1. Hypothesis Testing – Two Samples
Chapters 12 & 13

2. Chapter Topics Comparing Two Independent Samples
Independent samples Z test for the difference in two means
Pooled-variance t test for the difference in two means
F Test for the Difference in Two Variances
Comparing Two Related Samples
Paired-sample Z test for the mean difference
Paired-sample t test for the mean difference
Two-sample Z test for population proportions
Independent and Dependent Samples

3. Comparing Two Independent Samples
Different Data Sources
Unrelated
Independent
Sample selected from one population has no effect or bearing on the sample selected from the other population
Use the Difference between 2 Sample Means
Use Z Test or t Test for Independent Samples

4. Independent Sample Z Test (Variances Known)
Assumptions
Samples are randomly and independently drawn from normal distributions
Population variances are known
Test Statistic

5. t Test for Independent Samples (Variances Unknown)
Assumptions
Both populations are normally distributed
Samples are randomly and independently drawn
Population variances are unknown but assumed equal
If both populations are not normal, need large sample sizes

6. Developing the t Test for Independent Samples
Setting Up the Hypotheses
H0: m 1 = m H1: m 1 ¹ m 2
H0: m 1 -m 2 = 0 H1: m m 2 ¹ 0
Two Tail OR
H0: m 1 £ m 2 H1: m 1 > m 2
H0: m 1 - m 2 £ 0 H1: m 1 - m 2 > 0
Right Tail OR
H0: m 1 ³ m 2
H0: m 1 - m 2 ³ 0 H1: m 1 - m 2 < 0
Left Tail OR
H1: m 1 < m 2

7. Developing the t Test for Independent Samples
(continued)
Calculate the Pooled Sample Variance as an Estimate of the Common Population Variance

8. Developing the t Test for Independent Samples
(continued)
Compute the Sample Statistic
Hypothesized difference

9. t Test for Independent Samples: Example
You’re a financial analyst for Charles Schwab. Is there a difference in average 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 (a = 0.05)?
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10. Calculating the Test Statistic

11. Solution .025 .025 t H0: m1 - m2 = 0 i.e. (m1 = m2)
a = 0.05
df = = 44
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Reject H
Reject H
Reject at a = 0.05.
.025
.025
There is evidence of a difference in means.
2.0154 t
2.03

12. (p-Value is between .02 and .05) < (a = 0.05) Reject.
p -Value Solution
(p-Value is between .02 and .05) < (a = 0.05) Reject.
p-Value 2
is between .01 and .025
Reject
Reject
a 2
=.025 Z
2.0154
2.03
Test Statistic 2.03 is in the Reject Region

13. Example
You’re a financial analyst for Charles Schwab. You collect the following data:
NYSE NASDAQ Number
Sample Mean
Sample Std Dev
You want to construct a 95% confidence interval for the difference in population average yields of the stocks listed on NYSE and NASDAQ.
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14. Example: Solution

15. Independent Sample (Two Sample) t- Test in JMP
Independent Sample t Test with Variances Known
Analyze | Fit Y by X | Measurement in Y box (Continuous) | Grouping Variable in X box (Nominal) |  | Means/Anova/Pooled t

16. Comparing Two Related Samples
Test the Means of Two Related Samples
Paired or matched
Repeated measures (before and after)
Use difference between pairs
Eliminates Variation between Subjects

17. Z Test for Mean Difference (Variance Known)
Assumptions
Both populations are normally distributed
Observations are paired or matched
Variance known
Test Statistic

18. t Test for Mean Difference (Variance Unknown)
Assumptions
Both populations are normally distributed
Observations are matched or paired
Variance unknown
If population not normal, need large samples
Test Statistic

19. Dependent-Sample t Test: Example
Assume you work in the finance department. Is the new financial package faster (a=0.05 level)? You collect the following processing times:
User Existing System (1) New Software (2) Difference Di
C.B Seconds Seconds
T.F
M.H
R.K
M.O
D.S
S.S
C.T
K.T
S.Z

20. Dependent-Sample t Test: Example Solution
Is the new financial package faster (0.05 level)?
H0: mD £ 0
H1: mD > 0
Reject
a =.05
D =
.072
a =.05
Critical Value= df = n - 1 = 9
1.8331
3.66
Decision: Reject H0
t Stat. in the rejection zone.
Test Statistic
Conclusion: The new software package is faster.

21. Confidence Interval Estimate for of Two Dependent Samples
Assumptions
Both populations are normally distributed
Observations are matched or paired
Variance is unknown
Confidence Interval Estimate:

22. Example
Assume you work in the finance department. You want to construct a 95% confidence interval for the mean difference in data entry time. You collect the following processing times:
User Existing System (1) New Software (2) Difference Di
C.B Seconds Seconds
T.F
M.H
R.K
M.O
D.S
S.S
C.T
K.T
S.Z

23. Solution:

24. F Test for Difference in Two Population Variances
Test for the Difference in 2 Independent Populations
Parametric Test Procedure
Assumptions
Both populations are normally distributed
Test is not robust to this violation
Samples are randomly and independently drawn

25. The F Test Statistic F = Variance of Sample 1 = Variance of Sample 2
n1 - 1 = degrees of freedom
= Variance of Sample 2
n2 - 1 = degrees of freedom F

26. Developing the F Test Hypotheses Test Statistic
H0: s12 = s22
H1: s12 ¹ s22
Test Statistic
F = S12 /S22
Two Sets of Degrees of Freedom
df1 = n1 - 1; df2 = n2 - 1
Critical Values: FL( ) and FU( )
FL = 1/FU* (*degrees of freedom switched)
Reject H0
Reject H0
Do Not
Reject
a/2
a/2 FL FU F
n1 -1, n2 -1
n1 -1 , n2 -1

27. Developing the F Test Easier Way Test Statistic
Put the largest in the num.
Test Statistic
F = S12 /S22
Reject H0
Do Not
Reject a F F

28. F Test: An Example
Assume you are a financial analyst for Charles Schwab. 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 a = 0.05 level?
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29. F Test: Example Solution
Finding the Critical Values for a = .05

30. F Test: Example Solution
Test Statistic:
Decision:
Conclusion:
H0: s12 = s22
H1: s12 ¹ s22
a = .05
df1 = df2 = 24
Critical Value(s):
Reject
Do not reject at a = 0.05.
.05
There is insufficient evidence to prove a difference in variances. F
2.33
1.25

31. F Test: One-Tail H0: s12 ³ s22 H0: s12 £ s22 or H1: s12 < s22
Degrees of freedom switched
Reject
Reject
a = .05
a = .05 F F

32. F Test: One-Tail Easier Way Test Statistic Put the largest in the num.
F = S12 /S22
Reject H0
Do Not
Reject a F F

33. Z Test for Differences in Two Proportions (Independent Samples)
What is It Used For?
To determine whether there is a difference between 2 population proportions and whether one is larger than the other
Assumptions:
Independent samples
Population follows binomial distribution
Sample size large enough: np  5 and n(1-p)  5 for each population

34. Z Test Statistic

35. The Hypotheses for the Z Test
Research Questions
No Difference
Prop 1 
Prop 2
Prop 1 
Prop 2
Hypothesis
Any Difference
Prop 1 < Prop 2
Prop 1 > Prop 2 p - p
= 0 p - p  p - p  H 1 2 1 2 1 2 p - p
< 0 p - p
> 0 H p - p  1 1 2 1 2 1 2

36. Z Test for Differences in Two Proportions: Example
As personnel director, you want to test the perception of fairness of two methods of performance evaluation. 63 of 78 employees rated Method 1 as fair. 49 of 82 rated Method 2 as fair. At the 0.01 significance level, is there a difference in perceptions? 

37. Calculating the Test Statistic

38. Z Test for Differences in Two Proportions: Solution
H0: p1 - p2 = 0
H1: p1 - p2  0
 = 0.01
n1 = 78 n2 = 82
Critical Value(s):
Test Statistic:
Decision:
Conclusion: Z  2 . 90
Reject at  = 0.01.
Reject H
Reject H
.005
.005
There is evidence of a difference in proportions.
-2.58
2.58 Z

39. Confidence Interval for Differences in Two Proportions
The Confidence Interval for Differences in Two Proportions

40. Confidence Interval for Differences in Two Proportions: Example
As personnel director, you want to find out the perception of fairness of two methods of performance evaluation. 63 of 78 employees rated Method 1 as fair. 49 of 82 rated Method 2 as fair. Construct a 99% confidence interval for the difference in two proportions. 

41. Confidence Interval for Differences in Two Proportions: Solution
We are 99% confident that the difference between two proportions is somewhere between and

42. Z Test for Differences in Two Proportions (Dependent Samples)
What is It Used For?
To determine whether there is a difference between 2 population proportions and whether one is larger than the other
Assumptions:
Dependent samples
Population follows binomial distribution
Sample size large enough: np  5 and n(1-p)  5 for each population

43. Z Test Statistic for Dependent Samples
Sample Two
Category 1
Category 2
Sample One a b
a+b c d
c+d
a+c
b+d N
This Z can be used when a+d>10
If 10<a+d<20, use the correction in the text

44. Unit Summary Compared Two Independent Samples
Performed Z test for the differences in two means
Performed t test for the differences in two means
Performed Z test for differences in two proportions
Addressed F Test for Difference in Two Variances

45. Unit Summary Compared Two Related Samples
Performed dependent sample Z tests for the mean difference
Performed dependent sample t tests for the mean difference
Performed Z tests for proportions using dependent samples