chapter five (&9) decision making for two samples

Chapter Five Chapter Five (&9)(&9)

Decision Making for Two SamplesDecision Making for Two Samples

Horng-Chyi HorngHorng-Chyi Horng Statistics II_FiveStatistics II_Five 22

Chapter OutlinesChapter Outlines

Inference for a Difference in MeansInference for a Difference in Means• Variance KnownVariance Known

• Two Normal Distributions, Variance UnknownTwo Normal Distributions, Variance Unknown

• Paired t-TestPaired t-Test

Inference on the Variances of Two Normal PopulationsInference on the Variances of Two Normal Populations

Inference on Two Population ProportionsInference on Two Population Proportions

Summary TableSummary Table


IntroductionIntroduction


Inference for a Difference in MeansInference for a Difference in Means--Variance KnownVariance Known &5-2 (&9-2)


Inference for a Difference in MeansInference for a Difference in Means--Variance KnownVariance Known

1,0~)(

,

)()()( and

)()()(

since , means sample the

in sdifference theis ofestimator point The

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212121

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nn

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Therefore

nnXVXVXXV

XEXEXXE

XX


Hypothesis Tests for a Difference in MeansHypothesis Tests for a Difference in Means--Variance KnownVariance Known


Example 9-1Example 9-1

A product developer is interested in reducing the drying time of a priA product developer is interested in reducing the drying time of a primer paint. Two formulations of the paint are tested; formulation 1 is mer paint. Two formulations of the paint are tested; formulation 1 is the standard chemistry, and formulation 2 has a new drying ingrediethe standard chemistry, and formulation 2 has a new drying ingredient that should reduce the drying time. From experience, it is known nt that should reduce the drying time. From experience, it is known that the standard deviation of drying time is 8 minutes, and this inhethat the standard deviation of drying time is 8 minutes, and this inherent variability should be unaffected by the addition of the new ingrerent variability should be unaffected by the addition of the new ingredient. Ten specimens are painted with formulation 1,and another l0 sdient. Ten specimens are painted with formulation 1,and another l0 specimens are painted with formulation 2; the 20 specimens are paintpecimens are painted with formulation 2; the 20 specimens are painted in random order. The two sample average drying times are 121 ed in random order. The two sample average drying times are 121 min. and 112 min., for formulation 1 and 2 respectively. What conclmin. and 112 min., for formulation 1 and 2 respectively. What conclusions can the product developer draw about the effectiveness of the usions can the product developer draw about the effectiveness of the new ingredient, using α=0.05? new ingredient, using α=0.05?


The Sample Size (I)The Sample Size (I)Assume that HAssume that H00: : 11--22 = = 00 is false and the true difference is is false and the true difference is

Given values of Given values of and and , find the required sample size n to , find the required sample size n to achieve a particular level of achieve a particular level of ..

Then,

Let

0 when Since

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22/

2

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02/

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ZZn

nn

ZZ

Z

nn

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The Sample Size (II)The Sample Size (II) Two-sided and one-sided Hypothesis TestingsTwo-sided and one-sided Hypothesis Testings


Identifying the Cause and EffectIdentifying the Cause and Effect

In Example 9-1In Example 9-1 Factors, Treatments, and Response VariablesFactors, Treatments, and Response Variables Completely Randomized ExperimentsCompletely Randomized Experiments

• Randomly assigned 10 test specimens to one formulation, and 10 test Randomly assigned 10 test specimens to one formulation, and 10 test specimens to the other formulation.specimens to the other formulation.

Observational StudyObservational Study• Not randomizedNot randomized

• Maybe caused by other factors not considered in the studyMaybe caused by other factors not considered in the study

• ExamplesExamples


Confidence Interval on a Difference in MeansConfidence Interval on a Difference in Means- Variance Known- Variance Known


Example 9-4Example 9-4 Tensile strength tests were performed on two different Tensile strength tests were performed on two different

grades of aluminum spars used in manufacturing the wing grades of aluminum spars used in manufacturing the wing of a commercial transport aircraft. The test data is listed in of a commercial transport aircraft. The test data is listed in Table 5-1. Find a 90% C.I. on the difference of the tensile Table 5-1. Find a 90% C.I. on the difference of the tensile strength of these two aluminum spars.strength of these two aluminum spars.


Choice of Sample Size to Achieve Precision of EstimationChoice of Sample Size to Achieve Precision of Estimation

Where E is the error allowed in estimating Where E is the error allowed in estimating 11--22..


One-Sided C.I.s on the Difference in Means One-Sided C.I.s on the Difference in Means – Variance Unknown– Variance Unknown

A 100(1-A 100(1-) percent upper-confidence interval on ) percent upper-confidence interval on 11--22 is is

And a 100(1-And a 100(1-) percent lower-confidence interval is ) percent lower-confidence interval is

2

221

1

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2121 nnzXX

212

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1

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nnzXX

chapter five (&9) decision making for two samples

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