Stat 2000, Section 001, Homework Assignment 9 (Due 11/5/2001 11:59pm)

• *) Reminder: Quiz 2 will take place on Friday 11/2/2001
  
  Note that all the material discussed in class (that is not part of the textbook) is relevant for the quiz as well.
  
  
• 0) Reading: Section 4.3; Homework Solutions & Additional Handouts
  
  
• 1) The statistical package WebStat is freely accessible on the Web at http://www.stat.sc.edu/webstat. When you go to this Web site, wait until the orange button appears. Then click this button to fire up WebStat.
  
  Use the sample of 8 teams of the "Cost of Victories" data set I indicated in Section 3.1 of the lecture notes. Type this data into WebStat and conduct a "Simple Linear Regression" in the "Stat" menu. Make sure that Payroll is the explanatory variable and Wins is the response variable.
  
  Indicate the exact values for slope, y-intercept, the regression equation, and the correlation coefficient obtained from WebStat. (5 Points)
  
  
• 2) The output below has been obtained from WebStat for the data from Exercise 2.111 (see also Homework 8, Exercise 1d). (5 Points)
  
  

  Simple linear regression results: 
  
  Independent variable: var1
  Dependent variable: var2
  Sample size: 16
  Correlation coefficient: -0.4596
  (See fitted line plot in Graphics Panel.)
  Estimate of sigma: 8.006156
  
  Parameter   Estimate     Std. Err.   DF   Tstat        Pval
  Intercept   166.48334    2.734623    14   60.879814    0
  var1        -1.1102564   0.5733327   14   -1.9364958   0.0366
  

  
  
  
  • a) Indicate the exact values for slope, y-intercept, the regression equation, and the correlation coefficient obtained from WebStat. What is the explanatory variable and what is the response variable? Comment on the relationship.
    
    
  • b) Use this regression equation to predict the corn yield for 5, 10, and 100 weeds per meter. Which of these predictions are meaningful, which not? Explain your answer.
    
    
  • c) Below is a plot of the residuals versus Weeds. Does the residual plot show any unusual pattern or do you think that, based on this residual plot, our least squares regression line describes the data reasonably well?
    
    
    
• 3) (2 Points)
  
  
  
• 4) When we introduced micromaps in class, we looked at a statistical display that contains information on the percentage of urban census tracts in a region and hazardous air pollutants in the same region. My claim was that there is a very close relationship between these two factors (however, we couldn't express this in terms of a regression line and correlation at that time).
  
  Below is a scatterplot of the same data. Our explanatory x-variable in this case is the percentage of urban census tracts and our response y-variable is the average concentration of hazardous air pollutants in each state. We are looking at data of 48 states plus Washington, D.C., i.e, n = 49. (5 Points)
  
  
  
  
  • a) Fit a least squares (linear regression) line to the data. It might help to know that:
    
    
    
    
  • b) What is a possible interpretation of the slope and y-intercept you calculated in (a) above? Explain.
    
    
  • c) Calculate Pearson's correlation coefficient r between x and y. How can we interpret this value for our given data set?
    
    
  • d) Based on your calculations in (a), what is the predicted concentration of air pollutants for a region that has 0%, 40%, 60%, 100% of urban census tracts. Which of these 4 predictions are more reliable, which are less reliable? Why?
    
    
• 5) Look at these 4 residual plots from G.A.F. Seber's ``Linear Regression Analysis'' book and answer the following two questions: (4 Points)
  
  
  
  
  • Which of these residual plot(s) represent(s) satisfactory pattern(s) that show that your selected statistical model (e.g., a least squares regression line) describes the data reasonably well.
    
    
  • For each of the residual plots that does not represent a satisfactory pattern, indicate the type of problem that occurs with the data in that particular case.
    
    
• 6) For the following situations, state which of the following types of the sampling design was used: systematic, cluster, stratified or opportunity sample. Explain in a short sentence whether you think the sampling design would result in a biased sample. If you think that the sample would be biased, explain briefly what would cause this bias (4 Points).
  
  
  • a) To survey the opinions of its customers, an airline company made a list of all its flights and randomly selected 25 flights. All of the passengers on those flights were asked to fill out the survey.
    
    
  • b) A pollster interested in opinion on gun control divided a city into city blocks, then surveyed the third house to the west of the southeast corner of each block. If the house was divided into apartments, the westernmost ground floor apartment was selected. The pollster conducted the survey during the day, but left a notice for those who were not at home to phone her so she could interview them.
    
    
  • c) To learn how its employees felt about higher student fees imposed by the legislature, a university divided employees into three categories: staff, faculty, and student employees. A random sample was selected from each group and they were telephoned and asked for their opinion.
    
    
  • d) A large variety store wants to know if consumers would be willing to pay slightly higher prices to have computers available throughout the store to help them locate items. The store posted an interviewer at the door and told her to collect a sample of 100 opinions by asking the next person who came in the door each time she had finished an interview.