9.1-9.3 - 9.1 z Test and Confidence intervals for a...

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9.1 z Test and Confidence intervals for a Difference Between two population means Let μ 1 , σ 1 and μ 2 , σ 2 are the mean and standard deviation of two populations or processes. Let n, x and s 1 represent the sample size, sample mean and sample s.d. from the first population, and m, y and s 2 represent the sample size, sample mean and sample s.d. from the second population. The properties: 1 1. If both population distributions are normal, so is the sampling distribution of y x - 2. If both sizes n and m are large, then by Central Limit Theorem y x - is approximately normal 3. The standard deviation is m n y x 2 2 2 1 σ σ σ + = - 4. y x - is unbiased estimator for estimating μ 1 - μ 2

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Do Exercise 2/page334: Assumption 1. Both samples are large and independent on each other 2. Sample sizes may differ ) - (1 100 α % confidence interval for μ 1 - μ 2 is = 2 - 1 c α m s n s z y x c 2 2 2 1 + ± - 2
Example : Use the accompanying data to estimate with a 95% confidence interval the difference between true average compressive strength (N/mm 2 ) for 7 – day-old concrete specimens and true average strength for 28-day-old specimens. 7-day old: 1 n = 68 1 x = 26.99 1 s = 4.89 28-day old: 2 n = 74 2 x = 35.76 2 s = 6.43 Test whether there is a difference between the means…. Solution : A 95% confidence interval for the difference between the true average compressive strength for 7-day-old concrete specimens and the true average strength for 28-day-old concrete specimens is: ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 9 . 6 , 64 . 10 87 . 1 77 . 8 74 43 . 6 68 89 . 4 96 . 1 76 . 35 99 . 26 96 . 1 2 2 2 2 2 1 2 1 - - ± - + ± - + ± - m s n s x x HW assignment: 3

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Section 9.1: #3, 5ab, 7, 9, 11, 8* 9.2 Two-Sample t Test and CI for 2 1 μ μ - of Two Normal Populations We are interested in comparing two normal populations and in obtaining the CI for the difference in their true means. Sampling designs: independent vs. paired Confidence intervals for independent sample Confidence Intervals for matched samples Two independent samples n 1 - observations n 2 - observations x 1 : x 11, x 12, x 2 : x 21, x 21, ( The observations are independent). 4 μ 1 σ 1 μ 2 σ 2
Independent Samples : The sample values ) , ( 2 1 i i x x selected from one population are not related and somehow paired with the sample values selected from the other population. Dependent Samples : If the values in one sample are related to the values in the other sample, the samples are dependent . Such samples are often referred to as matched pairs or paired samples . Example: Independent samples: one group of subjects is treated with a drug, and a separate group is given a placebo. Matched pairs: (i) Blood pressure levels of subjects are taken before and after taking a drug. (Marks obtained by students in two tests).

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