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Section8

# Section8 - Anna Lee Jery Stedinger Fall 2006 CEE 304...

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Anna Lee & Jery Stedinger Fall 2006 CEE 304 Section # 8 Handout Confidence Intervals for Small Samples: 1. An environmental engineer is studying the concentration of organics in a groundwater source for a rural water supply. Twelve samples yielded a sample mean and variance of 24.3 μ g/l and 81 ( μ g/l) 2 , respectively. a.) Construct a 99% confidence interval for the true mean concentration of organics in this source of water assuming individual observations have a normal distribution. α = 1%, n = 12, 3 . 24 = x , 81 2 = X s σ unknown Æ use t test So, the 100(1- α )% CI: + n s t x n s t x n n 1 , 2 / 1 , 2 / , α 106 . 3 11 , 005 . 0 1 , 2 / = = t t n (use t table in Devore) So, 99% CI: + 12 9 106 . 3 3 . 24 , 12 9 106 . 3 3 . 24 = (16.23, 32.37) b.) What is the probability the true mean value is contained in the particular interval you just calculated? 0, 1 Æ it either is, or it isn’t Suppose you are asked before any data is gathered (before computing the interval as above), the probability is 0.99 that the random interval + n s t x n s t x n n 1 , 005 . 0 1 , 005 . 0 , includes the true value of population mean μ . In other words, 99% of such intervals would cover μ , and 1% would fail to cover μ . 2. The sample average ultimate tensile strength for a sample of 35 high-strength magnetic alloy steel rings used in turbine generators was 152.3 ksi, while the sample standard deviation was 4.8 ksi. Obtain a 99% confidence interval for the true average tensile

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Section8 - Anna Lee Jery Stedinger Fall 2006 CEE 304...

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