4/14/11 Lecture 20 1 STOR 155 Introductory Statistics Lecture 20: Comparing two means Section 7.2 (a modified case only) The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL4/14/11 Lecture 20 2 Comparing 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 both and are large (e.g. at least 40). ö1ö2ö1à ö2D = Xö1àXö2s1s2n1Xö1n2Xö2n1n2û1û24/14/11 Lecture 20 3 Useful probability facts The random variable D has approximately a normal distribution with mean and standard deviation SD(D) = An estimate of SD(D) (Note: This is the only expression, not like in Lecture 19 with two different formulas) ö1à ö2û21=n1+ û22=n2qSED= s21=n1+ s22=n2q4/14/11 Lecture 20 4 Confidence Interval for • Expression: where the margin of error – Confidence level C determines m = zãSEDzã[D à m; D + m ]ö1à ö24/14/11 Lecture 20 5 Hypothesis 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=SED4/14/11 Lecture 20 6 Hypothesis 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=ö2ë4/14/11 Lecture 20 7 Sampling survey by USDA Compare 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;4/14/11 Lecture 20 8 Wheat price increase ? (continued) • Test vs • Test statistic: • P-value = P(Z > 18.03) = 0.00 • Reject • 95% CI for is (0.59, 0.73), where z = (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) = 0:074/14/11 Lecture 20 9 Take 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-values ö1ö1à
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