1. Chapter 10 Two Sample Tests
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2. Chapter Goals After completing this chapter, you should be able to:
Test hypotheses or form interval estimates for
two independent population means
Standard deviations known
Standard deviations unknown
two means from paired samples
Use the F table to find critical F values
Yandell – Econ 216

3. Two-Sample Tests Two-Sample Tests
Population Means, Independent Samples
Population Means, Related Samples
Population Proportions
Population Variances
Examples:
Group 1 vs. Group 2
Same group before vs. after treatment
Proportion 1 vs. Proportion 2
Variance 1 vs.
Variance 2
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4. Difference Between Two Means
Population means, independent samples
Goal: Form a confidence interval for the difference between two population means, μ1 – μ2 *
σ1 and σ2 known
The point estimate for the difference is
σ1 and σ2 unknown,
assumed equal
X1 – X2
σ1 and σ2 unknown,
assumed unequal
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5. Population means, independent samples
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, pooled variance t test, or separate-variance t test
Population means, independent samples *
σ1 and σ2 known
σ1 and σ2 unknown,
assumed equal
σ1 and σ2 unknown,
assumed unequal
Yandell – Econ 216

6. Hypothesis tests for μ1 – μ2
Population means, independent samples
Use a Z test statistic
σ1 and σ2 known
Use S1 and S2 to estimate a pooled estimate for σ , use a t test statistic
σ1 and σ2 unknown,
assumed equal
σ1 and σ2 unknown,
assumed unequal
Use S1 and S2 with a separate-variance t test
Yandell – Econ 216

7. Population means, independent samples
σ1 and σ2 known
Population means, independent samples
Assumptions:
Samples are randomly and
independently drawn
population distributions are
normal or both sample sizes
are  30
Population standard
deviations are known *
σ1 and σ2 known
σ1 and σ2 unknown,
assumed equal
σ1 and σ2 unknown,
assumed unequal
Yandell – Econ 216

8. σ1 and σ2 known
(continued)
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…
Population means, independent samples *
σ1 and σ2 known
…and the standard error of
x1 – x2 is
σ1 and σ2 unknown,
assumed equal
σ1 and σ2 unknown,
assumed unequal
Yandell – Econ 216

9. Population means, independent samples
σ1 and σ2 known
Population means, independent samples
The test statistic for
μ1 – μ2 is: *
σ1 and σ2 known
σ1 and σ2 unknown,
assumed equal
σ1 and σ2 unknown,
assumed unequal
Yandell – Econ 216

10. σ1 and σ2 unknown, large samples
Assumptions:
Samples are randomly and
independently drawn
both sample sizes are  30, or the populations are both normally distributed if not
Population standard
deviations are unknown
Population means, independent samples
σ1 and σ2 known *
σ1 and σ2 unknown,
assumed equal
σ1 and σ2 unknown,
assumed unequal
Yandell – Econ 216

11. σ1 and σ2 unknown, large samples
(continued)
Population means, independent samples
Process:
The population variances are assumed equal, so use the two sample standard deviations S1 and S2 and pool them to estimate σ
the test statistic is a t value
with (n1 + n2 – 2) degrees
of freedom
σ1 and σ2 known *
σ1 and σ2 unknown,
assumed equal
σ1 and σ2 unknown,
assumed unequal
Yandell – Econ 216

12. σ1 and σ2 unknown, large samples
(continued)
Population means, independent samples
The pooled standard deviation is
σ1 and σ2 known *
σ1 and σ2 unknown,
assumed equal
σ1 and σ2 unknown,
assumed unequal
Yandell – Econ 216

13. σ1 and σ2 unknown, large samples
(continued)
The test statistic for μ1 – μ2 is:
Population means, independent samples
σ1 and σ2 known *
σ1 and σ2 unknown,
assumed equal
Where t/2 has (n1 + n2 – 2) d.f.,
and
σ1 and σ2 unknown,
assumed unequal
Yandell – Econ 216

14. σ1 and σ2 unknown, small samples
Population means, independent samples
Assumptions:
populations are normally
distributed
the populations do not have equal variances
samples are independent
σ1 and σ2 known
σ1 and σ2 unknown,
assumed equal *
σ1 and σ2 unknown,
assumed unequal
Yandell – Econ 216

15. σ1 and σ2 unknown, small samples
(continued)
Process:
The population variances are assumed unequal, so a pooled estimate is not appropriate
use a separate-variance t test, degrees of freedom must be calculated:
Population means, independent samples
σ1 and σ2 known
σ1 and σ2 unknown,
assumed equal *
σ1 and σ2 unknown,
assumed unequal
Yandell – Econ 216

16. σ1 and σ2 unknown, small samples
The test statistic for
H0: μ1 – μ2 = 0 is:
(with d.f. calculated using the formula on the previous slide)
Population means, independent samples
σ1 and σ2 known
σ1 and σ2 unknown,
assumed equal *
σ1 and σ2 unknown,
assumed unequal
Yandell – Econ 216

17. Hypothesis Tests for Two Population Means
Two Population Means, Independent Samples
Lower tail test:
H0: μ1  μ2
H1: μ1 < μ2
i.e.,
H0: μ1 – μ2  0
H1: μ1 – μ2 < 0
Upper tail test:
H0: μ1 ≤ μ2
H1: μ1 > μ2
i.e.,
H0: μ1 – μ2 ≤ 0
H1: μ1 – μ2 > 0
Two-tailed test:
H0: μ1 = μ2
H1: μ1 ≠ μ2
i.e.,
H0: μ1 – μ2 = 0
H1: μ1 – μ2 ≠ 0
Yandell – Econ 216

18. Hypothesis tests for μ1 – μ2
Two Population Means, Independent Samples
Lower tail test:
H0: μ1 – μ2  0
H1: μ1 – μ2 < 0
Upper tail test:
H0: μ1 – μ2 ≤ 0
H1: μ1 – μ2 > 0
Two-tailed test:
H0: μ1 – μ2 = 0
H1: μ1 – μ2 ≠ 0 a a
a/2
a/2
-ta ta
-ta/2
ta/2
Reject H0 if t < -ta
Reject H0 if t > ta
Reject H0 if t < -ta/2
or t > ta/2
Yandell – Econ 216

19. Pooled sp t Test: Example
You’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 NASDAQ Number
Sample mean
Sample std dev
Assuming equal variances, is there a difference in average yield ( = 0.05)?
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20. Calculating the Test Statistic
The test statistic is:
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21. 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.
Yandell – Econ 216

22. Related (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
Related samples
D = X1 - X2
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23. Related (Paired) Differences
The ith paired difference is Di , where
Related samples
Di = X1i - X2i
The point estimate for the population mean paired difference is D :
The sample standard deviation is
n is the number of pairs in the paired sample
Yandell – Econ 216

24. Hypothesis Testing for Paired Samples
The test statistic for D is
Related samples
n is the number of pairs in the paired sample
Where t/2 has n - 1 d.f.
and SD is:
Yandell – Econ 216

25. Hypothesis Testing for Paired Samples
(continued)
Related Samples
Lower tail test:
H0: μD  0
H1: μD < 0
Upper tail test:
H0: μD ≤ 0
H1: μD > 0
Two-tailed test:
H0: μD = 0
H1: μD ≠ 0 a a
a/2
a/2
-ta ta
-ta/2
ta/2
Reject H0 if t < -ta
Reject H0 if t > ta
Reject H0 if t < -ta/2
or t > ta/2
Where t has n - 1 d.f.
Yandell – Econ 216

26. Paired Samples Example
Assume you send your salespeople to a “customer service” training workshop. Is the training effective? You collect the following data: 
Number of Complaints: (2) - (1)
Salesperson Before (1) After (2) Difference, Di
C.B
T.F
M.H
R.K
M.O
-21 Di
D = n
= -4.2
Yandell – Econ 216

27. Critical Value = ± 4.604 d.f. = n - 1 = 4
Paired Samples: Solution
Has the training made a difference in the number of complaints (at the 0.01 level)?
Reject
Reject
H0: μD = 0
H1: μD  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.
Yandell – Econ 216

28. Two Population Proportions
Goal: test a hypothesis or form a confidence interval for the difference between two population proportions, π1 – π2
Population proportions
Assumptions:
n1 π1  5 , n1(1- π1)  5
n2 π2  5 , n2(1- π2)  5
The point estimate for the difference is
Yandell – Econ 216

29. Two Population Proportions
In the null hypothesis we assume π1 = π2 and pool the two sample estimates
Population proportions
The pooled estimate for the overall proportion is:
where X1 and X2 are the number of items of interest in samples 1 and 2
Yandell – Econ 216

30. Two Population Proportions
(continued)
The test statistic for
π1 – π2 is a Z statistic:
Population proportions
where
Yandell – Econ 216

31. Hypothesis Tests for Two Population Proportions
Lower-tail test:
H0: π1  π2
H1: π1 < π2
i.e.,
H0: π1 – π2  0
H1: π1 – π2 < 0
Upper-tail test:
H0: π1 ≤ π2
H1: π1 > π2
i.e.,
H0: π1 – π2 ≤ 0
H1: π1 – π2 > 0
Two-tail test:
H0: π1 = π2
H1: π1 ≠ π2
i.e.,
H0: π1 – π2 = 0
H1: π1 – π2 ≠ 0
Yandell – Econ 216

32. Hypothesis Tests for Two Population Proportions
(continued)
Population proportions
Lower-tail test:
H0: π1 – π2  0
H1: π1 – π2 < 0
Upper-tail test:
H0: π1 – π2 ≤ 0
H1: π1 – π2 > 0
Two-tail test:
H0: π1 – π2 = 0
H1: π1 – π2 ≠ 0 a a
a/2
a/2
-za za
-za/2
za/2
Reject H0 if ZSTAT < -Za
Reject H0 if ZSTAT > Za
Reject H0 if ZSTAT < -Za/2
or ZSTAT > Za/2
Yandell – Econ 216

33. Hypothesis Test 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 35 of 50 women indicated they would vote Yes
Test at the 0.05 level of significance
Yandell – Econ 216

34. Hypothesis Test Example: Two population Proportions
(continued)
The hypothesis test is:
H0: π1 – π2 = 0 (the two proportions are equal)
H1: π1 – π2 ≠ 0 (there is a significant difference between proportions)
The sample proportions are:
Men: p1 = 36/72 = 0.50
Women: p2 = 35/50 = 0.70
The pooled estimate for the overall proportion is:
Yandell – Econ 216

35. Hypothesis Test Example: Two population Proportions
(continued)
Reject H0
Reject H0
The test statistic for π1 – π2 is:
.025
.025
-1.96
1.96
-2.20
Decision: Reject H0
Conclusion: There is significant evidence of a difference in proportions between men and women who will vote yes.
Critical Values = ±1.96
For  = 0.05
Yandell – Econ 216

36. Confidence Interval for Two Population Proportions
The confidence interval for
π1 – π2 is:
Yandell – Econ 216

37. Two Sample Tests in EXCEL
For independent samples:
Independent sample Z test with variances known:
Data | data analysis | z-test: two sample for means
Independent sample Z test with large sample
If the population variances are unknown, use sample variances
For paired samples (t test):
Data | data analysis | t-test: paired two sample for means
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38. Two Sample Tests in PHStat
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39. Two Sample Tests in PHStat
Input
Output
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40. Hypothesis Tests for Variances
Tests for Two
Population Variances
F test statistic
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41. F Test for Difference in Two Population Variances*
Tests for Two
Population Variances
H0: σ12 – σ22 = 0
H1: σ12 – σ22 ≠ 0
Two tailed test
H0: σ12 – σ22  0
H1: σ12 – σ22 < 0
F test statistic
Lower tail test
H0: σ12 – σ22 ≤ 0
H1: σ12 – σ22 > 0
Upper tail test
Yandell – Econ 216

42. F Test for Difference in Two Population Variances
Tests for Two
Population Variances
The F test statistic is:
(Place the larger sample variance in the numerator) *
F test statistic
= Variance of Sample 1
n1 - 1 = numerator degrees of freedom
= Variance of Sample 2
n2 - 1 = denominator degrees of freedom
Yandell – Econ 216

43. The F Distribution The F critical value is found from the F table
The are two appropriate degrees of freedom: numerator and denominator
In the F table,
numerator degrees of freedom determine the row
denominator degrees of freedom determine the column
where df1 = n1 – 1 ; df2 = n2 – 1
Yandell – Econ 216

44. Finding the Critical Value
/2 F F F
F/2
Do not
reject H0
Reject H0
Do not
reject H0
Reject H0
rejection region for a two-tailed test is
rejection region for a one-tail test is
(when the larger sample variance in the numerator)
Yandell – Econ 216

45. F Test: An Example
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  = 0.05 level?
Yandell – Econ 216

46. F Test: Example Solution
Form the hypothesis test:
H0: σ21 – σ22 = 0 (there is no difference between variances)
H1: σ21 – σ22 ≠ 0 (there is a difference between variances)
Find the F critical value for  = 0.05:
Numerator:
df1 = n1 – 1 = 21 – 1 = 20
Denominator:
df2 = n2 – 1 = 25 – 1 = 24
F.05/2, 20, 24 = 2.327
Yandell – Econ 216

47. F Test: Example Solution
(continued)
The test statistic is:
H0: σ12 – σ22 = 0
H1: σ12 – σ22 ≠ 0
/2 = 0.025
F = is not greater than the critical F value of 2.327, so we do not reject H0
Do not
reject H0
Reject H0
F/2 =2.327
Conclusion: There is no evidence of a difference in variances at  = 0.05
Yandell – Econ 216

48. Using EXCEL and PHStat EXCEL 2013 F test for two variances: PHStat
Data | data analysis | F-test: two sample for variances
PHStat
PHStat | two-sample tests | F test for differences in two variances
Yandell – Econ 216

49. Chapter Summary Compared two independent samples
Performed Z test for the differences in two means
Performed t test for the differences in two means
Compared two related samples (paired samples)
Performed paired sample t tests for the mean difference
Yandell – Econ 216

50. Chapter Summary
(continued)
Performed F tests for the difference between two population variances
Used the F table to find F critical values
Yandell – Econ 216

51. Chapter 10 Review Problems
The problems below are representative of the types of problems discussed in Chapter 10. The answer to each is provided in the text (6th edition). Use these problems to gauge your understanding of the chapter material:
10.8 (page 343)
10.14 (page 344)
10.30 (page 359)
Yandell – Econ 216