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6/10/10 Lecture 20 1STOR 155 Introductory StatisticsLecture 20: Comparing two meansSection 7.2 (a modified case only)The UNIVERSITY of NORTH CAROLINAat CHAPEL HILL6/10/10 Lecture 20 2Comparing two population means• Population 1: with mean s.d. • Population 2: with mean s.d. • Interested in the difference• Sample 1: size mean s.d.• Sample 2: size mean s.d.• Consider the difference• Assume the two samples are independent, and bothand are large (e.g. at least 40). ö1ö2ö1à ö2D = Xö1à Xö2s1s2n1Xö1n2Xö2n1n2û1û26/10/10 Lecture 20 3Useful probability factsThe random variable D has approximately a normal distribution with mean and standard deviationSD(D) =An estimate of SD(D) (Note: This is the onlyexpression, not like in Lecture 19 with two different formulas)ö1à ö2û21=n1+ û22=n2qSED= s21=n1+ s22=n2q6/10/10 Lecture 20 4Confidence Interval for • Expression:where the margin of error – Confidence level C determines m = zãSEDzã[D à m; D + m ]ö1à ö26/10/10 Lecture 20 5Hypothesis testing for • We want to test versus some (1-sided or 2-sided) alternative. Recall the 4 steps …• Step 1: need to specify the alternative • Step 2: test statistic H0: ö1= ö2Haö1à ö2z = D=SED6/10/10 Lecture 20 6Hypothesis testing (continued)• Step 3: The P-value will be equal to P(Z > z) for 1-sided (upper tail) P(Z < z) for 1-sided (lower tail) 2 P(Z > |z|) for 2-sided • Step 4: Compare the P-value with the significance level and draw your conclusion.Ha: ö1> ö2Ha: ö1< ö2Ha: ö16=ö2ë6/10/10 Lecture 20 7Sampling survey by USDACompare average wheat prices ($ per bushel) in July and September using independent samples of wheat producers• Population 1: (September) • Population 2: (July) • Sample 1: • Sample 2: ö1ö2s1= 0:19s2= 0:22n1= 45;Xö1= 3:61;n2= 90;Xö2= 2:95;6/10/10 Lecture 20 8Wheat price increase ? (continued)• Test vs• Test statistic:• P-value = P(Z > 18.03) = 0.00• Reject• 95% CI for is (0.59, 0.73), wherez = (3:61 à2:95) ä (0:19)2=45 +(0:22)2=90p= 18:03ö1à ö2H0: ö1= ö2Ha: ö1> ö2H0SED= (0:19)2=45 +(0:22)2=90p= 0:0366;m = zãSED= (1:96)(0:0366) = 0:076/10/10 Lecture 20 9Take Home Message• CI for the difference• Hypothesis testing for comparing and … 4 steps• Assumption: both and are large• Several other cases in Section 7.2 not studied:--- small samples--- normal populations--- using t-distributions to calculate P-valuesö1ö1à


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UNC-Chapel Hill STOR 155 - Lecture 20- Comparing two means

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