Math 1710 Class 24 Examples Power 2-Sample CIs Dr. Allen Back - PowerPoint PPT Presentation

Lost Luggage Ch. 20 #29 Math 1710 Class 24 V1 Sample Size for a Given MOE Examples Hypotheses: Power H 0 : p = . 9 (or p ≥ . 9 ) 2-Sample CI’s and HT’s H a : p < . 9 2-Sample Examples Power Example Power Properties

Lost Luggage Ch. 20 #29 Math 1710 Class 24 V1 Sample Size for a Given MOE Examples Power Random Sampling: Hopefully. 2-Sample CI’s and HT’s 2-Sample Examples Power Example Power Properties

Lost Luggage Ch. 20 #29 Math 1710 Class 24 V1 Sample Size for a Given MOE Examples 10% Condition: Depends on definition of the population. Power Hopefully much less than 10% of population. 2-Sample CI’s and HT’s Certainly much less than total volume of luggage. 2-Sample Examples Power Example Power Properties

Lost Luggage Ch. 20 #29 Math 1710 Class 24 V1 Sample Size for a Given MOE Success/Failure: Examples � 103 � Power 122 ≥ 10 122 2-Sample CI’s � 19 and HT’s � 2-Sample 122 ≥ 10 Examples 122 Power Example Power Properties

Power, Type I/II Errors, α , and β Math 1710 Class 24 V1 Sample Size for a Given MOE Examples Power Given H 0 : p = p 0 , there are two ways an HT can report an 2-Sample CI’s inaccurate result: and HT’s 2-Sample Examples Power Example Power Properties

Power, Type I/II Errors, α , and β Math 1710 Class 24 V1 Sample Size Given H 0 : p = p 0 , there are two ways an HT can report an for a Given MOE inaccurate result: H 0 true H 0 false Examples Retain H 0 Good Type II Error Power 2-Sample CI’s probability = β and HT’s depends on value of p 2-Sample Examples Reject H 0 Type I Error Good Power probability = α probability = 1- β Example = power Power Properties

Power, Type I/II Errors, α , and β Math 1710 Class 24 V1 Given H 0 : p = p 0 , there are two ways an HT can report an inaccurate result: Sample Size for a Given Type I Error Examples: MOE Examples a: False Positive in a diagnosis; i.e. deciding a Power person is sick when they really are not. (H 0 : The 2-Sample CI’s person is well.) and HT’s 2-Sample b: Convicting an innocent person. (H 0 : The person Examples is innocent.) Power Example c: Producer Risk; the chance that a good good Power shipment erroneously fails a a test for quality. Properties (H 0 : A product meets a specification.)

Power, Type I/II Errors, α , and β Given H 0 : p = p 0 , there are two ways an HT can report an Math 1710 Class 24 inaccurate result: V1 Type I Error Examples: Sample Size a: False Positive in a diagnosis; i.e. deciding a for a Given MOE person is sick when they really are not. (H 0 : The Examples person is well.) Power b: Convicting an innocent person. (H 0 : The person 2-Sample CI’s is innocent.) and HT’s c: Producer Risk; the chance that a good good 2-Sample Examples shipment erroneously fails a a test for quality. Power Example (H 0 : A product meets a specification.) Power Type II Error Examples: Properties a: False Negative; missing a sick person. b: Letting a guilty person go free. c: Consumer Risk; the chance that a bad shipment erroneously passes a a test for quality.

300 stdnts, 60% approve; 200 faclty, 65% Math 1710 Class 24 V1 Sample Size for a Given MOE Examples Power A significant difference? CI for difference in rates of approval? 2-Sample CI’s and HT’s 2-Sample Examples Power Example Power Properties

300 stdnts, 60% approve; 200 faclty, 65% Math 1710 Class 24 V1 Sample Size for a Given MOE Examples A significant difference? CI for difference in rates of approval? Power Let p 1 and p 2 denote the true proportions of students and 2-Sample CI’s and HT’s faculty that approve. 2-Sample Examples Power Example Power Properties

300 stdnts, 60% approve; 200 faclty, 65% Math 1710 Class 24 V1 Sample Size for a Given MOE Examples A significant difference? CI for difference in rates of approval? Power 2-sample inference based on the sampling distribution of 2-Sample CI’s and HT’s p 1 − ˆ ˆ p 2 2-Sample Examples Power Example Power Properties

300 stdnts, 60% approve; 200 faclty, 65% Math 1710 Class 24 V1 A significant difference? CI for difference in rates of approval? Sample Size for a Given 2-sample inference based on the sampling distribution of MOE p 1 − ˆ ˆ p 2 Examples Power µ = p 1 − p 2 2-Sample CI’s and HT’s = p 1 q 1 + p 2 q 2 Var ( ˆ p 1 − ˆ p 2 ) = Var ( ˆ p 1 ) + Var ( ˆ p 2 ) 2-Sample n 1 n 2 Examples Power � p 1 q 1 + p 2 q 2 Example SD ( ˆ p 1 − ˆ p 2 ) = n 1 n 2 Power Properties

300 stdnts, 60% approve; 200 faclty, 65% A significant difference? CI for difference in rates of approval? Math 1710 Class 24 2-sample inference based on the sampling distribution of V1 p 1 − ˆ ˆ p 2 Sample Size for a Given Samp. Dist. of ˆ p 1 − ˆ p 2 : N ( p 1 − p 2 , SD ( ˆ p 1 − ˆ p 2 )) MOE Examples Power 2-Sample CI’s and HT’s 2-Sample Examples Power Example Power Properties

300 stdnts, 60% approve; 200 faclty, 65% Math 1710 Class 24 V1 Sample Size for a Given MOE Examples 2-sample inference based on the sampling distribution of Power p 1 − ˆ ˆ p 2 2-Sample CI’s and HT’s Find a CI for p 1 − p 2 : 2-Sample Examples Power Example Power Properties

300 stdnts, 60% approve; 200 faclty, 65% Math 1710 Class 24 V1 Sample Size for a Given MOE 2-sample inference based on the sampling distribution of Examples p 1 − ˆ ˆ p 2 Power Find a CI for p 1 − p 2 : 2-Sample CI’s Since we don’t know p 1 and p 2 , we can’t directly compute and HT’s SD ( ˆ p 1 − ˆ p 2 ). 2-Sample Examples So we use SE ( ˆ p 1 − ˆ p 2 ) instead. Power Example Power Properties

300 stdnts, 60% approve; 200 faclty, 65% Math 1710 Class 24 V1 Sample Size for a Given MOE 2-sample inference based on the sampling distribution of Examples p 1 − ˆ ˆ p 2 Power Find a CI for p 1 − p 2 : 2-Sample CI’s and HT’s � p 1 ˆ ˆ q 1 + ˆ p 2 ˆ q 2 2-Sample SE ( ˆ p 1 − ˆ p 2 ) = Examples n 1 n 2 Power Example Power Properties

300 stdnts, 60% approve; 200 faclty, 65% Math 1710 Class 24 V1 2-sample inference based on the sampling distribution of Sample Size p 1 − ˆ ˆ p 2 for a Given MOE Find a CI for p 1 − p 2 : Examples � Power p 1 ˆ ˆ q 1 + ˆ p 2 ˆ q 2 SE ( ˆ p 1 − ˆ p 2 ) = 2-Sample CI’s n 1 n 2 and HT’s 2-Sample Examples Same argument as in the 1-sample case gives a CI for p 1 − p 2 of Power Example p 1 − ˆ ˆ p 2 ± z ∗ SE ( ˆ p 1 − ˆ p 2 ) . Power Properties

300 stdnts, 60% approve; 200 faclty, 65% 2-sample inference based on the sampling distribution of Math 1710 Class 24 p 1 − ˆ ˆ p 2 V1 Find a CI for p 1 − p 2 : Sample Size for a Given � MOE p 1 ˆ ˆ q 1 + ˆ p 2 ˆ q 2 SE ( ˆ p 1 − ˆ p 2 ) = Examples n 1 n 2 Power 2-Sample CI’s Same argument as in the 1-sample case gives a CI for p 1 − p 2 of and HT’s 2-Sample Examples p 1 − ˆ ˆ p 2 ± z ∗ SE ( ˆ p 1 − ˆ p 2 ) . Power Example Power Here we have Properties � . 6 · . 4 300 + . 65 · . 35 SE ( ˆ p 1 − ˆ p 2 ) = = . 0440 . 200

300 stdnts, 60% approve; 200 faclty, 65% 2-sample inference based on the sampling distribution of Math 1710 Class 24 p 1 − ˆ ˆ p 2 V1 Find a CI for p 1 − p 2 : Sample Size Same argument as in the 1-sample case gives a CI for p 1 − p 2 of for a Given MOE p 1 − ˆ ˆ p 2 ± z ∗ SE ( ˆ p 1 − ˆ p 2 ) . Examples Power 2-Sample CI’s and HT’s Here we have 2-Sample Examples � . 6 · . 4 300 + . 65 · . 35 SE ( ˆ p 1 − ˆ p 2 ) = = . 0440 . Power 200 Example Power Properties A 95% CI for p 1 − p 2 is: ( . 6 − . 65) ± 1 . 96 · . 0440 = − . 05 ± . 0863 = ( − . 1363 , . 0363) .

300 stdnts, 60% approve; 200 faclty, 65% Math 1710 Class 24 V1 Sample Size for a Given MOE Examples Carry out an HT at a sig level of α = . 05 of whether faculty Power and student approval rates are different. Calculate the P-value 2-Sample CI’s and HT’s as well. 2-Sample Examples Power Example Power Properties

300 stdnts, 60% approve; 200 faclty, 65% Math 1710 Class 24 V1 Sample Size for a Given MOE Carry out an HT at a sig level of α = . 05 of whether faculty Examples and student approval rates are different. Calculate the P-value Power as well. 2-Sample CI’s and HT’s Without the request for P-value, we could use the CI above. 2-Sample But for the P-value we need to use “Method 1.” Examples Power Example Power Properties

300 stdnts, 60% approve; 200 faclty, 65% Math 1710 Class 24 V1 Sample Size for a Given MOE Carry out an HT at a sig level of α = . 05 of whether faculty Examples and student approval rates are different. Calculate the P-value Power as well. 2-Sample CI’s Our hypotheses are: and HT’s H 0 : p 1 = p 2 2-Sample Examples H a : p 1 � = p 2 Power Example Power Properties

300 stdnts, 60% approve; 200 faclty, 65% Math 1710 Class 24 V1 Carry out an HT at a sig level of α = . 05 of whether faculty Sample Size and student approval rates are different. Calculate the P-value for a Given as well. MOE Examples Our hypotheses are: Power H 0 : p 1 = p 2 2-Sample CI’s and HT’s H a : p 1 � = p 2 2-Sample A twist enters. We are only interested in the reasonableness of Examples our observed ˆ p 1 − ˆ p 2 with respect to the sampling dist if H 0 is Power Example true. There are many such distributions (since we don’t know Power the common value of p 1 = p 2 to use.) In particular what we did Properties with SE ( ˆ p 1 − ˆ p 2 ) above does not fit the p 1 = p 2 situation.

300 stdnts, 60% approve; 200 faclty, 65% Carry out an HT at a sig level of α = . 05 of whether faculty Math 1710 Class 24 and student approval rates are different. Calculate the P-value V1 as well. Sample Size Our hypotheses are: for a Given MOE H 0 : p 1 = p 2 Examples H a : p 1 � = p 2 Power We resolve this conflict by making our best estimate of the 2-Sample CI’s and HT’s common value of p 1 and p 2 , namely the weighted average 2-Sample Examples p pooled = n 1 ˆ p 1 + n 2 ˆ p 2 ˆ Power n 1 + n 2 Example Power and then Properties � p pooled � � + � p pooled � q pooled q pooled SE pooled ( ˆ p 1 − ˆ p 2 ) = . n 1 n 2

300 stdnts, 60% approve; 200 faclty, 65% Math 1710 Class 24 V1 Carry out an HT at a sig level of α = . 05 of whether faculty and student approval rates are different. Calculate the P-value Sample Size for a Given as well. MOE Here the weighted average is Examples Power p pooled = 300 · . 60 + 200 · . 65 2-Sample CI’s ˆ = . 6 · 300 + . 4 · . 65 = . 62 and HT’s 200 + 300 2-Sample Examples and then Power Example � . 62 · . 38 + . 62 · . 38 Power SE pooled ( ˆ p 1 − ˆ p 2 ) = = . 0443 . Properties 300 200

300 stdnts, 60% approve; 200 faclty, 65% Math 1710 Class 24 V1 Sample Size for a Given Carry out an HT at a sig level of α = . 05 of whether faculty MOE and student approval rates are different. Calculate the P-value Examples as well. Power Our z-statistic is 2-Sample CI’s and HT’s p 1 − ˆ ˆ p 2 p 2 ) = − . 05 2-Sample z = . 0443 = − 1 . 12 . Examples SE pooled ( ˆ p 1 − ˆ Power Example Power Properties

300 stdnts, 60% approve; 200 faclty, 65% Carry out an HT at a sig level of α = . 05 of whether faculty Math 1710 Class 24 and student approval rates are different. Calculate the P-value V1 as well. Sample Size for a Given Approx Samp. Dist. of ˆ p 1 − ˆ p 2 : N (0 , . 0443) if H 0 is true. MOE Examples Power 2-Sample CI’s and HT’s 2-Sample Examples Power Example Power Properties

300 stdnts, 60% approve; 200 faclty, 65% Carry out an HT at a sig level of α = . 05 of whether faculty Math 1710 Class 24 and student approval rates are different. Calculate the P-value V1 as well. Sample Size for a Given Tail Prob. is P ( Z < − 1 . 12) = . 1314. MOE Examples Power 2-Sample CI’s and HT’s 2-Sample Examples Power Example Power Properties

300 stdnts, 60% approve; 200 faclty, 65% Carry out an HT at a sig level of α = . 05 of whether faculty Math 1710 Class 24 and student approval rates are different. Calculate the P-value V1 as well. Sample Size for a Given P-value=2(Tail Prob.) = 2(.1314) = .2628 MOE Examples Power 2-Sample CI’s and HT’s 2-Sample Examples Power Example Power Properties

300 stdnts, 60% approve; 200 faclty, 65% Math 1710 Class 24 V1 Sample Size for a Given MOE Examples Carry out an HT at a sig level of α = . 05 of whether faculty Power and student approval rates are different. Calculate the P-value 2-Sample CI’s as well. and HT’s Our P-value is larger than α = . 05, so we retain H 0 . 2-Sample Examples Power Example Power Properties

Chicken Contamination Math 1710 Class 24 V1 Sample Size for a Given MOE Suppose: Examples Power 1 33% of 75 Perdue chickens contaminated. 2-Sample CI’s 2 45% of 75 Store Brand chickens contaminated. and HT’s 2-Sample 3 56% of 75 Tyson chickens contaminated. Examples Power Example Power Properties

Chicken Contamination Math 1710 Class 24 V1 1 33% of 75 Perdue chickens contaminated. Sample Size for a Given 2 45% of 75 Store Brand chickens contaminated. MOE Examples 3 56% of 75 Tyson chickens contaminated. Power Questions: 2-Sample CI’s and HT’s 1 Purdue safer than Store Brand? 2-Sample Examples 2 Tyson safer than Store Brand? Power 3 Tyson different in safety than Store Brand? Example Power 4 Confidence interval for difference in safety between Store Properties Brand and Tyson?

Chicken Contamination Math 1710 Class 24 V1 Sample Size for a Given MOE Examples 1 33% of 75 Perdue chickens contaminated. Power 2 45% of 75 Store Brand chickens contaminated. 2-Sample CI’s and HT’s 3 56% of 75 Tyson chickens contaminated. 2-Sample Examples Question: Purdue safer than Store Brand? Power Example Power Properties

Chicken Contamination Math 1710 Class 24 V1 Sample Size for a Given MOE 1 33% of 75 Perdue chickens contaminated. Examples 2 45% of 75 Store Brand chickens contaminated. Power 3 56% of 75 Tyson chickens contaminated. 2-Sample CI’s and HT’s Question: Purdue safer than Store Brand? 2-Sample Examples Notation: Let p 1 denote the proportion of Purdue which are Power contaminated and p 2 the proportion for Store Brand. Example Power Properties

Chicken Contamination Math 1710 Class 24 V1 1 33% of 75 Perdue chickens contaminated. Sample Size for a Given 2 45% of 75 Store Brand chickens contaminated. MOE Examples 3 56% of 75 Tyson chickens contaminated. Power Question: Purdue safer than Store Brand? 2-Sample CI’s and HT’s Notation: Let p 1 denote the proportion of Purdue which are 2-Sample contaminated and p 2 the proportion for Store Brand. Examples Hypotheses: Power Example H 0 : p 1 = p 2 (or p 1 ≥ p 2 ) Power Properties H a : p 1 < p 2

Chicken Contamination Math 1710 Class 24 V1 1 33% of 75 Perdue chickens contaminated. Sample Size for a Given 2 45% of 75 Store Brand chickens contaminated. MOE Examples 3 56% of 75 Tyson chickens contaminated. Power Hypotheses: 2-Sample CI’s and HT’s H 0 : p 1 = p 2 (or p 1 ≥ p 2 ) 2-Sample Examples H a : p 1 < p 2 Power Example p pooled = . 33 · 75 + . 45 · 75 Power ˆ = . 39 Properties 75 + 75

Chicken Contamination Math 1710 Class 24 1 33% of 75 Perdue chickens contaminated. V1 2 45% of 75 Store Brand chickens contaminated. Sample Size 3 56% of 75 Tyson chickens contaminated. for a Given MOE Hypotheses: Examples Power H 0 : p 1 = p 2 (or p 1 ≥ p 2 ) 2-Sample CI’s H a : p 1 < p 2 and HT’s 2-Sample Examples p pooled = . 33 · 75 + . 45 · 75 ˆ = . 39 Power 75 + 75 Example Power Properties � 1 � 75 + 1 � SE pooled ( ˆ p 1 − ˆ p 2 ) = . 39 · . 61 = . 0796 . 75

Chicken Contamination Math 1710 Class 24 V1 1 33% of 75 Perdue chickens contaminated. Sample Size for a Given 2 45% of 75 Store Brand chickens contaminated. MOE Examples 3 56% of 75 Tyson chickens contaminated. Power Hypotheses: 2-Sample CI’s and HT’s H 0 : p 1 = p 2 (or p 1 ≥ p 2 ) 2-Sample Examples H a : p 1 < p 2 Power Example z = ˆ p 1 − ˆ p 2 = − . 12 Power . 0796 = − 1 . 51 . Properties SE pooled

Chicken Contamination Math 1710 1 33% of 75 Perdue chickens contaminated. Class 24 2 45% of 75 Store Brand chickens contaminated. V1 3 56% of 75 Tyson chickens contaminated. Sample Size for a Given MOE z = ˆ p 1 − ˆ p 2 = − . 12 . 0796 = − 1 . 51 . Examples SE pooled Power 2-Sample CI’s and HT’s 2-Sample Examples Power Example Power Properties N(0,1)

Chicken Contamination Math 1710 Class 24 1 33% of 75 Perdue chickens contaminated. V1 2 45% of 75 Store Brand chickens contaminated. Sample Size for a Given 3 56% of 75 Tyson chickens contaminated. MOE Examples Hypotheses: Power H 0 : p 1 = p 2 (or p 1 ≥ p 2 ) 2-Sample CI’s and HT’s H a : p 1 < p 2 2-Sample Examples z = ˆ p 1 − ˆ p 2 = − . 12 Power . 0796 = − 1 . 51 . Example SE pooled Power Properties P-value = tail probability = P ( Z < − 1 . 51) = . 0655 . At a level of α = . 05, we’d retain H 0 . Purdue might not be safer.

Chicken Contamination Math 1710 Class 24 V1 Sample Size for a Given MOE Examples 1 33% of 75 Perdue chickens contaminated. Power 2 45% of 75 Store Brand chickens contaminated. 2-Sample CI’s and HT’s 3 56% of 75 Tyson chickens contaminated. 2-Sample Examples Question: Tyson safer than Store Brand? Power Example Power Properties

Chicken Contamination Math 1710 Class 24 V1 Sample Size for a Given MOE 1 33% of 75 Perdue chickens contaminated. Examples 2 45% of 75 Store Brand chickens contaminated. Power 3 56% of 75 Tyson chickens contaminated. 2-Sample CI’s and HT’s Question: Tyson safer than Store Brand? 2-Sample Examples Notation: Let p 2 denote the proportion of Store Brand which Power are contaminated and p 3 the proportion for Tyson. Example Power Properties

Chicken Contamination Math 1710 Class 24 V1 1 33% of 75 Perdue chickens contaminated. Sample Size for a Given 2 45% of 75 Store Brand chickens contaminated. MOE Examples 3 56% of 75 Tyson chickens contaminated. Power Question: Tyson safer than Store Brand? 2-Sample CI’s and HT’s Notation: Let p 2 denote the proportion of Store Brand which 2-Sample are contaminated and p 3 the proportion for Tyson. Examples Hypotheses: Power Example H 0 : p 3 = p 2 (or p 3 ≥ p 2 ) Power Properties H a : p 3 < p 2

Chicken Contamination Math 1710 Class 24 V1 Sample Size for a Given MOE 1 33% of 75 Perdue chickens contaminated. Examples 2 45% of 75 Store Brand chickens contaminated. Power 3 56% of 75 Tyson chickens contaminated. 2-Sample CI’s and HT’s 2-Sample p pooled = . 45 · 75 + . 56 · 75 Examples ˆ = . 505 75 + 75 Power Example Power Properties

Chicken Contamination Math 1710 Class 24 V1 1 33% of 75 Perdue chickens contaminated. Sample Size 2 45% of 75 Store Brand chickens contaminated. for a Given MOE 3 56% of 75 Tyson chickens contaminated. Examples Power p pooled = . 45 · 75 + . 56 · 75 2-Sample CI’s ˆ = . 505 and HT’s 75 + 75 2-Sample Examples Power Example � 1 � Power 75 + 1 � Properties SE pooled ( ˆ p 2 − ˆ p 3 ) = . 505 · . 495 = . 0816 . 75

Chicken Contamination Math 1710 Class 24 V1 Sample Size for a Given 1 33% of 75 Perdue chickens contaminated. MOE Examples 2 45% of 75 Store Brand chickens contaminated. Power 3 56% of 75 Tyson chickens contaminated. 2-Sample CI’s and HT’s 2-Sample z = ˆ p 2 − ˆ p 3 = − . 11 Examples . 0816 = − 1 . 35 . SE pooled Power Example Power Properties

Chicken Contamination Math 1710 1 33% of 75 Perdue chickens contaminated. Class 24 2 45% of 75 Store Brand chickens contaminated. V1 3 56% of 75 Tyson chickens contaminated. Sample Size for a Given MOE z = ˆ p 2 − ˆ p 3 = − . 11 . 0816 = − 1 . 35 . Examples SE pooled Power 2-Sample CI’s and HT’s Which side provides as much or more support for H a of 2-Sample p 3 < p 2 ? Examples Power Example Power Properties

Chicken Contamination Math 1710 1 33% of 75 Perdue chickens contaminated. Class 24 2 45% of 75 Store Brand chickens contaminated. V1 3 56% of 75 Tyson chickens contaminated. Sample Size for a Given MOE z = ˆ p 2 − ˆ p 3 = − . 11 . 0816 = − 1 . 35 . Examples SE pooled Power 2-Sample CI’s and HT’s Which side provides as much or more support for p 3 < p 2 ? 2-Sample Examples Power Example Power Properties

Chicken Contamination Math 1710 Class 24 V1 Sample Size for a Given 1 33% of 75 Perdue chickens contaminated. MOE Examples 2 45% of 75 Store Brand chickens contaminated. Power 3 56% of 75 Tyson chickens contaminated. 2-Sample CI’s and HT’s Our statistic provides no support for H a so we immediately 2-Sample retain H 0 . Examples It is a matter of convention whether we’d view the p-value as Power Example . 5 or even larger. Power Properties

Chicken Contamination Math 1710 Class 24 V1 Sample Size for a Given MOE Examples 1 33% of 75 Perdue chickens contaminated. Power 2 45% of 75 Store Brand chickens contaminated. 2-Sample CI’s and HT’s 3 56% of 75 Tyson chickens contaminated. 2-Sample Examples Question: Tyson different in safety than Store Brand? Power Example Power Properties

Chicken Contamination Math 1710 Class 24 V1 Sample Size for a Given MOE 1 33% of 75 Perdue chickens contaminated. Examples 2 45% of 75 Store Brand chickens contaminated. Power 3 56% of 75 Tyson chickens contaminated. 2-Sample CI’s and HT’s Question: Tyson different in safety than Store Brand? 2-Sample Examples Notation: Let p 2 denote the proportion of Store Brand which Power are contaminated and p 3 the proportion for Yson. Example Power Properties

Chicken Contamination Math 1710 Class 24 V1 1 33% of 75 Perdue chickens contaminated. Sample Size for a Given 2 45% of 75 Store Brand chickens contaminated. MOE Examples 3 56% of 75 Tyson chickens contaminated. Power Question: Tyson different in safety than Store Brand? 2-Sample CI’s and HT’s Notation: Let p 2 denote the proportion of Store Brand which 2-Sample are contaminated and p 3 the proportion for Yson. Examples Hypotheses: Power Example H 0 : p 2 = p 3 Power Properties H a : p 2 � = p 3

Chicken Contamination Math 1710 Class 24 V1 Sample Size for a Given MOE 1 33% of 75 Perdue chickens contaminated. Examples 2 45% of 75 Store Brand chickens contaminated. Power 2-Sample CI’s 3 56% of 75 Tyson chickens contaminated. and HT’s 2-Sample Question: Tyson different in safety than Store Brand? Examples Still ˆ p pooled = . 505, SE pooled ( ˆ p 2 − ˆ p 3 ) = . 0816, z = − 1 . 35 . Power Example Power Properties

Chicken Contamination Math 1710 Class 24 1 33% of 75 Perdue chickens contaminated. V1 2 45% of 75 Store Brand chickens contaminated. Sample Size for a Given 3 56% of 75 Tyson chickens contaminated. MOE Examples Question: Tyson different in safety than Store Brand? Power 2-Sample CI’s and HT’s 2-Sample Examples Power Example Power Properties

Chicken Contamination Math 1710 Class 24 V1 Sample Size 1 33% of 75 Perdue chickens contaminated. for a Given MOE 2 45% of 75 Store Brand chickens contaminated. Examples 3 56% of 75 Tyson chickens contaminated. Power 2-Sample CI’s and HT’s tail probability = P ( Z < − 1 . 35) = . 0885 . 2-Sample P-value = 2(tail probability)=2(.0885)=.177 Examples At a level of α = . 05, we’d retain H 0 . Power Example Tyson might not have a different level of safety than Store Power Brand. Properties

Chicken Contamination Math 1710 Class 24 V1 Sample Size for a Given MOE 1 33% of 75 Perdue chickens contaminated. Examples 2 45% of 75 Store Brand chickens contaminated. Power 2-Sample CI’s 3 56% of 75 Tyson chickens contaminated. and HT’s 2-Sample Confidence interval for difference in safety between Store Brand Examples and Tyson? Power Example Power Properties

Chicken Contamination Math 1710 Class 24 V1 Sample Size 1 33% of 75 Perdue chickens contaminated. for a Given MOE 2 45% of 75 Store Brand chickens contaminated. Examples 3 56% of 75 Tyson chickens contaminated. Power 2-Sample CI’s Confidence interval for difference in safety between Store Brand and HT’s and Tyson? 2-Sample Examples Power � . 45 · . 55 + . 56 · . 44 Example SE pooled = = . 0812 75 75 Power Properties

Chicken Contamination Math 1710 Class 24 1 33% of 75 Perdue chickens contaminated. V1 2 45% of 75 Store Brand chickens contaminated. Sample Size for a Given 3 56% of 75 Tyson chickens contaminated. MOE Examples Confidence interval for difference in safety between Store Brand Power and Tyson? 2-Sample CI’s and HT’s � . 45 · . 55 + . 56 · . 44 2-Sample SE pooled = = . 0812 Examples 75 75 Power Example A 95% CI for p 2 − p 3 would be Power Properties − . 11 ± 1 . 96 · . 0812 = − . 11 ± . 159 = ( − . 269 , . 049)

Chicken Contamination Math 1710 1 33% of 75 Perdue chickens contaminated. Class 24 V1 2 45% of 75 Store Brand chickens contaminated. 3 56% of 75 Tyson chickens contaminated. Sample Size for a Given MOE Confidence interval for difference in safety between Store Brand Examples and Tyson? Power 2-Sample CI’s � . 45 · . 55 + . 56 · . 44 and HT’s SE pooled = = . 0812 75 75 2-Sample Examples Power A 95% CI for p 2 − p 3 would be Example Power Properties − . 11 ± 1 . 96 · . 0812 = − . 11 ± . 159 = ( − . 269 , . 049) The fact that this CI contains 0 is another way of doing the last 2 HT’s.

Power Example Math 1710 Class 24 V1 Sample Size for a Given MOE Examples 40% of employees are women. Power Woman under-represented as executives? 2-Sample CI’s What would it take in ˆ p for the company to prove that women and HT’s are as well represented among executives? 2-Sample Examples Power Example Power Properties

Power Example Math 1710 Class 24 V1 Sample Size for a Given 40% of employees are women. MOE Woman under-represented as executives? Examples What would it take in ˆ p for the company to prove that women Power 2-Sample CI’s are as well represented among executives? and HT’s H 0 : p = . 4 2-Sample Examples H a : p > . 4 Power Example Power Properties

Power Example Math 1710 Class 24 V1 Sample Size 40% of employees are women. for a Given MOE Woman under-represented as executives? Examples What would it take in ˆ p for the company to prove that women Power are as well represented among executives? 2-Sample CI’s and HT’s H 0 : p = . 4 2-Sample H a : p > . 4 Examples Power One could also do a HT to see if a given ˆ p demonstrates Example women are under represented. Then H a would be p < . 4. Power Properties

Power Example Math 1710 Class 24 V1 Sample Size H 0 : p = . 4 for a Given MOE H a : p > . 4 Examples n = 420, SD (ˆ p ) = . 0239 . Power 2-Sample CI’s z = ˆ p − . 4 and HT’s 2-Sample . 0239 Examples z > z ∗ means ˆ p > . 0239 z ∗ + . 4. Power Example α = . 05 ⇒ z ∗ = 1 . 645; rejection means ˆ p > . 439 . Power α = . 01 ⇒ z ∗ = 2 . 326; rejection means ˆ Properties p > . 456 .

Power Example Math 1710 Class 24 V1 How a ˆ p will be dealt with in the HT Sample Size for a Given MOE Examples Power 2-Sample CI’s and HT’s 2-Sample Examples Power Example Power Properties

Power Example Math 1710 Class 24 V1 Sample Size for a Given MOE Examples How does the power depend on the effect size? Power 2-Sample CI’s i.e. on the actual value of p and HT’s 2-Sample Examples Power Example Power Properties

Power Example How does the power depend on the effect size? Math 1710 Class 24 V1 Sampling Dist of ˆ p when effect size is large. Sample Size for a Given Power nearly 1 MOE Examples Power 2-Sample CI’s and HT’s 2-Sample Examples Power Example Power Properties

Power Example How does the power depend on the effect size? Math 1710 Class 24 V1 Sampling Dist of ˆ p when effect size is small. Sample Size for a Given Power nearly α MOE Examples Power 2-Sample CI’s and HT’s 2-Sample Examples Power Example Power Properties

Power Example How does the power depend on the effect size? Math 1710 Class 24 Sampling Dist of ˆ p when effect size is middle size. V1 Sample Size Power in middle between 0 and 1; you could calulate it, but we for a Given MOE won’t ask you to. Examples Power 2-Sample CI’s and HT’s 2-Sample Examples Power Example Power Properties

Power Example Math 1710 Class 24 V1 Sample Size How does the power depend on the effect size? for a Given MOE α = . 05 case: Examples power p β Power 2-Sample CI’s .4001 .95 .05 and HT’s .41 .89 .11 2-Sample Examples .45 .33 .67 Power .5 .01 .99 Example Power Properties

Power Example Math 1710 Class 24 V1 Sample Size How does the power depend on the effect size? for a Given MOE α = . 01 case: Examples power p β Power 2-Sample CI’s .4001 .99 .01 and HT’s .41 .97 .03 2-Sample Examples .45 .59 .41 Power .5 .04 .96 Example Power Properties

Power Example How does the power depend on the effect size? Math 1710 Class 24 V1 Power vs. alternative value of p in 2-sided case. Sample Size for a Given A Power Curve MOE Examples Power 2-Sample CI’s and HT’s 2-Sample Examples Power Example Power Properties

Power Poperties Math 1710 Class 24 Prob of type I error = α . V1 Sample Size for a Given ( H 0 1-sides, α = . 05 below) MOE Examples Power 2-Sample CI’s and HT’s 2-Sample Examples Power Example Power Properties

Power Poperties Math 1710 Class 24 V1 Power = 1 - β always. Sample Size for a Given MOE Examples Power 2-Sample CI’s and HT’s 2-Sample Examples Power Example Power Properties