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Hypothesis Tests on the Mean H0: = 0 H1: 0 Reject H 0 if Fail to reject H 0 if Z0 X 0 n Z0 z / 2 or Z0 z / 2 z / 2 Z0 z / 2 1 Hypothesis Tests (one side) H0: = 0 Reject H0 if H1: > 0 Z0 z H0: = 0 Reject H0 if H1: < 0 Z0 z 2 Example: two-sided test Suppose Guido takes a random sample of n=25 and obtains an average burn rate of 51.3 cm/s. Specs require that burn rate must be 50 cm/s, and the standard deviation is known to be 2 cm/s. He decides to specify a type I error probability (significance level) of 0.05. What conclusions can be drawn? 3 P-value The P-value is the smallest level of significance that would lead to rejection of the null hypothesis H0 with the given data. 2 [ 1 (| z0 |)] for test : H 0 : 0 H 1 : 0 P 1 ( z0 ) for test : H 0 : 0 H 1 : 0 ( z ) for test : H 0 : 0 H 1 : 0 0 4 Probability of Type II Error The probability of the type II error is the probability that Z0 falls between -z/2 and z/2 given that H1 is true. n n z / 2 z / 2 5 Requirement Sample Size For two - tailed tests ( z / 2 z ) 2 n 2 2 , 0 For one - tailes tests ( z z ) 2 n 2 2 , 0 6 Example (cont.) Suppose Guido wants to design the burn rate test so that if the true mean burn rate differs from 50 cm/s by as much as 1 cm/s, the test will detect this (i.e. reject the null hypothesis) w.p. 0.90. Determine the sample size required to detect this departure. 7 Confidence Intervals If x is the sample mean of a random sample of size n from a population with variance , a 100(1 )% CI on is given by 2 , x z / 2 x z / 2 n n 8 Example Construct a 95% confidence interval for the burn rate. 25 samples were taken. The sample mean is 51.3 and the standard deviation is 2. What is the relationship between hypothesis testing and confidence intervals? 9 Error Bound Sample Size If x is an estimate of , we can be 100(1 )% confident that the error |x | will not exceed a specified amount E when the sample size is: z / 2 n E 2 10 Example Suppose that we wanted the error in estimating the mean burn rate of the rocket propellant to less than 1.5 cm/s, with 95% confidence. What is the required sample size? 11 One-Sided CIs The 100(1 )% upper-CI for is x z n The 100(1 )% lower-CI for is x z n 12 General CIs Let ˆ be an estimator for . The 100(1 )% CI, (L,U ), is given by: P L ˆ U 1 13