Chapter 8 Testing Hypotheses Part 2 Tests for Proportions and Two Sample Problems J C Wang Outline One Proportion Z Test One Proportion Z Test One Proportion Z Test Wald s Method One Proportion Z Test Scores Method Two Sample Problems Two Sample t Test Paired Samples t Test Paired Samples vs Two Independent Samples Types of Errors Two Types of Decision Errors An Example Prescription Drug Dosage Example I Suppose you are concerned if doctors were consistently adjusting dosages for weight of elderly patients You studied 200 prescriptions and found that 56 prescriptions the doctors failed to adjust the dosage I Let s consider the given information to answer the above question I Is there evidence that the proportion of doctors who failed to adjust dosage is less than 13 of the written prescription I Given x 56 n 200 p0 1 3 Prescription Drug Dosage Example I How many tails is this test A One Note the keyword less than I What are the hypotheses H0 p 13 vs H1 p 13 I What is the significance level 0 05 I What is the sample size n 200 I Now there are two methods to determine the SE Prescription Drug Dosage Example Wald s method I What is the SE using the Wald s Method r p 1 p 56 p 0 28 SE 0 03175 200 n I What is the test statistic 0 28 13 pt est H0 value z 1 6798 SE 0 03175 I What is the p value ofTails normalCDF TS 9999 1 normalCDF 1 6798 9999 0 0465 Prescription Drug Dosage Example Wald s method continued I Conclusion Is the p value significance level i e Is 0465 0 05 Yes Reject H0 there is statistically significant evidence that the proportion of doctors who failed to adjust dosage is less than 13 of the written prescriptions Prescription Drug Dosage Example Scores method I What is the SE using Scores method s r 1 1 p0 1 p0 3 1 3 0 03333 SE n 200 I What is the test statistic 0 28 13 pt est H0 Value z 1 6 SE 0 03333 I What is the p value ofTails normalCDF TS 9999 1 normalCDF 1 6 9999 0 0548 Prescription Drug Dosage Example Scores method continued I Conclusion Is the p value significance level i e Is 0537 0 05 No Do not reject H0 there is not enough evidence to conclude that the proportion of doctors who failed to adjust dosage is less than 13 of the written prescriptions Note Wald s method is statistically significant while the Scores method is not significant Prescription Drug Dosage Example Scores method using TI calculators Note The TI calculators use scores method I Do this STAT TESTS 1 PropZTest p0 33333 x 56 n 200 prop p0 CALCULATE I Readout 1 PropZTest prop 33333 z 1 6 P 0548 P 28 n 200 I Conclusion Do not reject H0 eg 1 prop Z test A well known business newspaper reported in March 2000 that a consultant had gathered data on how companies delivered their human resources services to employees It found that 233 of 295 companies surveyed use their intranet The article also revealed that in 1998 it was believed that 50 of companies used their intranet to deliver these services At the 5 level of significance is there evidence that the proportion has changed from the previous value of 50 STAT TESTS 1 PropZTest p0 5 x 233 n 295 prop 6 p0 CALCULATE 1 PropZTest prop 6 5 z 9 956004917 P 2 420455E 23 i e 0 P 7898305085 n 295 Since the p value 05 reject H0 and conclude that there is strong evidence that the proportion has changed from the previous value of 50 iClicker Question 8 3 iClicker Question 8 3 Gasoline Example An independent testing agency has been contracted to determine whether there is any difference in the gasoline mileage output of two different gasoline on the same model car Gas A is tested on 25 cars and produced a sample mean of 18 5 mpg with a sample SD of 4 6 mpg Gas B is tested on 24 cars and produced a sample mean of 19 3 mpg with a sample SD of 5 2 mpg At the 0 05 level of significance Is there evidence of a significant difference between the averages of the two gasoline performances Gasoline Example continued Stats x s n Gas A 18 5 4 6 25 Gas B 19 3 5 2 24 I How many tails in this test A two Note keyword is difference I What are the hypotheses H0 A B vs H1 A 6 B Gasoline Example continued I What is the significance level 0 05 I What are the sample sizes nA 25 and nB 24 I Should we pool the variances SD Yes if larger SD smaller SD 2 Since 5 2 4 6 1 13 is less than 2 we should indicate Pooled YES on our TI 83 calculator Gasoline Example continued using TI calculators I Do this STAT TESTS 2 SampTTEST x 1 18 5 Sx1 4 6 n1 25 x 2 19 3 Sx2 5 2 n2 24 1 6 2 pooled yes CALCULATE I Readout 2 SampTTest 1 6 2 t 5709832971 P 5707311948 df 47 Sxp 4 90279989 Gasoline Example continued I What is the conclusion Is p value i e is 0 5707 0 05 No do not reject H0 there is not enough evidence of a significant performance difference between the averages of the two gasolines Gasoline Example continued Suppose in addition to the above information as obtained from TI calculators you are interested in finding the SE Remember this equation x 1 x 2 1 2 t SE So SE x 1 x 2 1 2 0 8 0 1 4011 t 5710 This is much easier than finding the pooled SD and using q SE SE12 SE22 iClicker Question 8 4 iClicker Question 8 4 Paired Samples t Test blood pressure gauge example A digital blood pressure gauge manufactured for home use was tested recently in the following manner Eight individuals had their blood pressure taken first by a highly respected physician then by the home device Is there evidence that there is difference between results from the digital device and from physicians The data are shown below Person Phys Dev Diff 1 112 126 14 2 109 108 1 3 139 116 23 4 141 123 18 5 120 138 18 6 99 123 24 Blood Pressure Gauge Example continued I What are the hypotheses H0 D 0 vs H1 D 6 0 I What is the significance level 0 05 I What is the sample size n 8 7 128 119 9 8 118 122 4 Blood Pressure Gauge Example continued I Data entry editing STAT EDIT and enter pairs into L1 physicians and L2 home device then place cursor on L3 and do 2nd 2 2nd 1 i e L2 L1 I Do this STAT TESTS T TEST DATA 0 0 LIST L3 FREQ 1 6 0 CALCULATE I …