2 = σy2 (there is no difference between variances)
H1: σx2 ≠ σy2 (there is a difference between variances)
Find the F critical values for  = .10/2:
Degrees of Freedom:
Numerator
(NYSE has the larger standard deviation):
nx – 1 = 21 – 1 = 20 d.f.
Denominator:
ny – 1 = 25 – 1 = 24 d.f.

38. F Test: Example Solution
The test statistic is:
H0: σx2 = σy2
H1: σx2 ≠ σy2
/2 = .05 F
Do not
reject H0
Reject H0
F = is not in the rejection region, so we do not reject H0
Conclusion: There is not sufficient evidence of a difference in variances at  = .10

39. Two-Sample Tests in EXCEL
For paired samples (t test):
Tools | data analysis… | t-test: paired two sample for means
For independent samples:
Independent sample Z test with variances known:
Tools | data analysis | z-test: two sample for means
For variances…
F test for two variances:
Tools | data analysis | F-test: two sample for variances

40. Two-Sample Tests in PHStat

41. Sample PHStat Output
Input
Output

42. Sample PHStat Output
Input
Output