1. Chapter 9: Inferences for Two –Samples
Yunming Mu
Department of Statistics
Texas A&M University
Chapter9

2. Outline
1 Overview
2 Inferences about Two Means: Independent      and Small Samples
3 Inferences about Two Means: Independent and Large Samples
4 Inferences about Two Proportions
5 Inferences about Two Means: Matched Pairs

3. Overview
There are many important and meaningful situations in which it becomes necessary to compare two sets of sample data.
page 438 of text
Examples in the discussion

4. Two Samples: Independent
Definitions
Two Samples: Independent
The sample values selected from one population are not related or somehow paired with the sample values selected from the other population.
If the values in one sample are related to the values in the other sample, the samples are dependent. Such samples are often referred to as matched pairs or paired samples.
Text will use the wording ‘matched pairs’.
Example at bottom of page

5. Example
Do male and female college students differ with respect to their fastest reported driving speed?
Population of all male college students
Population of all female college students
Sample of n1 = 17 males report average of mph
Sample of n2 = 21 females report average of 85.7 mph

6. Graphical summary of sample data

7. Numerical summary of sample data
Gender N Mean Median TrMean StDev
female
male
Gender SE Mean Minimum Maximum Q1 Q3
female
male
The difference in the sample means is = mph

8. The Question in Statistical Notation
Let M = the average fastest speed of all male students.
and F = the average fastest speed of all female students.
Then we want to know whether M  F.
This is equivalent to knowing whether M - F  0

9. All possible questions in statistical notation
In general, we can always compare two averages by seeing
how their difference compares to 0:

10. Set up hypotheses Null hypothesis: Alternative hypothesis:
H0: M = F [equivalent to M - F = 0]
Alternative hypothesis:
Ha: M  F [equivalent to M - F  0]

11. Inferences about Two Means: Independent and Small Samples

12. Pooled Two-Sample T Test
and T Interval
Assumptions:
1. The two samples are independent.
2. Both samples are normal or the two sample sizes are small, n1 < 30 and n2 < 30
3. Both variances are unknown but equal. Assume variances are equal only if neither sample standard deviation is more than twice that of the other sample standard deviation.
page 439

13. Confidence Intervals Normal Samples w/ Unknown Equal Variance
(x1 - x2) - E < (µ1 - µ2) < (x1 - x2) + E
where

14. Leaded vs Unleaded
Each of the cars selected for the EPA study was tested and the number of miles per gallon for each was obtained and recorded (Leaded=1 and Unleaded=2).
Leaded (1) Unleaded(2) n x S

15. 95% Confidence Interval

16. Pooled Two-Sample T Tests
Normal Samples w/ unknown Variance
P-value: Use t distribution with n1+n2-2 degrees of freedom and find the P-value by following the same procedure for t tests summarized in Ch 8.
Critical values: Based on the significance level , use for upper tail tests, use
for lower tail tests and use for two tailed tests.

17. Leaded vs Unleaded Claim: 1 < 2 Ho : 1 = 2 H1 : 1 < 2
 = 0.01
Reject H0
Fail to reject H0
Further explanation of interpretation is given in text.
The magnitude of the difference is the weights is not anything that consumers would notice.
Also this test simply indicates the Coke ingredients weigh less which does not indicate that there is less volume of the product.
-1.729

18. Pooled Two-Sample T Test
Leaded vs Unleaded
Pooled Two-Sample T Test
Claim: 1 < 2
Ho : 1 =  H1 : 1 < 2  = 0.05

19. Leaded vs Unleaded Claim: 1 < 2 Ho : 1 = 2 H1 : 1 < 2
There is significant evidence to support the claim that the leaded cars have a lower mean mpg than unleaded cars
Claim: 1 < 2
Ho : 1 = 2
H1 : 1 < 2
 = 0.01
Reject H0
Fail to reject H0
Further explanation of interpretation is given in text.
The magnitude of the difference is the weights is not anything that consumers would notice.
Also this test simply indicates the Coke ingredients weigh less which does not indicate that there is less volume of the product.
Reject Null
-1.729
sample data:
t =
P-value=0.0077(=area of red region)

20. Two-Sample T Test and T Interval
Assumptions:
1. The two samples are independent.
2. Both samples are normal or the two sample sizes are small, n1 < 30 and n2 < 30
3. Both variances are unknown but unequal
page 439

21. Confidence Intervals Normal Samples w/ Unknown Unequal Variance
(x1 - x2) - E < (µ1 - µ2) < (x1 - x2) + E
where
(round v down to the nearest integer)

22. Unpooled Two Sample T-Test
Normal Samples w/ Unknown Variance
P-value: Use t distribution with v degrees of freedom and find the P-value by following the same procedure for t tests summarized in Ch 8.
Critical values: Based on the significance level , use for upper tail tests, use
for lower tail tests and use for two tailed tests.

23. Example
We compare the density of two different types of brick. Assuming normality of the two densities distributions and unequal unknown variances, test if there is a difference in the mean densities of two different types of brick.
Type I brick Type 2 brick n x S

24. Unpooled Two-Sample T-Test
Ho : 1 =  H1 : 1  2  = 0.05
P-Value = 0.001; Reject the null and conclude that there is significant difference in the mean densities of the two types of brick

25. Two-sample t-test in Minitab
Select Stat. Select Basic Statistics.
Select 2-sample t to get a Pop-Up window.
Click on the radio button before Samples in one Column. Put the measurement variable in Samples box, and put the grouping variable in Subscripts box.
Specify your alternative hypothesis.
If appropriate, select Assume Equal Variances.
Select OK.

26. Pooled two-sample t-test
Two sample T for Fastest
Gender N Mean StDev SE Mean
female
male
95% CI for mu (female) - mu (male ): ( -25.2, -7.5)
T-Test mu (female) = mu (male ) (vs not =): T = -3.75
P =
DF = 36
Both use Pooled StDev = 13.4

27. (Unpooled) two-sample t-test
Two sample T for Fastest
Gender N Mean StDev SE Mean
female
male
95% CI for mu (female) - mu (male ): ( -25.9, -6.8)
T-Test mu (female) = mu (male ) (vs not =): T = -3.54
P =
DF = 23