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

nd

d :difference Average

Hypothesis Tests and Confidence Hypothesis Tests and Confidence Intervals for Matched PairsIntervals for Matched Pairs

• Hypothesis Test: H0: D = dHA: D > d

Test statistic:

• Confidence Interval:

Both the hypothesis test and the confidence interval have nD-1 degrees of freedom.

ErrorStandard

dEstimate

Point

t

DD n/sdd t

Error

Standardt

EstimatePoint

DFα/2, nstd

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 Irvine15-Dec 9000 30-Nov 670025-Nov 8500 8-May 490030-Jun 4000 13-Mar 480022-Jul 5000 6-Mar 360015-Aug 5000 15-Jun 65001-Feb 6000 20-Oct 420015-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 Irvine25-Nov 8500 82002-Feb 2800 27005-May 4200 4000

25-Aug 5600 490025-Apr 5700 530012-Jun 7300 700021-Dec 10000 9200

Difference300100200700400300800

CalculateDifferences

_Calculate statistics: d =400 sD = 258.2

Hypothesis TestHypothesis TestH0: D = 200HA: 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 -4007/s

200 -d t D

95% Confidence Interval95% Confidence Interval

D

D.025,6

nstd

7258.22.447 400

400 ± 238.8161.2 638.8

Excel For Matched PairsExcel 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 TestGo 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 1Column C for Range 2

200 for Hypothesized Mean Difference

CheckLabels Designate first cell

for output.

Hypothesis Test (Cont’d)Hypothesis Test (Cont’d)

p-value forone-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 attwo-tail “” test

95% Confidence Interval for 95% Confidence Interval for Matched PairsMatched Pairs

=B2-C2Drag to D3:D8

=I3+I16

=I3-I16

Go toTools/Data AnalysisDescriptive Statistics

On Column D.Store outputbeginning 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