Math 1710 Class 26 Inference Coffee Machine Dr. Allen Back Using - PowerPoint PPT Presentation

Math 1710 Class 26 V2c 2-Sample Examples Approx Mean Math 1710 Class 26 Inference Coffee Machine Dr. Allen Back Using Table T t-CI’s and HT’s A Sample Size Problem Oct. 26, 2016 t-Conditions Robustness

Chicken Contamination Math 1710 Class 26 V2c 2-Sample Examples Approx Mean Suppose: Inference Coffee 1 33% of 75 Perdue chickens contaminated. Machine 2 45% of 75 Store Brand chickens contaminated. Using Table T t-CI’s and 3 56% of 75 Tyson chickens contaminated. HT’s A Sample Size Problem t-Conditions Robustness

Chicken Contamination Math 1710 Class 26 V2c 1 33% of 75 Perdue chickens contaminated. 2-Sample Examples 2 45% of 75 Store Brand chickens contaminated. Approx Mean Inference 3 56% of 75 Tyson chickens contaminated. Coffee Machine Questions: Using Table T 1 Purdue safer than Store Brand? t-CI’s and HT’s 2 Tyson safer than Store Brand? A Sample Size 3 Tyson different in safety than Store Brand? Problem t-Conditions 4 Confidence interval for difference in safety between Store Robustness Brand and Tyson?

Chicken Contamination Math 1710 Class 26 V2c 2-Sample Examples Approx Mean Inference 1 33% of 75 Perdue chickens contaminated. Coffee 2 45% of 75 Store Brand chickens contaminated. Machine Using Table T 3 56% of 75 Tyson chickens contaminated. t-CI’s and HT’s Question: Purdue safer than Store Brand? A Sample Size Problem t-Conditions Robustness

Chicken Contamination Math 1710 Class 26 V2c 2-Sample Examples 1 33% of 75 Perdue chickens contaminated. Approx Mean Inference 2 45% of 75 Store Brand chickens contaminated. Coffee Machine 3 56% of 75 Tyson chickens contaminated. Using Table T Question: Purdue safer than Store Brand? t-CI’s and HT’s Notation: Let p 1 denote the proportion of Purdue which are A Sample Size contaminated and p 2 the proportion for Store Brand. Problem t-Conditions Robustness

Chicken Contamination Math 1710 Class 26 V2c 1 33% of 75 Perdue chickens contaminated. 2-Sample Examples 2 45% of 75 Store Brand chickens contaminated. Approx Mean Inference 3 56% of 75 Tyson chickens contaminated. Coffee Machine Question: Purdue safer than Store Brand? Using Table T Notation: Let p 1 denote the proportion of Purdue which are t-CI’s and contaminated and p 2 the proportion for Store Brand. HT’s Hypotheses: A Sample Size Problem H 0 : p 1 = p 2 (or p 1 ≥ p 2 ) t-Conditions Robustness H a : p 1 < p 2

Chicken Contamination Math 1710 Class 26 V2c 1 33% of 75 Perdue chickens contaminated. 2-Sample Examples 2 45% of 75 Store Brand chickens contaminated. Approx Mean Inference 3 56% of 75 Tyson chickens contaminated. Coffee Machine Hypotheses: Using Table T H 0 : p 1 = p 2 (or p 1 ≥ p 2 ) t-CI’s and HT’s H a : p 1 < p 2 A Sample Size Problem p pooled = . 33 · 75 + . 45 · 75 t-Conditions ˆ = . 39 75 + 75 Robustness

Chicken Contamination Math 1710 Class 26 1 33% of 75 Perdue chickens contaminated. V2c 2 45% of 75 Store Brand chickens contaminated. 2-Sample 3 56% of 75 Tyson chickens contaminated. Examples Approx Mean Hypotheses: Inference Coffee H 0 : p 1 = p 2 (or p 1 ≥ p 2 ) Machine H a : p 1 < p 2 Using Table T t-CI’s and HT’s p pooled = . 33 · 75 + . 45 · 75 ˆ = . 39 A Sample Size 75 + 75 Problem t-Conditions Robustness � 1 � 75 + 1 � SE pooled ( ˆ p 1 − ˆ p 2 ) = . 39 · . 61 = . 0796 . 75

Chicken Contamination Math 1710 Class 26 V2c 1 33% of 75 Perdue chickens contaminated. 2-Sample Examples 2 45% of 75 Store Brand chickens contaminated. Approx Mean Inference 3 56% of 75 Tyson chickens contaminated. Coffee Machine Hypotheses: Using Table T H 0 : p 1 = p 2 (or p 1 ≥ p 2 ) t-CI’s and HT’s H a : p 1 < p 2 A Sample Size Problem z = ˆ p 1 − ˆ p 2 = − . 12 t-Conditions . 0796 = − 1 . 51 . SE pooled Robustness

Chicken Contamination Math 1710 1 33% of 75 Perdue chickens contaminated. Class 26 2 45% of 75 Store Brand chickens contaminated. V2c 3 56% of 75 Tyson chickens contaminated. 2-Sample Examples z = ˆ p 1 − ˆ p 2 = − . 12 Approx Mean . 0796 = − 1 . 51 . Inference SE pooled Coffee Machine Using Table T t-CI’s and HT’s A Sample Size Problem t-Conditions Robustness N(0,1)

Chicken Contamination Math 1710 Class 26 1 33% of 75 Perdue chickens contaminated. V2c 2 45% of 75 Store Brand chickens contaminated. 2-Sample Examples 3 56% of 75 Tyson chickens contaminated. Approx Mean Inference Hypotheses: Coffee Machine H 0 : p 1 = p 2 (or p 1 ≥ p 2 ) Using Table T H a : p 1 < p 2 t-CI’s and HT’s z = ˆ p 1 − ˆ p 2 = − . 12 A Sample Size . 0796 = − 1 . 51 . Problem SE pooled t-Conditions Robustness 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 26 V2c 2-Sample Examples Approx Mean Inference 1 33% of 75 Perdue chickens contaminated. Coffee 2 45% of 75 Store Brand chickens contaminated. Machine Using Table T 3 56% of 75 Tyson chickens contaminated. t-CI’s and HT’s Question: Tyson safer than Store Brand? A Sample Size Problem t-Conditions Robustness

Chicken Contamination Math 1710 Class 26 V2c 2-Sample Examples 1 33% of 75 Perdue chickens contaminated. Approx Mean Inference 2 45% of 75 Store Brand chickens contaminated. Coffee Machine 3 56% of 75 Tyson chickens contaminated. Using Table T Question: Tyson safer than Store Brand? t-CI’s and HT’s Notation: Let p 2 denote the proportion of Store Brand which A Sample Size are contaminated and p 3 the proportion for Tyson. Problem t-Conditions Robustness

Chicken Contamination Math 1710 Class 26 V2c 1 33% of 75 Perdue chickens contaminated. 2-Sample Examples 2 45% of 75 Store Brand chickens contaminated. Approx Mean Inference 3 56% of 75 Tyson chickens contaminated. Coffee Machine Question: Tyson safer than Store Brand? Using Table T Notation: Let p 2 denote the proportion of Store Brand which t-CI’s and are contaminated and p 3 the proportion for Tyson. HT’s Hypotheses: A Sample Size Problem H 0 : p 3 = p 2 (or p 3 ≥ p 2 ) t-Conditions Robustness H a : p 3 < p 2

Chicken Contamination Math 1710 Class 26 V2c 2-Sample Examples 1 33% of 75 Perdue chickens contaminated. Approx Mean Inference 2 45% of 75 Store Brand chickens contaminated. Coffee Machine 3 56% of 75 Tyson chickens contaminated. Using Table T t-CI’s and p pooled = . 45 · 75 + . 56 · 75 HT’s ˆ = . 505 75 + 75 A Sample Size Problem t-Conditions Robustness

Chicken Contamination Math 1710 Class 26 V2c 1 33% of 75 Perdue chickens contaminated. 2-Sample 2 45% of 75 Store Brand chickens contaminated. Examples 3 56% of 75 Tyson chickens contaminated. Approx Mean Inference Coffee Machine p pooled = . 45 · 75 + . 56 · 75 ˆ = . 505 Using Table T 75 + 75 t-CI’s and HT’s A Sample Size Problem � 1 � t-Conditions 75 + 1 � SE pooled ( ˆ p 2 − ˆ p 3 ) = . 505 · . 495 = . 0816 . Robustness 75

Chicken Contamination Math 1710 Class 26 V2c 2-Sample Examples 1 33% of 75 Perdue chickens contaminated. Approx Mean Inference 2 45% of 75 Store Brand chickens contaminated. Coffee Machine 3 56% of 75 Tyson chickens contaminated. Using Table T t-CI’s and z = ˆ p 2 − ˆ p 3 = − . 11 HT’s . 0816 = − 1 . 35 . SE pooled A Sample Size Problem t-Conditions Robustness

Chicken Contamination Math 1710 1 33% of 75 Perdue chickens contaminated. Class 26 2 45% of 75 Store Brand chickens contaminated. V2c 3 56% of 75 Tyson chickens contaminated. 2-Sample Examples z = ˆ p 2 − ˆ p 3 = − . 11 Approx Mean . 0816 = − 1 . 35 . Inference SE pooled Coffee Machine Using Table T Which side provides as much or more support for H a of t-CI’s and p 3 < p 2 ? HT’s A Sample Size Problem t-Conditions Robustness

Chicken Contamination Math 1710 1 33% of 75 Perdue chickens contaminated. Class 26 2 45% of 75 Store Brand chickens contaminated. V2c 3 56% of 75 Tyson chickens contaminated. 2-Sample Examples z = ˆ p 2 − ˆ p 3 = − . 11 Approx Mean . 0816 = − 1 . 35 . Inference SE pooled Coffee Machine Using Table T Which side provides as much or more support for p 3 < p 2 ? t-CI’s and HT’s A Sample Size Problem t-Conditions Robustness

Chicken Contamination Math 1710 Class 26 V2c 2-Sample Examples 1 33% of 75 Perdue chickens contaminated. Approx Mean Inference 2 45% of 75 Store Brand chickens contaminated. Coffee 3 56% of 75 Tyson chickens contaminated. Machine Using Table T Our statistic provides no support for H a so we immediately t-CI’s and retain H 0 . HT’s It is a matter of convention whether we’d view the p-value as A Sample Size Problem . 5 or even larger. t-Conditions Robustness

Chicken Contamination Math 1710 Class 26 V2c 2-Sample Examples Approx Mean Inference 1 33% of 75 Perdue chickens contaminated. Coffee 2 45% of 75 Store Brand chickens contaminated. Machine Using Table T 3 56% of 75 Tyson chickens contaminated. t-CI’s and HT’s Question: Tyson different in safety than Store Brand? A Sample Size Problem t-Conditions Robustness