1. Hypothesis Testing and Confidence Intervals
Two Population Means
Hypothesis Testing and Confidence Intervals
For Matched Pairs

2. Matched Pairs
Sometimes experiments are conducted in such a way that samples from two populations are matched with something in common so that the i-th sample taken from the first population has something in common with the i-th sample of the second population.
It is the “common element” (same date, same weight, etc.) that is chosen at random and dictates the corresponding observations from each population.
Differences between the sample values (dictated by the “common element”) from each population are computed.
If it can be assumed that the differences have a normal distribution, t-tests can then be performed or t-intervals constructed for the average value of the differences.
Pairing, in general, reduces the variability in the problem.

3. Hypothesis Tests and Confidence Intervals for Matched Pairs
Suppose there a random sample of n elements is taken. For each a corresponding sample from each population is observed. The difference is denoted di. So there are n observations of differences, di’s.
Statistics calculated:

4. Distribution of average difference _ d
Distribution: t distribution

5. Hypothesis Tests and Confidence Intervals for Matched Pairs
H0: D = v
HA: D > v
Test statistic:

6. Hypothesis Tests and Confidence Intervals for Matched Pairs
Both the hypothesis test and the confidence interval have n-1 degrees of freedom: DF=n-1

7. Example
Objective: Compare sales at two branch stores, one in Anaheim, the other in Irvine.
Can it be concluded that average daily sales in Anaheim is at least $200 greater than average daily sales in Irvine?
Construct a 95% confidence interval for the average difference in daily sales between the Anaheim and Irvine branches.

8. Approach 1
Records of sales on seven random dates in Anaheim are selected and seven random dates in Irvine are selected.
There is nothing in common between the Anaheim and Irvine samples.
Would have to use Difference in Means approach.
Probably not the best approach.
Date
Anaheim
Irvine
15-Dec
9000
30-Nov
6700
25-Nov
8500
8-May
4900
30-Jun
4000
13-Mar
4800
22-Jul
5000
6-Mar
3600
15-Aug
15-Jun
6500
1-Feb
6000
20-Oct
4200
15-Mar
7000
15-Apr
3100

9. Approach 2
Do not choose the receipts at random, but choose the dates at random and observe the sales at the Anaheim and Irvine branch stores on these dates.
These data are paired by the random dates.
Date
Anaheim
Irvine
25-Nov
8500
8200
2-Feb
2800
2700
5-May
4200
4000
25-Aug
5600
4900
25-Apr
5700
5300
12-Jun
7300
7000
21-Dec
10000
9200
Difference
300
100
200
700
400
800
Calculate
Differences _
Calculate statistics: d =400 sD = 258.2

10. Hypothesis Test H0: D = 200 HA: D > 200 Select α = .05.
Reject H0 (Accept HA) if t > t.05,6 = 1.943
2.049 > 1.943; thus it can be concluded that average daily sales in Anaheim > $200 more than average daily sales in Irvine.

11. 95% Confidence Interval
400 ± 238.8
161.2  638.8

12. Excel For Matched Pairs
Hypothesis Tests
Go to Tools/Data Analysis and select t-Test Paired Two Sample for Means.
Look at p-value for the test.
Confidence Intervals
Create a column of differences.
Go to Tools/Data Analysis and select Descriptive Statistics: Mean ± Confidence

13. Excel - Hypothesis Test
Go Tools
Select Data Analysis
Select t-Test: Paired Two Sample for Means

14. Excel: t-Test for Matched Pairs
Since HA is D > 200, enter
Column B for Range 1
Column C for Range 2
200 for Hypothesized Mean Difference
Check
Labels
Designate first cell
for output.

15. Hypothesis Test (Cont’d)
p-value for
one-tail test
p-value for at
two-tail “” test
Low p-value for 1-tail test (compared to α =.05)!
Can conclude average daily sales in Anaheim exceed those in Irvine by > $200

16. 95% Confidence Interval for Matched Pairs
Go to
Tools/Data Analysis
Descriptive Statistics
On Column D.
Store output
beginning in
cell H1.
=B2-C2
Drag to D3:D8
=I3-I16
=I3+I16

17. Review Summary What constitutes “matched pairs”
Matched pairs normally reduces variability from difference in means tests
Create a set of differences
Hypothesis Tests/Confidence Intervals for average difference
By hand
By Excel
Summary