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# Class Note for STAT 528 at OSU 35

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Date Created: 02/06/15
Stat 528 Autumn 2008 Elly Kaizar Power calculations Reading Section 64 o The power of a test 0 Cola bottles example 0 Power calculations the game plan 0 Power for a onesided Z test 0 Power for the two sided Z test o lncreasing the power in theory and in practice 0 Power calculations in MlNlTAB 0 Male executive blood pressure example The power of a test c We saw earlier that in tests of signi cance there are two types of error Each error has a given probability of occurring 1 The signi cance level type I error is oz Pltreject H0 when H0 is true 2 The probability of a type ll error is 6 Pfail to reject H0 when H0 is false 0 The power of the test is related to the probability of a type ll error lt is de ned by 1 6 1 Pfail to reject H0 when H0 is false Pltreject H0 when H0 is false 0 We need to be speci c about what 7when H 0 is false7 Ineans Cola bottles example Bottles of a popular cola drink are supposed to contain 300 ml of cola There is some variation from bottle to bottle because the lling machine is not perfectly precise Suppose that distribution of the contents is normal With stdev a 3 ml Suppose we want to carry out the following hypothesis test at signi cance level 005 based on a sample of six bottles of cola H0 u 300 versus Ha u lt 300 Calculate the power of this test against the alternative of u 298 What is the probability of type ll error in this case Power calculations the game plan 1 State the hypotheses 2 Work out the conditions to reject H 0 at signi cance level oz 3 Choose the particular case of when H0 is false 4 Now calculate the probability of rejecting H0 when H0 is false We will need to know the distribution of the test statistic when H0 is false A onesided ztest 0 Consider the following set of hypotheses Hozun0 versus HAzultu0 0 3 04 02 01 00 3 1 0 1 2 3 2 Zia 0 We reject H0 at signi cance level oz if z za E Ho UxW o Equivalently since our test statistic is z we reject H0 at signi cance level oz if E g L where L 0 za a Calculating the power c We Choose a speci c case of 7when H 0 is false Let u Ma 0 The power is the probability of rejecting H0 when H0 is false ie when u ya in this case PltY g L when H0 is false 0 Under this case of the alternative hypothesis Y has a Nma a distribution PW Lwhenuuagt Plt aNltua0 gt RV L PltZ ms Ma HO PZlt a lt Z ONE where Z is a standard normal RV In summary for the onesided ztest o Hypotheses H0 n 0 versus Ha n lt MO 0 Let the signi cance level be oz the population standard de viation be a and the sample size be 71 0 Then the power to detect a difference in the mean of d Ma no units is given by P Z S Za ad l 0 Note d is always negative for this onesided test Answering the cola bottles example a Calculate the power of this test against the alternative of u 298 What is the probability of type ll error in this case b Calculate the power of this test against the alternative of u 294 C ls the power against H 296 higher or lower than the power when u 294 Explain why this result makes sense The other onesided ztest 0 Now consider the hypotheses H0 n 0 versus Ha n gt MO 0 Let the signi cance level be oz the population standard de viation be a and the sample size be 71 o The power to detect a difference in the mean of d Ma no units is given by Plzzzradm 0 Note d is always positive for this onesided test Power for the twosided ztest 0 Now suppose we carry out a two sided Z test of H0 n 0 versus Ha n y MO 0 Let the signi cance level be oz the population standard de viation be a and the sample size be 71 0 Then the power to detect a difference in the mean of d Ma no units is given by d d This one takes a little more InatheInatical work to show 0 Note d can be negative or positive for the two sided test The power is symmetric about d O 10 Increasing the power in theory c We always want to maximize our power 0 We can increase the power in a number of ways 1 Increase oz 2 Move Ma further away from the null hypothesis value 0 Larger differences are easier to detect This increases d and hence the power of the test 3 lncrease the sample size More data gives more infor mation about in and thus we better discriminate between different values of u 4 Decrease a 11 Increasing the power in practice 0 Can be hard to increase the power in practice 1 We may not want to increase oz and make more type l errors 2 We often have a speci c distance between Ma and no in mind before we collect the data 3 Increasing the sample size can be expensive 4 Can be hard to decrease a 0 Keep a close eye on experimental procedure Beware 0 Example ln a medical study you are often required to com pute a speci c sample size n so that you can detect im portant differences 80 of the time using a 5 test The process to choose 71 is called a sample size calculation 12 Power calculations in MINITAB o Stat gt Power and Sample Size gt 1Sample Z 0 Under Options select the Alternative Hypothesis and Signi cance Level 0 Then enter any two of the following three items 1 Sample sizes 2 Differences 3 Power values 0 Enter the Standard deviation and click OK 13 Male executive blood pressure example Do middleaged male executives have different average blood pres sure than the general population The National Center for Health Statistics reports that the mean systolic blood pressure for males 35 to 44 years of age is 128 with standard deviation 15 The medical director of a company wants to investigate whether the company executives in this agegroup have blood pressures differ ing from the national average He wants to test H0 21 128 versus Ha 21 y 128 a What sample size is required to detect a difference in 2 units 80 of the time using an or 005 test b What sample size is required to detect a difference in 3 units 80 of the time using an or 005 test 14

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