1. Two Sample Tests Ho Ho Ha Ha TEST FOR EQUAL VARIANCES
TEST FOR EQUAL MEANS Ho Ho
Population 1
Population 1
Population 2
Population 2 Ha Ha
Population 1
Population 2
Population 1
Population 2

2. Hypothesis Tests for Two Population Variances
Two-Tailed Test
Upper One-Tailed Test
Lower One-Tailed Test

3. Hypothesis Tests for Two Population Variances
F-TEST STATISTIC FOR TESTING WHETHER TWO POPULATIONS HAVE EQUAL VARIANCES
where:
ni = Sample size from ith population
nj = Sample size from jth population
si2= Sample variance from ith population
sj2= Sample variance from jth population

4. Hypothesis Tests for Two Population Variances (Example 9-2)
df: Di = 10, Dj =12
a = .10
Rejection Region /2 = 0.05
F = 1.47 F
Since F=1.47  F/2= 2.76, do not reject H0

5. Independent Samples Independent samples
Selected from two or more populations
Values in one sample have no influence on the values in the other sample(s).

6. Hypothesis Tests for Two Population Means
Format 1
Two-Tailed Test
Upper One-Tailed Test
Lower One-Tailed Test

7. Hypothesis Tests for Two Population Means
Format 2
Two-Tailed Test
Upper One-Tailed Test
Lower One-Tailed Test

8. Hypothesis Tests for Two Population Means
T-TEST STATISTIC
(EQUAL POPULATION VARIANCES)
where:
Sample means from populations 1 and 2
Hypothesized difference
Sample sizes from the two populations
Pooled standard deviation

9. Hypothesis Tests for Two Population Means
POOLED STANDARD DEVIATION
Where:
s12 = Sample variance from population 1
s22 = Sample variance from population 2
n1 and n2 = Sample sizes from populations 1 and 2 respectively

10. Hypothesis Tests for Two Population Means
(Unequal Variances)
t-TEST STATISTIC
where:
s12 = Sample variance from population 1
s22 = Sample variance from population 2

11. Hypothesis Tests for Two Population Means (Example 9-4)
Rejection Region  /2 = 0.025
Rejection Region  /2 = 0.025
Since t < 2.048, do not reject H0

12. Hypothesis Tests for Two Population Means
DEGREES OF FREEDOM FOR t-TEST STATISTIC WITH UNEQUAL POPULATION VARIANCES

13. Confidence Interval Estimates for 1 - 2
STANDARD DEVIATIONS UNKNOWN
AND 12 = 22
where:
= Pooled standard deviation
t/2 = critical value from t-distribution for desired confidence level and degrees of freedom equal to n1 + n2 -2

14. Confidence Interval Estimates for 1 - 2 (Example 9-5)
- $330.46
$1,458.34

15. Confidence Interval Estimates for 1 - 2
STANDARD DEVIATIONS UNKNOWN
AND 12  22
where:
t/2 = critical value from t-distribution for desired confidence level
and degrees of freedom equal to:

16. Confidence Interval Estimates for 1 - 2
LARGE SAMPLE SIZES
where:
z/2 = critical value from the standard normal distribution for desired confidence level

17. Paired Samples Hypothesis Testing and Estimation
Paired samples are samples that selected such that each data value from one sample is related (or matched) with a corresponding data value from the second sample. The sample values from one population have the potential to influence the probability that values will be selected from the second population.

18. Paired Samples Hypothesis Testing and Estimation
PAIRED DIFFERENCE
where:
d = Paired difference
x1 and x2 = Values from sample 1 and 2, respectively

19. Paired Samples Hypothesis Testing and Estimation
MEAN PAIRED DIFFERENCE
where:
di = ith paired difference
n = Number of paired differences

20. Paired Samples Hypothesis Testing and Estimation
STANDARD DEVIATION FOR PAIRED DIFFERENCES
where:
di = ith paired difference
= Mean paired difference

21. Paired Samples Hypothesis Testing and Estimation
t-TEST STATISTIC FOR PAIRED DIFFERENCES
where:
= Mean paired difference
d = Hypothesized paired difference
sd = Sample standard deviation of paired differences
n = Number of paired differences

22. Paired Samples Hypothesis Testing and Estimation (Example 9-6)
Rejection Region  = 0.05
Since t= < 1.833, do not reject H0

23. Paired Samples Hypothesis Testing and Estimation
PAIRED CONFIDENCE INTERVAL ESTIMATE

24. Paired Samples Hypothesis Testing and Estimation (Example 9-7)
95% Confidence Interval
4.927
9.273

25. Hypothesis Tests for Two Population Proportions
Format 1
Two-Tailed Test
Upper One-Tailed Test
Lower One-Tailed Test

26. Hypothesis Tests for Two Population Proportions
Format 2
Two-Tailed Test
Upper One-Tailed Test
Lower One-Tailed Test

27. Hypothesis Tests for Two Population Proportions
POOLED ESTIMATOR FOR OVERALL PROPORTION
where: x1 and x2 = number from samples 1 and 2 with desired characteristic.

28. Hypothesis Tests for Two Population Proportions
TEST STATISTIC FOR DIFFERENCE IN POPULATION PROPORTIONS
where:
(1 - 2) = Hypothesized difference in proportions from populations 1 and 2, respectively
p1 and p2 = Sample proportions for samples selected from population 1 and 2
= Pooled estimator for the overall proportion for both populations combined

29. Hypothesis Tests for Two Population Proportions (Example 9-8)
Rejection Region  = 0.05
Since z =-2.04 < , reject H0

30. Confidence Intervals for Two Population Proportions
CONFIDENCE INTERVAL ESTIMATE
FOR 1- 2
where:
p1 = Sample proportion from population 1
p2 = Sample proportion from population 2
z = Critical value from the standard normal table

31. Confidence Intervals for Two Population Proportions (Example 9-10)
-0.034
0.104

32. Key Terms
Independent Samples
Paired Samples