BIOSTAT
mid1practicesoln

mid1practicesoln - Midterm 1 Practice 1 Solutions THE STORY...

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Midterm 1 Practice 1 Solutions THE STORY BEHIND THE EXAM R.A. Fisher, the brilliant British statistician who is responsible for our using α = . 05 for confidence intervals and hypothesis tests, did most of his applied work in agricuture. In this exam, his descendant, S.B. “Fish” Fisher is running an organic fruit and vegetable farm, but he continues the family tradition by using statistics to try to improve his produce production. Your job is to help him out ..... Question 1: Farmer Fisher’s Fertilizer (45 points, 50 minutes) Farmer Fisher has recently been trying out a new organic fertilizer to increase the amount of fruit on his apple trees. He has used different amounts of fertilizer, X (in cups per square yard) in n=12 different fields and recorded Y, the amount of apples produced (in pounds per square yard). He has tried to fit a simple linear regression of Y on X, but unfortunately his computer package seems to have a bug (perhaps a fruit fly?) and some of the numbers are missing. He does know that the average amount of fertilizer he used was ¯ X = 2 cups per square yard, and that his average yield of apples was ¯ Y = 50 pounds per square yard. He also remembers that when he didn’t use any fertilizer he had a yield of 45 pounds of apples per square yard and that this point fell exactly on his regression line. Use this information plus the partial printout below to answer the following questions. The regression equation is Apples = __45__ + _2.5__ Fertilizer Root MSE = 3.00 R-sq = __80%__ R-sq(adj) = 78% Analysis of Variance SOURCE DF SS MS F Regression __1__ _360_ _360_ 40.0 Error _10__ __90_ __9__ Total _11__ _450_ Part a (14 points) Use the information provided to fill in the missing elements of the printout. You must show your work or reasoning for b 0 , b 1 , and R 2 below but you don’t need to do so for any of the other numbers unless you are unsure of your answers. 1

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Solution: First, we are told that when Farmer Fisher used no fertilizer (X=0) he had a yield of Y=45 pounds of apples per square yard, and that this data point fell on the estimated regression line. Thus we know b 0 = 45. Further we know that the regression coefficients are related through the equation b 0 = ¯ Y - b 1 ¯ X . We are given ¯ Y = 50 , ¯ X = 2 , and we now know b 0 = 45. Solving, we get b 1 = 2 . 5. Thus our estimated regression equation is ˆ Y = 45 + 2 . 5 X . To fill in the ANOVA table we can proceed as follows. First, we do the degrees of freedom. In a simple linear regression, the degrees of freedom for the regression, error, and total, are 1, n-2, and n-1 respectively. We are given that n=12 since Farmer Fisher has data for 12 fields. Thus our degrees of freedom are 1, 10, and 11. Next, we are given Root MSE = 3. Thus we must have MSE = 9. We fill this in to the mean squares column. Next note that F = MSR/MSE. We know F=40 from the printout and we have just found MSE = 9. It follows that MSR = 360. This completes the mean squares column. We also know that the sum of squares column divided by the degrees of freedom gives the mean squares column. Working backwards, this means SSR/1 = MSR = 360 or SSR = 360 and SSE/10 = MSE = 9 or SSE = 90. We then note that SST = SSR +SSE = 360+90 = 450. This completes the sums of squares column. The last missing number is
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• Winter '07
• Sugar
• Regression Analysis, Null hypothesis, Statistical hypothesis testing, Square yard, Farmer Fisher

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