# CU-Boulder MCEN 3037 - 6 (14 pages)

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Mcen 3037 - Data Analysis
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Going back to the z score z x Calculate z Table We can also write x z Probability P Gives the range over which we expect the next value to fall with the given probability x z What is z 95 Central limit theorem Population with a mean and a standard deviation Sample size n Sample mean x We can use the z score to find the probability of the mean falling within a given range x z n Standard deviation of the mean Example Let X denote the number of flaws in a 1 length of copper wire The probability mass function is given as P X 0 0 48 P X 3 0 01 P X 1 0 39 P X 2 0 12 We sample 100 wires from this population What is the probability that the average number of flaws per wire is less than 0 5 n Mean xi f xi 0 66 i 1 n 2 Variance xi f xi 0 5244 2 i 1 Large n average of flaws has a normal distribution with a mean Of 0 66 and a standard deviation of 2 n 0 0724 0 50 0 66 z 2 21 0 0724 P x 0 5 0 5 0 468 0 0136 We sample n points 1 Our best guess at the mean is the sample mean xs N N x i 1 i 1 2 xi xs The sample standard deviation is s x N 1 i 1 N 1 2 What do we want to find 1 Estimate where a single value will fall with a given probability 2 Estimate the true mean with a given probability CASE A n large take a lot of samples Let s shoot for a 95 confidence interval on the mean Sx x x z n 95 Steps 1 Calculate sample mean and standard deviation 2 Look up z value for 0 95 2 one sided table 0 475 Sx x x 1 96 n 95 95 Confidence Interval on Mean CASE B n small not a lot of data Let s shoot for a 95 confidence interval on the mean x x tv P Sx n 95 Degrees of freedom n 1 Data follows a t distribution rather than a normal distribution Confidence interval depends on the number of samples Huge range at small n Goes to z value at large n In a sample of 50 microdrills drilling a low carbon alloy steel the average lifetime expressed as the number of holes drilled before failure was 12 68 with a standard deviation of 6 83 Find an 80 confidence interval on the mean lifetime of microdrills used

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