worksheet5 solution.pdf - STAB22 Worksheet 5 Tutorial 1...

Unformatted text preview: STAB22 Worksheet 5 Tutorial 1 Last Name: Student number: First Name: 1. Many high school students take either the SAT or the ACT. However, some students take both. Data was collected from 60 students who took both college entrance exams. The average SAT score was 912 with a standard deviation of 180. The average ACT score was 21 with a standard deviation of 5. The correlation between the two variables equals 0.817. (a) Calculate the least squares regression equation for predicting SAT scores from ACT scores (i.e. the regression of SAT score on ACT score)? Show your work clearly. (b) What proportion of the variation in SAT scores is explained by the linear regression model with ACT scores? Show your work clearly. (c) Predict the SAT score of a student whose ACT score is 23. Show your work clearly. Solution: = 29.412 slope = 0.817 × 180 5 intercept 912 − 29.412 × 21 = 294.348 Reg eq: SAT = 294.348 + 29.412 ACT b) The proportion of the variation in rate is explained by the linear regression model with ACT = R-sq = = 0.817*0.817 = 0.6675 = 66.75%. c) SAT = 294.348 + 29.412 * 23 = 970.824 STAB22 Worksheet 5 Tutorial 2 Last Name: Student number: First Name: 1. Metabolic rate, the rate at which the body consumes energy, is important in studies of weight gain, dieting, and exercise. Researchers collected data on the lean body mass and resting metabolic rate for 12 women who are subjects in a study of dieting. Lean body mass, given in kilograms, is a person's weight leaving out all fat. Metabolic rate is measured in calories burned per 24 hours, the same calories used to describe the energy content of foods. The researchers believe that lean body mass is an important inuence on metabolic rate. Some summary statistics of the data are given below: Variable Mass Rate N 12 12 Mean 43.03 1235.1 StDev 6.87 188.3 Minimum Median Maximum 33.10 42.00 54.60 913.0 1230.0 1502.0 Pearson correlation of Mass and Rate = 0.876 (a) Calculate the least squares regression equation of rate on mass. Show your work clearly. (b) What proportion of the variation in rate is explained by the linear regression model with mass? Show your work clearly. (c) Predict the rate (i.e. metabolic rate) of an individual with a lean body mass of 50 kg. Show your work clearly. Solution: = 24.0 slope = 0.876 × 188.3 6.87 intercept 1235.1 − 24 × 43.03 = 202.38 Reg eq: Rate = 202.38 + 24.0 mass b) The proportion of the variation in rate is explained by the linear regression model with mass = R-sq = 0.876*0.876 = 0.767376 = 76.7%. c) predicted rate = 202.38+24*50 = 1402.38 STAB22 Worksheet 5 Tutorial 3 Last Name: Student number: First Name: 1. A real estate agent is interested in knowing whether there is a relationship between the age of house and the selling price. Listed below are the ages (in years) and selling prices (in $1,000) of a sample of 5 houses the agent has sold in the past year: Descriptive Statistics: Age, Price Variable N Mean StDev Minimum Age 5 8.000 2.121 5.000 Price 5 90.00 17.68 75.00 Q1 6.000 77.50 Median Q3 8.000 10.000 85.00 105.00 Maximum 10.000 120.00 Pearson correlation of Age and Price = -0.933 (a) Find least squares regression equation for predicting price based on age.Show your work clearly. (b) Use your regression line to predict the expected price of a house that is 9 years old. Show your work clearly. (c) What is the expected price of a new (i.e. Age=0) house? (d) What proportion of the variability in price is explained by age? Solution: (a) b = -0.933 x 17.68/2.121 = -7.78 a = 90 - (-7.78) x 8 = 152.24 Least squares regression equation is: y = 152.24 - 7.78 x (b) y = (152.24 - 7.78 x 9)x 1000 (c) 152.24 x 1000 (d) 0.933 x 0.933 STAB22 Worksheet 5 Tutorial 4 Last Name: Student number: First Name: 1. Ecologists sometimes nd rather strange relationships in our environment. One study seems to show that beavers benet beetles. The researchers laid out 23 circular plots each four meters in diameter, in an area where beavers were cutting down cottonwood trees. In each plot they counted the number (x) of stumps from trees cut by beavers and the number (y) of clusters of beetle larvae. Some descriptive statistics of the data are given below: Descriptive Statistics: x, y Variable N x 23 y 23 Mean 2.217 25.09 StDev 1.204 15.64 Minimum Q1 Median Q3 Maximum 1.000 1.000 2.000 3.000 5.000 6.00 12.00 21.00 40.00 56.00 Correlations: x, y Pearson correlation of x and y = 0.916 (a) Calculate the least squares regression equation for predicting the number (y) of clusters of beetle larvae from the number (x) of stumps from trees cut by beavers. Show your work clearly. (b) What proportion of the variation in the number of clusters of beetle larvae is explained by the linear regression model with the number of stumps from trees cut by beavers. Show your work clearly. (c) Predict the number of clusters of beetle larvae (i.e. y) for a plot with 2 stumps (i.e. x = 2). Show your work clearly. Solution: = 11.9 slope = 0.916 × 15.64 1.204 intercept 25.091 − 11.9 × 2.217 = −1.3 Reg eq: Rate = -1.3 + 11.9x b) The proportion of the variation in rate is explained by the linear regression model with 0.916*0.916 = 0.839 = 83.9%. c) ) predicted y = -1.3 + 11.9 * 2 = 22.5 STAB22 Worksheet 5 Tutorial 5 Last Name: Student number: First Name: 1. Many high school students take either the SAT or the ACT. However, some students take both. Data was collected from 60 students who took both college entrance exams. The average SAT score was 912 with a standard deviation of 180. The average ACT score was 21 with a standard deviation of 5. The correlation between the two variables equals 0.817. (a) Calculate the least squares regression equation for predicting SAT scores from ACT scores (i.e. the regression of SAT score on ACT score)? Show your work clearly. (b) What proportion of the variation in SAT scores is explained by the linear regression model with ACT scores? Show your work clearly. (c) Predict the SAT score of a student whose ACT score is 23. Show your work clearly. Solution: = 29.412 slope = 0.817 × 180 5 intercept 912 − 29.412 × 21 = 294.348 Reg eq: SAT = 294.348 + 29.412 ACT b) The proportion of the variation in rate is explained by the linear regression model with ACT = R-sq = = 0.817*0.817 = 0.6675 = 66.75%. c) SAT = 294.348 + 29.412 * 23 = 970.824 STAB22 Worksheet 5 Tutorial 6 Last Name: Student number: First Name: 1. Many high school students take either the SAT or the ACT. However, some students take both. Data was collected from 60 students who took both college entrance exams. The average SAT score was 912 with a standard deviation of 180. The average ACT score was 21 with a standard deviation of 5. The correlation between the two variables equals 0.817. (a) Calculate the least squares regression equation for predicting SAT scores from ACT scores (i.e. the regression of SAT score on ACT score)? Show your work clearly. (b) What proportion of the variation in SAT scores is explained by the linear regression model with ACT scores? Show your work clearly. (c) Predict the SAT score of a student whose ACT score is 23. Show your work clearly. Solution: = 29.412 slope = 0.817 × 180 5 intercept 912 − 29.412 × 21 = 294.348 Reg eq: SAT = 294.348 + 29.412 ACT b) The proportion of the variation in rate is explained by the linear regression model with ACT = R-sq = = 0.817*0.817 = 0.6675 = 66.75%. c) SAT = 294.348 + 29.412 * 23 = 970.824 STAB22 Worksheet 5 Tutorial 7 Last Name: Student number: First Name: 1. For human women, a pregnancy lasts about 9 months. For other animals, the gestation period (average length of pregnancy) is dierent. A researcher believes that longer-lived animals generally have longer gestation periods. The researcher measured life expectancy in years and gestation period in days, and did a regression for predicting gestation period from life expectancy. The regression line had intercept -39.5 and slope 15.5, with an R-squared of 72.2%. (a) Is the association between between gestation period and life expectancy positive or negative? Explain. (b) What is the correlation coecient between gestation period and life expectancy? (c) What is the predicted gestation period, in days, for an animal with life expectancy 10 years? Show your work clearly. √ (a) positive because the slope is positive. (b) 0.722 = 0.85 (c)Plug 10 into the regression equation: y = -39.5 + 15.5(10), which is about 115. Solution: STAB22 Worksheet 5 Tutorial 8 Last Name: Student number: First Name: 1. Marine biologists warn that the growing number of powerboats registered in Florida threatens the existence of manatees. A study was made of the the relationship between powerboat registrations (x, in hundreds) and the number of manatees killed (y). The data are summarized below: Descriptive Statistics: x, y Variable N N* Mean x 19 0 615.1 y 19 0 35.16 SE Mean StDev Minimum Q1 Median 26.9 117.0 447.0 512.3 613.5 3.83 16.68 13.00 21.00 34.00 Q3 716.0 47.00 Correlations: x, y Pearson correlation of x and y = 0.924 (a) Calculate the slope and intercept of the regression line for predict- ing the number of manatees killed from the number of powerboat registrations. Show your work clearly. (b) Use your regression line to predict the number of manatees killed in a year where there are 510 hundred powerboat registrations. Show your work clearly. (c) 510 hundred powerboat registrations is below the mean number. Was the predicted number of manatees killed above or below its mean? Explain briey why this is not a surprise. (a) Slope b = (0.924)(16.68)=117 = 0.13; intercept 35.16 - (0.3)(615.1) = -45.87. (b) Use x = 510 to get predicted y = -45.87 + (0.13)(510) = 20.43. (c) Below the mean also. Because the correlation is positive, values of x below the mean go with values of y below the mean. Solution: Maximum 860.0 81.00 ...
View Full Document