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ASSIGNMENT Chapter 11 Statistical Methods NAME: M&S 575, 577, 589, 600, 608, 616 11.23 (4 points) Redshifts of Quasi Stellar Objects. Astronomers call a shift in the spectrum of galaxies a “redshift.” A correlation between redshift level and apparent magnitude (i.e., brightness on a logarithmic scale) of a quasi-stellar object was discovered and reported in the Journal of Astrophysics & Astronomy (Mar./Jun. 2003). Physicist D. Basu (Carleton University, Ottawa) applied simple linear regression to data collected for a sample of over 6,000 quasi-stellar objects with confirmed redshifts. The analysis yielded the following results for a specific range of magnitudes: y = 18.13+ 6.21x, where y = magnitude and x = redshift level. a. Graph the least squares lines. Is the slope of the line positive or negative? ANSWER b. Interpret the estimate of the y-intercept in the words of the problem. ANSWER c. Interpret the estimate of the slope in the words of the problem. ANSWER 11.28 (4 points) Sweetness of orange juice. The quality of the orange juice produced by a manufacturer is constantly monitored. There are numerous sensory and chemical components that combine to make the best-testing orange juice. For example, one manufacturer has developed a quantitative index of the “sweetness” of orange juice. (The higher the index, the sweeter is the juice.) Is there a relationship between the sweetness index and a chemical measure such as the amount of water-soluble pectin (parts per million) in the orange juice? Data collected on these two variables during 24 production runs at a juice-manufacturing plant are shown in the table. Suppose a manufacturer wants to use simple linear regression to predict the sweetness (y) from the amount of pectin (x). OJUICE Run Sweetness Index Pectin (ppm) 1 5.2 220 2 5.5 227 3 6.0 259 4 5.9 5 5.8 6 6.0 7 5.8 8 5.6 9 5.6 10 5.9 11 5.4 12 5.6 13 5.8 14 5.5 15 5.3 16 5.3 17 5.7 18 5.5 19 5.7 20 5.3 21 5.9 22 5.8 23 5.8 24 5.9 a. Find the least squared line for the data. ANSWER 210 224 215 231 268 239 212 410 256 306 259 284 383 271 264 227 263 232 220 246 241 b. Interpret 0 and 1 in the words of the problem. ANSWER c. Predict the sweetness index if the amount of pectin in the orange juice is 300 ppm. [Note: A measure of reliability of such a prediction is discussed in Section 11.6.] ANSWER 11.63 (3 points) Relation of eye and head movements. How do eye and head movements relate to body movements when a person reacts to a visual stimulus? Scientists at the California Institute of Technology designed an experiment to answer this question and reported their results in Nature (Aug 1998). Adult male rhesus monkeys were exposed to a visual stimulus (i.e., a panel of light-emitting diodes), and their eye, head, and body movements were electronically recorded. In one variation of the experiment, two variables were measured: active head movement (x, percent per degree) and body-plus-head rotation (y, percent per degree). The data for n = 39 trials were subjected to a simple linear regression analysis, with the following result: 1 = .88, s = .14 1 a. Conduct a test to determine whether the two variables, active head movement x and body-plus-head rotation y are positively linearly related. Use = .05. ANSWER b. Construct and interpret a 90% confidence interval for 1 . ANSWER c. The scientists want to know whether the trueslope of the line differs significantly from 1. On the basis of your answer to part b, make the appropriate inference. ANSWER 1.80 (6 points) Salary linked to height. Are short people shortchanged when it comes to salary? According to business professors T. A. Judge (University of Florida) and D. M. Cable (University of North Carolina), tall people tend to earn more money over their career than short people earn. (Journal of Applied Psychology, June 2004.) Using data collected from participants in the National Longitudinal Surveys begun in 1979, the researchers computed the correlation between average earnings (in dollars) from 1985 to 2000 and height (in inches) for several occupations. The results are given in the following table: Occupation Sales Managers Blue Collar Service Workers Professional/Technical Clerical Crafts/Forepersons Correlation, r Sample Size, n .41 .35 .32 .31 .30 .25 .24 117 455 349 265 453 358 250 a. Interpret the value of r for people in sales occupations. ANSWER b. Compute r2 for people in sales occupations. Interpret the result. ANSWER c. Given H0 and Ha for testing whether average earnings and height are positively correlated. ANSWER d. The test statistic for testing H0 and Ha in part c is t = r n 2 1 r 2 . Compute the value of t for people in sales occupations. ANSWER e. Use the result you obtained in part d to conduct the test at = .01. State the appropriate conclusion. ANSWER 11.98 (2 points) Sweetness of orange juice. Refer to the simple linear, regression of sweetness index y and amount of pectin, x, for n = 24 orange juice samples, presented in Exercise 11.28 (p. 577). The SPSS printout of the analysis is shown below. A 90% confidence interval for the mean sweetness index E(y) for each value of x is shown on the SPSS spreadsheet. Select an observation and interpret this interval. run 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 ANSWER 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 sweet 5.2 5.5 6.0 5.9 5.8 6.0 5.8 5.6 5.6 5.9 5.4 5.6 5.8 5.5 5.3 5.3 5.7 5.5 5.7 5.3 5.9 5.8 5.8 5.9 pectin 220 227 259 210 224 215 231 268 239 212 410 256 306 259 284 383 271 264 227 263 232 220 246 241 lower90m 5.64898 5.63898 5.57819 5.66194 5.64337 5.65564 5.63284 5.55553 5.61947 5.65946 5.05526 5.58517 5.43785 5.57819 5.50957 5.15725 5.54743 5.56591 5.63898 5.56843 5.63125 5.64898 5.60640 5.61587 upper90m 5.83848 5.81613 5.72904 5.87173 5.82560 5.85493 5.80379 5.71011 5.78019 5.86497 5.55416 5.73592 5.65219 5.72904 5.68213 5.57694 5.70434 5.71821 5.81613 5.72031 5.80075 5.83848 5.76091 5.77454 11.112 (4 points) Predicting sale prices of homes. Real-estate investors, home buyers, and homeowners often use the appraised value of property as a basis for predicting the sale of that property. Data on sale prices and total appraised value of 78 residential properties sold in 2006 in an upscale Tampa, Florida, neighborhood named Hunter's Green are saved in the HUNGREEN file. Selected observations are listed in the following table. HUNGREEN (selected observations) Property Sale Price Appraised Value 1 $489,900 $418,601 2 1,825,000 1,577,919 3 890,000 687,836 4 250,00 191,620 5 1,275,000 1,063,901 : : : 74 325,000 292,702 75 516,000 407,449 76 309,300 272,275 77 370,000 347,320 78 580,000 511,359 a. Propose a straight-line model to relate the appraised property value (x) to the sale price (y) for residential properties in this neighborhood. ANSWER b. A MNITAB scatterplot of the data with the least squared lines is shown (p. 616, top). Does it appear that a straight-line model will be an appropriate fit to the data? ANSWER c. A MINITAB simple linear regression printout is also shown (p. 616, bottom). Find the equation of the least squared line. Interpret the estimated slop and yinterpret in the words of the problem. ANSWER Total points: 23