6. We are going to consider data which shows the number of bee colonies and the number of juvenile arrests for marijuana possession for each year in the 1990s. a) Before you see any of the data or make a scatterplot (in other words, reading from the sentence above what the variables represent without peeking at the numbers) would you expect r, the correlation coefficient to be closer to 0, 1, or -1? EXPLAIN your reasoning, don't just give me your guess. (2 points ) There should be a negative relationship between X and y I should be close to - 1 Let x = hundreds of bee colonies in the US. Let y = thousands of juvenile arrests for marijuana possession. The data are in the table below. If you entered the data I sent you in advance, these are in L3 and L4. year 1990 1991 1992 1993 1994 1995 1996 X 1997 32 1998 32 1999 30 28 y 28 21 26 16 25 26 25 27 26 38 61 82 88 94 91 90 b) Find the sample mean and standard deviation for each variable. (8 points ) x = 28 J = 40. 6 S x = 2.539 SV = 32.401 X = EX = 28 n y= 27: 60.4 Sx= (xi-x) n - 1 = 2.539 Sy = = ( Yi - 7) = 32.401 n - 1c) Make a scatterplot of the data. Number the x-axis go from 24-34, counting each square as 1. Number the y-axis from 0-100, counting each square as 10. Place an asterisk (*) on your scatterplot at the point (x, J). (3 points ) 100 90 70 Y 50 40 30 20 10 24 25 26 29 30 31 32 33 34 X d) Examine your scatterplot and, without performing any calculations, EXPLAIN whether you expect the correlation coefficient to be closer to -1, 0, or 1 based on your scatterplot, and WHY. (3 points ) Based on the scatterplot, It seems to be a negative correlation I should be closer to - 1 el) What is the equation of the least squares regression line (the line found by your calculator as the ' best fit' line) for this data? (3 points ) 9= 393.2207- 11.8793x nexy - sxay e2) What is r, the correlation coefficient for this data? DEXY- ( Ex)7 .[ney-( Ex ) - - 0. 9307 ( 2 points ) y= - 11.879x + 393.22 f) Plot the regression line from part e) on the scatterplot. (4 points ) To receive credit, show the coordinates of two points used to help sketch the line.g) Find the slope, and y-intercept, and x-intercept of the regression line (you may use the equation from part e 'or the x-intercept, no need to recalculate a new line with the variables reversed). Give the "real world" interpretations for this example. If the interpretations are not valid here, explain why. (9 points ) slope = - 1.879 y-intercept = 313. 22 oulas do you x-intercept = h) In 1992, there were 3000 bee colonies (x = 30) and 25,000 arrests (y = 25). Calculate the residual (error) for this data point. SHOW YOUR WORK. (3 points ) i) Circle the data point referred to in part h) on the scatterplot and indicate the residual on the scatterplot. ( 1 point ) j) Give a "real world" interpretation of the residual calculated in part h). (2 points ) I've given you data where we would expect there to be no correlation (because there is no plausible explanation of why the number of bee colonies and the number of marijuana arrests would be related), but the scatterplot indicates a strong negative correlation and our calculation of r confirms this. One possibility is that this is an example of a 'spurious correlation'. If you examine enough bizarre, unrelated variables, eventually you'll find a pair with a strong correlation- even though they're not related. Google "spurious correlations" for a few laughs if you need to see more Statistics after the exam. Another possibility is that there is something else causing the variables to both move, but one does not 'cause' the other. Here I believe that environmental factors (pesticide use?) was hurting the bee population while 'tough on crime' laws resulted in more arrests. Forget for now that these variables are likely unrelated (correlation does not prove causation) and continue with the problem as if there is a plausible explanation as to why these are related.Make the following predictions, where appropriate, or explain why such a prediction is not appropriate. ( 2 points each ) If there are 2900 bee colonies (x = 29) in a given year, how many juvenile arrests for marijuana possession would you expect? If there are 900 bee colonies (x = 9) in a given year, how many juvenile arrests for marijuana possession would you expect? L) If there are 69,000 juvenile arrests for marijuana possession in a given year, how many bee colonies do you think there were? (2 points ) You will soon perform a hypothesis test with Ho: B = 0 H1: B