Exam3ReviewSp2011

# Exam3ReviewSp2011 - Review for Final Exam Stat 205...

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Review for Final Exam Stat 205: Statistics for the Life Sciences Tim Hanson, Ph.D. University of South Carolina T. Hanson (USC) Stat 205: Statistics for the Life Sciences 1 / 20

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Final Exam Overview of Final Exam Logistics... * Open book, not open notes . Bring a calculator. * You can put post-it notes in your book. * Thursday, April 28, 9am–noon. Be on time . * 3 regular problems, each worth 20 points: 60 total. * Rest of today’s lecture is reviewing this material. * 1 problem is short answer spanning entire course. * JeanMarie’s office hours next week: Monday 9-11, Tuesday/Thursday 11-12, & Wednesday 11-1 in LeConte 215A. Dr. Hanson’s office hours: Monday/Wednesday 1:30-2:30, & Tuesday 1-2. T. Hanson (USC) Stat 205: Statistics for the Life Sciences 2 / 20
Final Exam Linear regression 12.1, 12.2, 12.3, 12.4: Linear regression * Have scatterplot of n paired values ( x 1 , y 1 ), ( x 2 , y 2 ), . . . ,( x n , y n ). * Theoretical model: Y i |{z} observed = β 0 + β 1 x i | {z } unknown trend + ² i |{z} slop * We use data ( x 1 , y 1 ), ( x 2 , y 2 ), . . . ,( x n , y n ) to obtain the fitted line Y = b 0 + b 1 x , where b 0 and b 1 are the least squares estimates of β 0 and β 1 : b 1 = ( x i - ¯ x )( y i - ¯ y ) ( x i - ¯ x ) 2 b 0 = ¯ y - b 1 ¯ x T. Hanson (USC) Stat 205: Statistics for the Life Sciences 3 / 20

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Final Exam Linear regression Linear regression * Estimate of var( ² i ) is s y | x = r SSE ( resid ) n - 2 , where SS ( resid ) = X ( y i - b 0 - b 1 x i ) 2 = X ( y i - ˆ y i ) 2 . * s y = q 1 n - 1 ( y i - ¯ y ) 2 is overall variability of y 1 , . . . , y n around the mean ¯ y . s y | x = q 1 n - 2 ( y i - ˆ y i ) 2 is overall variability of y 1 , . . . , y n around the line b 0 + b 1 x . * s y | x < s y . How much smaller tells you how “well” the line is working to explain the data. * HW: 12.3, 12.7, 12.9. T. Hanson (USC) Stat 205: Statistics for the Life Sciences 4 / 20
Final Exam Linear regression Linear regression * If we assume slop ² 1 , . . . , ² n are normal then SE b 1 = s y | x p ( x i - ¯ x ) 2 . * Inference for β 1 : (a) (1 - α )100% confidence interval for β 1 has endpoints b 1 ± t 1 - α/ 2 SE b 1 , where df = n - 2. (b) To test H 0 : β 1 = 0 at level α , see if confidence interval from (a) includes zero. If not, reject H 0 : β 1 = 0. * If you reject H 0 : β 1 = 0 then x and y are significantly, linearly related . * 5 statistics needed: ¯ x , ¯ y , ( x i - ¯ x ) 2 , ( x i - ¯ x )( y i - ¯ y ), SS ( resid ) = ( y i - b 0 - b 1 x i ) 2 . * HW: 12.16, 12.17, 12.19, 12.23(a). T. Hanson (USC) Stat 205: Statistics for the Life Sciences 5 / 20

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Final Exam Linear regression Example A criminologist studying the relationship between level of education and crime rate in medium-sized U.S. counties collected data from a random sample of n = 84 counties. Y is the crime rate (crimes per 100 people) and X is the percentage of individuals in the county having at least a high-school diploma. Here’s a scatterplot: 60 65 70 75 80 85 90 2 4 6 8 10 14 highschool crimes T. Hanson (USC) Stat 205: Statistics for the Life Sciences 6 / 20
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