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 under these conditions n 50 xs 12 68 S x 6 83 n large we can assume that the mean follows a normal distribution and use z table Sx x xs z n 80 So what do we need to do to find z 10 10 z 1 28 n 50 xs 12 68 S x 6 83 Plug in we get x 12 68 1 24 80 In your quest to design a new lightweight golf club you find what you think is an ideal material design combination You decide to quantify the performance of the club by firing golf balls at the head and measuring the ratio of the outgoing velocity of the ball to the incoming velocity coefficient of restitution COR 15 measurements of the COR give xs 8 735 S x 0 02456 Determine a 95 confidence interval on the mean COR n 15 x x tv P Sx n P x x t14 95 Sx n 95 What is this t14 95 2 145 n 15 S x 0 02456 xs 8 735 0 02456 x 8 735 2 145 15 8 735 0 014 95
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