Question #1: The file Bears.csv contains data collected on bears in a field study: chest is chest girth in inches, weight is bear weight in pounds, and sex is a numerical coding of the animal's sex (male = 1, female = 0). Interest lies in the relationship between weight and chest girth.

a) Make a scatterplot of weight (y-axis) versus chest girth (x-axis) using a different color or symbol for the two sexes.

b) What is the form, strength, and direction of the relationship? Does the pattern of the relationship differ between sexes? How do the male bears as a group differ from the female bears as a group?

Question #2: Tree height is time-consuming to measure accurately. Tree diameter is easy to measure accurately. The file Trees.csv contains observations on height (Ht in feet) and diameter (DBH in inches) for a sample of trees. Interest lies in the relationship between height (as a response) and DBH (as an explanatory variable or predictor).

a) Graph the data appropriately given the interest specified (correct variables on axes). Can the relationship be considered, at least tentatively, linear? Explain.

b) Compute the correlation coefficient between height and DBH (if you rejected linearity in a), just "play along").

c) Obtain the least squares (simple linear) regression of height on DBH (again, "play along" if necessary). Identify the estimates of slope and intercept and state their units.

d) What is the coefficient of determination for the regression in c)? What does the value mean? Verify the known relationship between the coefficient of determination and the correlation coefficient (given this is a simple linear regression) by comparing the two numbers.

e) Plot the standardized residuals from the regression in c) against DBH (or equivalently fitted values). Do you see any issues?

Question #3: Continuing question #2, it has been suggested that transforming height to the natural logarithm of height and DBH to inverse of DBH results in a "more linear" relationship. Using ln(Ht) and (1/DBH) for the response and explanatory variables, respectively, answer the following questions:

f) Obtain the least squares (simple linear) regression of ln(Ht) on 1/DBH (Hint: make two new columns, one labeled "ln(Ht)" and the other "1/DBH"; use these new columns when fitting your regression).

g) What is the coefficient of determination for the regression in f)?

h) Plot the standardized residuals from the regression in f) against 1/DBH (or equivalently fitted values). Do you see any issues?

i) Explain why the slope of this relationship is negative.

j) Do you agree or disagree that transforming the variables is the appropriate thing to do in this case? Explain.

Extra Credit: Prove, formally using symbols, that the estimated least-squares regression line passes through the point (32,37): that is, forx = 3?, show that j? = 37. Hint: See the box on page 112 of your text and utilize the expression for be. \fTrees DBH Ht 12.6 63 13 66 46.8 137 30.8 120 56.5 146 14.2 62 44.1 135 60.1 160 16.5 76 45.1 122 56.2 142 26.4 104 36.5 116 58.8 161 36.8 122 45.8 133 18.5 84 27.7 109 12.2 62 30.8 116 22.5 97 38.1 129 56.7 146 44.2 133 60.2 132 60.2 162 30.5 109 15.3 78 36.2 137 51.5 144 21.3 95 34.3 131 20.8 83 48.6 145 57.4 154 13.6 71 45.1 149 52.5 153 30.7 111 38.6 133 25.4 102 18.1 91 35.1 123 24.2 113 43.5 141 25.1 89 42.2 137 52.1 131 59.7 147 17.5 72 54 137 40.1 131 23.5 93 45.1 127