# Lecture09 - 23 May 2003 Biostatistics 6650-L9 1 Todays...

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23 May 2003 Biostatistics 6650--L9 1

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23 May 2003 Biostatistics 6650--L9 2 Today’s Schedule Hypothesis Testing: one sample Normal test review 3 known(one sample z-test) 3 unknown (one sample t-test) Paired t-test Binomial proportions Making inferences from hypothesis tests
23 May 2003 Biostatistics 6650--L9 3 Hypothesis Testing: Normal X~N(3,3 2 )--- One Sided Test Test Ho: 3=3 o against Ha: 3 > 3 o Assume 3 is known , take a sample of size n Reject Ho if: or P-value < α , P=Pr(Z> z obs | 3 o ) = 1-3(z obs ) G G z <-/ /S / q 1 , where z , 1-sample z-test o obs obs X z Z n α μ σ - - = 1 Z α - α FTR o 1- Cμ +z σ/ n α = o μ α Dist’n of Z ' Dist n of X Reject obs z P Reject 0

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23 May 2003 Biostatistics 6650--L9 4 Hypothesis Testing: Normal X~N(3,3 2 )--- Two Sided Test Test Ho: 3=3 o against Ha: 3 = 3 o Assume 3 is known , take a sample of size n Reject Ho if: G G <- / / S / 1 / 2 | | , where z o obs obs X z Z n α μ σ - - = 1 / 2 Z α - / 2 α FTR Dist’n of Z Reject obs z Reject / 2 α obs z - P/2 P/2 /2 Z α 0 P-value Calculation: z obs < 0: P= 2 x Pr(Z< z obs )=2x3(z obs ) z obs >0: P= 2xPr(Z>z obs ) =2x(1-3(z obs ) ) General: P=2x(1- Φ ( |z obs | ) )
23 May 2003 Biostatistics 6650--L9 5 Hypothesis Testing How likely is it that you will know σ? What if you don’t know σ?

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23 May 2003 Biostatistics 6650--L9 6 Hypothesis Testing: 1 -sample t-test One-sided : Ho: 3=3 o Ha: 3 > 3 o P-value=Pr( t n-1 > t obs | 3=3 o ) Reject Ho if: t n-1 0 t obs One-sided : Ho: 3=3 o Ha: 3 < 3 o P-value=Pr( t n-1 < t obs | 3=3 o ) Reject Ho if: t n-1 t obs 0 1,1 n-1 obs or P=Pr( t > t ) α o obs n X t t s n α μ - - - = 1, n-1 obs or P=Pr( t < t ) α o obs n X t t s n α μ - - = <
23 May 2003 Biostatistics 6650--L9 7 Hypothesis Testing: 1-sample t-test X~N(3,3 2 ), 3 is unknown : Two-sided test Test Ho: 3=3 o against Ha: 3= 3 o Reject Ho if: t n-1 0 t obs P/2 P/2 α n-1,1- 2 t α n-1, 2 t 1, 1 / 2 n-1 obs obs n-1 obs obs n-1 obs | | , where or if Pα, where P = 2 Pr( t < t ), if t 0 = 2 Pr( t > t ), if t >0 = 2 Pr( t > | t | ), symmetry o obs n obs X t t t s n α μ - - - = g g g

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23 May 2003 Biostatistics 6650--L9 8 Hypothesis Testing: 1-sample t-test Example: Body Mass Index in Male Office Workers BMI= wt(kg)/ht(m) 2 Upper limit of normal BMI for adult males with good health =25 (Royal College of Physicians, 1983) Sample of 25 male office workers, Ho: 3 o =25 Ha: 3 o > 25 3=.05 Reject Ho if: t obs = (28.12-25) / (3.18/5) > t 24,.95 =1.71 = 4.91 > 1.71 , Reject Ho or equivalently if P= Pr(t 24 >t obs ) < .05 P=Pr(t 24 >4.91), P<.0001 Conclude that male office workers have significantly (P<.0001, 1-sided) higher BMI’s than adult males in good health . 28.12, s=3.18 x =
23 May 2003 Biostatistics 6650--L9 9 Hyp. Testing: 1-sample t-test Distributions bmi 20 25 30 35 Moments Mean 28.12000 Std Dev 3.17962 Std Err Mean 0.63592 upper 95% Mean 29.43247 lower 95% Mean 26.80753 N 25.00000 Test Mean=value Hypothesized Value 25 Actual Estimate 28.12 df 24 Std Dev 3.17962 t Test Test Statistic 4.9062 Prob > |t| <.0001 Prob > t <.0001 Prob < t 1.0000 In JMP: analyze distribution specify BMI(conts) •test mean(supply 3 o )

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23 May 2003 Biostatistics 6650--L9 10 Hypothesis Testing: Paired t-test Paired versus Two-sample tests Two samples are independent if the data for one
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