STAT1 - Homework Due Friday, June 14 Instructions: In this assignment, you will learn about the Pearson Correlation Coefficient and least-squares regression equations. Unless you are specifically asked to do a problem by hand, use Minitab. 1. Standardized tests, such as the Comprehensive Test of Basic Skills (CTBS), are used by school systems to evaluate the performance of their students. CTBS scores for two different areas, such as math and reading, can be plotted in a scatterplot for a group of randomly to see if the scores show evidence of a linear relationship. Construct scatterplot and calculate Pearson correlation coefficient. Interpret the results! Student Reading Score Math Score 1 47 4 71 81 3 64 68 4 35 43 5 43 50 6 60 75 7 38 47 8 59 59 9 67 69 10 56 57 11 67 57 1 57 54 13 69 75
14 38 38 15 54 59 16 76 63 17 53 57 18 40 40 19 47 5 0 3. Explain why a strong correlation would be found between weekly sales of firewood and weekly sales of cough drops over a 1-year period. Would it imply that fires cause coughs? 3. If you peruse the bookshelves of a typical college professor, you will find a variety of books ranging from textbooks to esoteric technical publications to paperback novels. In order to determine whether or not the price of a book can be determined by the number of pages it contains, a college professor recorded the number of pages and price for 15 books on one shelf. The data are shown below. Pages Price Pages Price Pages Price 104* 3.95 34* 49.95 436 5.95 188* 4.95 378 4.95 458* 60.00 0* 49.95 385 5.99 466* 49.95 64* 79.95 417 4.95 469 5.99 336 4.50 417* 39.75 585 5.95 *Denotes Hardback Book a. Relating # of pages and price, would you expect a positive or negative correlation? b. Construct scatterplot and calculate Pearson correlation coefficient. Depict type of book on your scatterplot.
c. Construct scatterplot and calculate Pearson correlation coefficient using just the data for the hardback books. d. Construct scatterplot and calculate Pearson correlation coefficient using just the data for the paperback books. e. What do you learn from parts b, c, and d above. 4. The accompanying data on fish survival and ammonia concentration is taken from the paper Effects of Ammonia on Growth and Survival of Rainbow Trout in Intensive Static-Water Culture (Trans. of Amer. Fisheries Soc. (1983):448-54). Let x= ammonia exposure (mg/l) and y= percent survival. x y x y xy 10 85 100 75 850 10 9 100 8464 90 0 85 400 75 1700 0 96 400 916 190 5 87 65 7569 175 7 80 79 6400 160 7 90 79 8100 430 31 59 961 3481 189 50 6 500 3844 3100 0 736 6544 6154 17084 Do the following parts by hand and on Minitab. a. Construct a scatter plot. b. Calculate the Pearson correlation coefficient. c. Determine least squares equation that can be used for predicting a value of y based on a value of x. d. Compute ( y yˆ) for the least squares line. e. Calculate r using the following formula: r ( y y) ( y yˆ) ( y y) =. Interpret the r value. f. Using your equation in part c, draw the least squares line on the scatterplot you constructed in part a. g. Use your prediction equation to predict percent survival if ammonia exposure is 8 mg/l. 5. The paper "The Relation Between Freely Chosen Meals and Body Habits" (Amer. J. Clinical Nutrition (1983): 3-40) reported results of an investigation into the relationship between body build and energy intake of an individuals diet. A measure of body build is the Quetelet index (x) with a high value of x indicating a thickset individual. The variable reflecting energy intake is y=dietary energy density. There were nine subjects in the investigation. Calculate r with and without subject 9 included. What do you learn? Subject x Y 1 1.67 8.86 3 3.78 4 11.54
5 31.91 6 15.44 7 4.90 8 33.94 9 68.93 6. Underinflated or overinflated tires can increase tire wear and decrease gas mileage. A manufacturer of a new tire tested the tire for wear at different pressures. Construct scatterplot and calculate Pearson correlation coefficient. The researcher for this study calculated a Pearson correlation coefficient and concluded no relationship existed between the two variables. Was he correct in his conclusion? Pressure Mileage(thous) 30 9.5 30 30. 31 3.1 31 34.5 3 36.3 3 35 33 38. 33 37.6 34 37.7 34 36.1
35 33.6 35 34. 36 6.8 36 7.4 7. Suppose a fire insurance company wants to relate the amount of fire damage in major residential fires to the distance between the residence and the nearest fire station. The study is to be conducted in a large suburb or a major city; a sample of fifteen recent fires in this suburb is selected. The amount of damage, y, and the distance, x, between the fire and the nearest fire station are recorded for each fire. Distance from fire station x, miles Fire Damage y, thousands of dollars 3.4 6. 1.8 17.8 4.6 31.3.3 3.1 3.1 7.5 5.5 36.0.7 14.1 3.0.3.6 19.6 4.3 31.3.1 4.0 1.1 17.3 6.1 43. 4.8 36.4 3.8 6.1 a. Construct a scatter plot. b. Calculate the Pearson correlation coefficient. c. Determine equation of least squares line that can be used for predicting a value of y based on a value of x. d. Compute SSE = ( y yˆ) for the least squares line. e. Calculate r and interpret. f. Using your equation in part c, draw the least squares line on the scatterplot you constructed in part a. g. Suppose a major residential fire occurs 4.5 miles from the nearest fire station. Using your prediction equation, predict the fire damage.