Linear Regression and Correlation In 1990 the number of driver deaths per 100,000 for the different age groups was as follows (Source: The National Highway Traffic Safety Administration's National Center for Statistics and Analysis): Age Number of Driver Deaths per 100,000 15-24 28 25-39 15 40-69 10 70-79 15 80+ 25 Complete the following using the table above: a. For each age group, pick the midpoint of the interval for the x value. (For the 80+ group, use 85.) b. Using ages as the independent variable and Number of driver deaths per 100,000 as the dependent variable, make a scatter plot of the data. c. Calculate the least squares (best-fit) line. Put the equation in the form of: ^y= a + bx d. Find the correlation coefficient. Is it significant? e. Pick two ages and find the estimated fatality rates. f. Use the two points in (e) to plot the least squares line on your graph from (b). g. Based on the above data, is there a linear relationship between age of a driver and driver fatality rate? h. What is the slope of the least squares (best-fit) line? Interpret the slope. Least Squares Below is the life expectancy for an individual born in the United States in certain years. (Source: National Center for Health Statistics) Year of Birth Life Expectancy 1930 59.7 1940 62.9 1950 70.2 1965 69.7 1973 71.4 1982 74.5 1987 75 1992 75.7 Using the table above, answer the following: a. Decide which variable should be the independent variable and which should be the dependent variable. b. Draw a scatter plot of the ordered pairs. c. Calculate the least squares line. Put the equation in the form of: ^y= a + bx d. Find the correlation coefficient. Is it significant? e. Find the estimated life expectancy for an individual born in 1950 and for one born in 1982. f. Why aren't the answers to part (e) the values on the above chart that correspond to those years? g. Use the two points in (e) to plot the least squares line on your graph from (b). h. Based on the above data, is there a linear relationship between the year of birth and life expectancy? i. Are there any outliers in the above data? j. Using the least squares line, find the estimated life expectancy for an individual born in 1850 Does the least squares line give an accurate estimate for that year? Explain why or why not. k. What is the slope of the least squares (best-fit) line? Interpret the slope. Decision Tree Analysis Purchase Behavior for Big-Screen Televisions Actually Purchased Yes No Total Yes 200 50 250 No 100 650 750 Total 300 700 1000 Planned To Purchase In the above table, households are classified according to whether they planned to purchase and whether they actually purchased big-screen televisions. Draw the Decision Tree for this example. Purchase Behavior Regarding Plasma Screen Televisions and DVR. Purchased DVR Yes No Total Plasma Screen 38 42 80 Not Plasma Screen 70 150 220 Total 108 192 300 Purchased Plasma Screen 1. Using the cross-classified data above, construct the decision tree. Use your Decision Tree to find the probability that a household purchased a DVR, given that the household purchased a plasma-screen television. (Note: the probability formulas for question 2 are listed below) The probability of A given B is equal to the probability of A and B divided by the probability of B. P(A/B) = P(A and B)/P(B) The probability of B given A is equal to the probability of A and B divided by the probability of A. P(B/A) = P(A and B)/P(A) Where: P(A and B) = joint probability of A and B P(A) = marginal probability of A P(B) = marginal probability of B