Tests for a population proportion What is a population proportion No more than 10 of people in Colorado thought that the Colts would beat the Broncos Mean of a bunch of 0 s and 1 s 0 won t beat them 1 will be them The probability of success p 0 10 Is this claim true What are the mean and variance of a Bernoulli trial p 0 10 2 p 1 p From the central limit theorem proportions drawn from the population will have a normal distribution p 0 10 n p 1 p 0 3 n n Ok say we sample 400 fans and get 12 5 thinking that the colts will win H 0 10 How can we test our original claim H1 10 Let s calculate the z value 0 125 0 10 z 1 666 0 3 400 What is the P value 0 0475 What do we conclude Here this is close it is just below 0 05 and may be sufficient to reject the null hypothesis Let s try an example Spacer collars for a transmission countershaft have a thickness specification of 38 98 39 02 mm The process that manufactures the collars is supposed to be calibrated to give a mean thickness of 39 mm A sample of 6 collars is drawn and measured for thickness giving 39 030 38 997 39 012 39 008 39 019 and 39 002 Can you conclude that the process needs recalibration What do we do What hypothesis do we want to test H 0 39 H1 39 We have a small sample size so let s use the t statistic From our sample we have What is the t value t 5 x 39 0113 S x 0 011928 39 00113 39 2 327 0 011928 6 This is a 2 sided Hypothesis Test What is the P value t 5 2 327 What is the P value It looks like it s between 0 05 and 0 10 Should we recalibrate We may want to recalibrate We do not have sufficient evidence to reject the null hypothesis but we re also not too confident that it is in calibration It s best to consider an example We re worried that ambient temperature may have an effect on out manufacturing process for ball bearings Morning cool we sample 120 ball bearings x 5 068mm S x 0 011mm Afternoon hot we sample 65 ball bearings y 5 072mm S y 0 007mm Can we conclude that the bearings in the morning have smaller diameters Let s do a hypothesis test How do we set it up H 0 x y 0 H1 x y 0 Ok the difference in the sample means is 0 004 We use a combined standard deviation of the mean of x2 Nx y2 Ny 0 001327 Now we can calculate a z value 0 0040 0 0 z 3 01 0 001327 0 0040 0 0 z 3 01 0 001327 What is the P value H 0 x y 0 H1 x y 0 The P value is 0 001 What can we conclude We can reject the null hypothesis and conclude that the diameters are larger in the afternoon Measures plausibility of null hypothesis through P value What is the correct P value below which we reject H0 From a practical standpoint we really want a firm cutoff Why Here we may define a cutoff point for the P value fixed level test Choose a value for a If P a 0 a 1 Significance level Null hypothesis rejected Alternate hypothesis taken as truth Common choice a 0 05 Critical point value of test statistic that produces P a dividing line for test statistic One side H0 rejected rejection region Other side H0 not rejected Let s look at an example A new concrete mix is being evaluated The plan is to sample 100 blocks made with the new mix compute the sample mean compressive strength and then test H 0 1350 MPa H1 1350 MPa We assume from other similar tests that the sample standard deviation is 70 MPa Find the critical point and the rejection region if the test will be conducted at a significance level of 5 We need to find the critical point x x We have P 0 05 S x 70 N 100 What is our test statistic Critical value Rejection region z 1 65 H 0 1350 MPa H1 1350 MPa Rejection region x z Sx N x 1350 1 65 7 x 1361 55 P 0 05 S x 70 N 100 Determine x x z Sx N x 1350 1 65 7 x 1361 55
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