Chapter 3 HW

Bivariate Data and Linear Regression 43 See Appendix A for details on creating and running an m-file. When this m-file is run in the command window, the output looks like the following: Command Window >> LSR Eqn for LSR: yhat = 1.406250 x + 1.656250 rho = 0.918559 The regression line accounts for 91.86% of the variance in the data The graph is shown in Figure 3.3. 4 X Y 5 6 7 1 2 3 4 5 6 7 8 9 10 11 12 13 2 3 Figure 3.3 Graph produced by the m-file LSR.m. 3.6 Exercises 3.1 How correlated are the x - and y -values in Exercise 2.10? 3.2 Suppose that the regression line for a set of data is y = 2.1 x + b and that the line passes through the point ( 1, 6 ) . What is the relationship between the mean of the x -values and the mean of the y -values? 3.3 For the eastern bluebird data from Exercise 2.1, make a scatter plot of years and number of bluebirds, draw a freehand line, and find the equation of your freehand line. For each of the problems 3.4 to 3.9, do the following: (a) Make a scatter plot. (b) Draw a freehand line. (c) Find the equation of your freehand line. (d) Calculate the least-squares regression line for the data. Do this by hand for the first problem and use Matlab for the remainder of the problems. (e) Compute the correlation coefficient. Again, do this by hand for the first problem and use Matlab for the remainder of the problems. Using the correlation coefficient, state

44 Chapter 3 how much confidence you place in a prediction based on the regression line. State whether you think the data are strongly or weakly related. (f) Choose a value that falls in the range of the data that you plotted on the horizontal axis but that is not one of the data points. Interpolate the corresponding vertical axis value using the equation for the least-squares regression line. (g) Choose a value that falls outside the range of the data you plotted on the horizontal axis. Extrapolate the corresponding vertical axis value using the equation for the least-squares regression line. 3.4 The average length and width of various bird eggs are given in the following table. Bird Name Width (cm) Length (cm) Canada goose 5.8 8.6 Robin 1.5 1.9 Turtledove 2.3 3.1 Hummingbird 1.0 1.0 Raven 3.3 5.0 3.5 An investigator was interested in examining the effect of different doses of a new drug on pulse rate in humans. Four doses were used in the experiment. Three people were randomly assigned each of the four doses. After a pre study pulse rate was recorded for each individual, subjects were injected with the appropriate drug dose. Pulse rates were again recorded an hour later. The changes in pulse rates in beats per minute (bpm) are listed below. Dose (mL/kg of 1.5 1.5 1.5 2.0 2.0 2.0 2.5 2.5 2.5 3.0 3.0 3.0 body weight) Change in 20 21 19 16 17 17 15 13 14 8 10 8 pulse rate (bpm) 3.6 Suppose that we are interested in the relation between carbon monoxide (CO) concentra- tions and the density of cars in some geographical area. The hundreds of cars per hour to the nearest 500 cars and the concentration of CO in parts per million (ppm) at a particular street corner are measured. The results are as follows: Cars/hour 1.0 1.0 1.0 1.5 1.5 1.5 2.0 2.0 3.0 3.0 3.0 3.0 [CO] 9.0 6.8 7.7 9.6 6.8 11.3 12.3 11.8 20.7 19.2 21.6 20.6 3.7 (From [60]) In a study of a free-living population of the snake Vipera bertis , researchers caught and measured nine adult females. Their body lengths x and weights y are shown in the table below. Length (cm), x 60 69 66 64 54 67 59 65 63 Weight (g), y 136 198 194 140 93 172 116 174 145 3.8 The data below give the infant mortality rate (MR) per 1000 live births in the United States for the period of 1960–1979. Year 1960 1965 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 MR 26.0 24.7 20.0 19.1 18.5 17.7 16.7 16.1 15.2 14.1 13.8 13.0