Math, statistics

58. T Crowdedness In a Chance magazine article (Summer 2005), Danielle Vasilescu and Howard Wainer used data from the United Nations Center for Human Settlements to investigate aspects of living conditions for several countries. Among the variables they looked at were the country's per capita gross domestic product (GDP, in $) and Crowdedness, defined as the average number of persons per room living in homes there. This scatterplot displays these data for 56 countries: 50,000 37,500 + ++ GDP 25,000 12,500 7 0 0.5 1.0 1.5 2.0 2.5 3.0 Crowdedness a. Explain why you should re-express these data before trying to fit a model. b. What re-expression of GDP would you try as a starting point?R22. Togethemess Are good grades in high school associated with family togetherness? A random sample of 142 high school students was asked how many meals per week their families ate together. Their responses produced a mean of 3.?8 meals per week, with a standard deviation of 2.2. Researchers then matched these responses against the students' grade point averages (GPAs). The scatterplot appeared to be reasonably linear. so they created a line of regression. No apparent pattern emerged in the residuals plot. The equation of the line was so = 2.73 + 0.11 Meats. a. Interpret the yintercept in this context. b. Interpret the slope in this context. c. What was the mean GPA for these students? d. If a student in this study had a negative residual. what did that mean? e. Upon hearing of this study, a counselor recommended that parents who want to improve the grades their children get should get the family to eat together more often. Do you agree with this interpretation? Explain. R2.12. Final exam A statistics instructor created a linear regression equation to predict students' final exam scores from their midterm exam scores. The regression equation was Fin = 10 + 0.9 Mid. a. If Susan scored a 70 on the midterm, what did the instructor predict for her score on the final? b. Susan got an 80 on the final. How big is her residual? c. If the standard deviation of the final was 12 points and the standard deviation of the midterm was 10 points, what is the correlation between the two tests? d. How many points would someone need to score on the midterm to have a predicted final score of 100? e. Suppose someone scored 100 on the final. Explain why you can't estimate this student's midterm score from the information given. f. One of the students in the class scored 100 on the midterm but got overconfident, slacked off, and scored only 15 on the final exam. What is the residual for this student? g. No other student in the class "achieved" such a dramatic turnaround. If the instructor decides not to include this student's scores when constructing a new regression model, will the R2 value of the regression increase, decrease, or remain the same? Explain. h. Will the slope of the new line increase or decrease