CHSPROBAND STATS 1CHAPTER 9

CHAPTER 99-3: INFERENCES ABOUT TWO MEANS

INDEPENDENT SAMPLES

Goal: To be able to use two sample MEANSfor constructing a confidence interval estimate of thedifference between the corresponding population means, or testing a claim made about the twopopulation means.

DEFINITIONS

Two samples are INDEPENDENT if the sample values selected from one population are not relatedto or somehow paired or matched with the sample ~ from the other population.

Two samples are DEPENDENT (or consist f MATCHED PAIRSUf the members of one sample canbe used to determine the members of the 0 e . les consisting of matched pairs (suchas husband/wife data) are dependent. In addition to matched pairs of sample data, dependencecould also occur with samples related through associations such as family members.

EXAMPLE: Determine whether each of the following is an independent sample or a matched pair.

a) The effectiveness of a diet is tested using weights of subjects measured before and after thediet treatment. /

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b) One group of subjects is treated with the cholesterol-reducing drug Lipitor, while anothergroup is given a placebo. [I NDe:P5-N"D€NT) Tt<.€ATM~T GtI2-OUP.

rl,..kC...e::Bo 6,R-Ou P.REQUIREMENTS

1. 0"1 and 0"2 are unknown and no assumption is made about the equality of 0"1 and 0"2'

2. The two samples are independent.3. Both samples are simple random samples.4. Either or both of these conditions is satisfied: The two sample sizes are both large (>3cCrJ

both samples come from populations having normal distributions. (For small samples~~normality requirement is loose in the sense that the procedures perform well as long as thereare no outliers and departure from normality are not too extreme.)

HYPOTHESIS TEST STATISTIC FOR TWO MEANS: INDEPENDENT SAMPLESTHE. VA~IANCE:' o r- THE. DIFF-6"~gNGGS 'Bt::"'WE.€.N do IND.Sp..MI'L..~

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CHS PROB AND STATSCHAPTER 9

2

Degrees of Freedom: When finding critical values or P-values, use the followingfor determining thenumber of degrees of freedom, denoted by ~;~_(~~~es~me!lw typically result in differentnumbers of degrees of freedom, the c l'lCllli~rH~1('Cct~ 'ls"i-arely ected by the choice.)

df = smaller Of~l -1'f-n/....n2-1)

P-values: Table A-3with the procedure summarized in Figure 8-6, Pg. 396.Critical values: Table A-3.

CONFIDENCE INTERVAL ESTIMATE OF #1 - #2: INDEPENDENT SAMPLES

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And the number of degrees of freedom df is described above for hypothesis tests.

where

E La *"2

Because the hypothesis test and confidence interval use the same distribution and standard error,they are equivalent in the sense that they result in the same conclusions. Consequently the nullhypothesis of 111 = 112 (or /11 - /12 = 0) can be tested by determining whether the confidence intervalincludes O. For a two-tailed hypothesis test with a 0.05 significance level, use a 95%confidence interval. For a one-tailed test with a significance level of 0.05, use a 90%confidence Interval. r'l.. ",O.(J<" ~.os

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MAKING INFERENCES ABOUT TWO POPULATION MEANS

Before making an inference explore the two samples to describe center, variation, distribution,outliers, and whether the population appears to be changing over time (CVDOT). It could be helpfulto do the following:

• Find descriptive statistics for both data sets. n..,) ~ l..s• Create boxplots or stemplots of both data sets, drawn on the same scale so they can be

compared.• Create histograms of both data sets, so that their distributions can be seen and compared.• Identify any outliers.

3-* TEST A ~l..AIM/'. tty-rol rr£.SIS IE 5T 1-EXAMPLE2: The Revenue Commissioners in Ireland conducted a contest for promotion. The agesof the unsuccessful and successful applicants are given below. Some of the applicants who wereunsuccessful in getting the promotion charged that the competitiOnSd discrimination basedon age. Treat the data as samples from larger populations and use 0.05 Igniftcance level to testthe claim that the unsuccessful applicants are from a population wi reater mean age than themean age of successful applicants. Based on the result. does there appear to be discriminationbased on age?

CHS PROB AND STArsCHAPfER9

I ,rL~~~ ~ IA~esof Unsuccessful Applicants

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CHS PROB AND STATSCHAPfER9

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EXAMPLE 3: Using the sam Ie data iven in the preceding example, construct a 90% confidenceinterval estimate of th difference between e mean age of unsuccessful applicants and the meanage of successful applicants. Remember, a one-sided hypothesis test with significance level 0.05can be tested by using a 90% confidence interval.) &,=1'd- "'I= Lf'1.0 SiJ.- .:: 5".9 :x;.::. 4-3.~rt.....~~ Ages of Unsuccessful Applicants

34 37 37 38 41 42 43 44 44 4545 45 46 48 49 53 53 54 54 5556 57 60

IL~OO Ages of Successful Applicants27 33 36 37 38 38 39 42 42 4343 44 44 44 45 45 45 45 46 4647 47 48 48 49 49 51 51 52 54

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HOMEWORK: Pg. 479- 481 (#4-8. 9.11. 12. 17)