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 Type I and Type II Errors Fixed level testing results in a firm decision In what way can we be wrong H 0 1350 MPa H1 1350 MPa We reject H0 when in fact it is TRUE TYPE I error We fail to reject H0 when in fact it is FALSE TYPE II error For a successful experimental plan we want to make the probability of having type I and type II errors small We can now look at the probabilities of each type of error TYPE I error We reject H0 when in fact it is TRUE Let s use H 0 1350 MPa a 0 05 H1 1350 MPa Mean lies in here x Probability of a type I always less than or equal to a Rejection region TYPE II error We fail to reject H0 when in fact it is FALSE To do this we compute the POWER of a test Power 1 P type II We want Power to be large Why H 0 1350 MPa H1 1350 MPa If the power is large the probability of a type II error is small Steps for computing the power 1 Select a specific value for the alternate hypothesis 2 Compute the rejection region 3 Compute the probability that the test statistic falls within the rejection region if the alternate hypothesis is true Let s consider the test We ll use a level of 5 H 0 80 MPa H1 80 MPa S X 5 N 50 First find the rejection region critical point z x 80 5 50 What is z z x 80 1 645 5 50 So the critical point is x 81 16 What does this mean If the value is greater than or equal to 81 16 the null hypothesis is rejected Here s how this looks Next Compute the probability that the test statistic falls within the rejection region if the alternate hypothesis is true Now we have to choose a specific value assume alternate hypothesis true to compute the power Select alternative hypothesis of 81 Now we want to figure out If the true mean is 81 what is the probability that we reject the null hypothesis Draw a distribution around 81 alternate distribution and determine the probability of being in the rejection region z 81 16 81 0 23 5 50 The power is 0 5 0 0910 0 409 H 0 80 MPa H1 80 MPa If the true mean is 81 Ho false we only reject the null hypothesis 40 of the time TYPE II error 60 of the time Steps for computing the power 1 Select a specific value for the alternate hypothesis 2 Compute the rejection region 3 Compute the probability that the test statistic falls within the rejection region if the alternate hypothesis is true A tire company claims that the lifetimes of its tires average 50 000 miles The standard deviation of tire lifetimes is known to be 5000 miles You sample 100 tires and will test the hypothesis that the mean tire lifetime is at least 50 000 miles against the alternative that it is less Assume in fact that the true mean lifetime is 49 500 miles A tire company claims that the lifetimes of its tires average 50 000 miles The standard deviation of tire lifetimes is known to be 5000 miles You sample 100 tires and will test the hypothesis that the mean tire lifetime is at least 50 000 miles against the alternative that it is less Assume in fact that the true mean lifetime is 49 500 miles a State the null and alternate hypothesis Which is true b It is decided to reject Ho if the sample mean is less than 49 400 Find the level and power of this test b If the test is made at the 5 level what is the power See Class 18 slides for solution