Two Population MeansTwo Population Means
Hypothesis Testing and Hypothesis Testing and Confidence IntervalsConfidence IntervalsFor Matched PairsFor Matched Pairs
Matched PairsMatched Pairs• Sometimes experiments are conducted in such a
way that samples from two populations are matchedmatched 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.
Hypothesis Tests and Confidence Hypothesis Tests and Confidence Intervals for Matched PairsIntervals for Matched Pairs
• Suppose there a random sample of nD elements is taken. For each a corresponding sample from each population is observed. The difference is denoted di. So there are nD observations of differences, di’s.
• Statistics calculated:
2DD
D
2i2
D
D
i
ss :difference average theofDeviation Standard Sample
1n
)d(d s :difference average theof Variance Sample
n
dd :difference Average
Hypothesis Tests and Confidence Hypothesis Tests and Confidence Intervals for Matched PairsIntervals for Matched Pairs
• Hypothesis Test: H0: D = d
HA: D > d
Test statistic:
• Confidence Interval:
Both the hypothesis test and the confidence interval have nD-1 degrees of freedom.
Error
Standard
dEstimate
Point
t
DD n/s
dd t
Error
Standardt
Estimate
PointDFα/2,
n
std
D
DDFα/2,
ExampleExampleObjective: 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.
Approach 1Approach 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 Date 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 5000 15-Jun 6500
1-Feb 6000 20-Oct 4200
15-Mar 7000 15-Apr 3100
Approach 2Approach 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 These data are pairedpaired by the random dates 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
300
800
CalculateDifferences
_
Calculate statistics: d =400 sD = 258.2
Hypothesis TestHypothesis 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.
049.27/258.2
200 -400
7/s
200 -d t
D
95% Confidence Interval95% Confidence Interval
D
D.025,6
n
std
7
258.22.447 400
400 ± 238.8161.2 638.8
ExcelExcel For Matched Pairs For Matched Pairs
• Hypothesis Tests– Go to Tools/Data Analysis and select
t-Test Paired Two Sample for Means.• Look at p-valuep-value for the test.
• Confidence Intervals– Create a column of differences.
• Go to Tools/Data Analysis and select Descriptive Statistics: Mean Mean ± Confidence± Confidence
Excel - Hypothesis TestExcel - Hypothesis Test
Go Tools
Select Data Analysis
Select t-Test: Paired Two Sample for Means
Excel: t-Test for Matched PairsExcel: t-Test for Matched PairsSince HA is D > 200, enter
Column B for Range 1
Column C for Range 2
200 for Hypothesized Mean Difference
Check
LabelsDesignate first cell
for output.
Hypothesis Test (Cont’d)Hypothesis Test (Cont’d)
p-value for
one-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
p-value for at
two-tail “” test
95% Confidence Interval for 95% Confidence Interval for Matched PairsMatched Pairs
=B2-C2
Drag to D3:D8
=I3+I16
=I3-I16
Go to
Tools/Data Analysis
Descriptive Statistics
On Column D.
Store output
beginning in
cell H1.
ReviewReview
• 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
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