# MATHEMATICS

11. Bird species from temperate regions must cope with relatively short breeding seasons. A study examined the relationship between blood testosterone level (ng/ml) and the duration of the egg-laying period (months) in temperate bird species. The scatterplot below displays this relationship, after taking the logarithm of each variable. A plausible value for the correlation between the logarithm of egg-laying duration and the logarithm of testosterone level is –0.9. –0.4. +0.2. +0.8. 12. When water flows across farm land, some of the soil is washed away, resulting in erosion. An experiment was conducted to investigate the effect of the rate of water flow (in liters per second) on the amount of soil washed away (in kilograms). The data are given in the following table. Flow rate 0.31 0.85 1.26 2.47 3.75 Eroded soil 0.82 1.95 2.18 3.01 6.07 The association between flow rate and amount of eroded soil is neither positive nor negative. positive. impossible to determine since both variables are categorical. negative. 13. Which of the following is TRUE of the correlation coefficient r? If r is the correlation between X and Y, then –r is the correlation between Y and X. All of the above –1 r 1. It is a resistant measure of association. 14. Researchers want to know if reading skills can explain IQ test scores in children with dyslexia. The following scatterplot examines the relationship between reading skill score and IQ test score for 22 dyslexic children. The least-squares regression line is displayed on the plot, along with the value of r2. The intercept of the least-squares regression line cannot be determined from the graph, even approximately. is greater than 120. is less than 50. is about 80, approximately. 15. Researchers want to know if reading skills can explain IQ test scores in children with dyslexia. The following scatterplot examines the relationship between reading skill score and IQ test score for 22 dyslexic children. The least-squares regression line is displayed on the plot, along with the value of r2. What percent of the variation in IQ test scores is explained by this regression model? 5% 48% 23% 52% 16. Before surgical removal of a diseased parathyroid gland, two tests are often performed: the standard intact test and the turbo test. Both tests measure parathyroid hormone (PTH, in ng/l), but the turbo test is very expensive. Researchers obtained data from both tests in a sample of 48 patients to predict turbo test results (y) from standard intact test results (x). The data ranged from roughly 0 to 500 ng/l, and a scatterplot showed a clear linear relationship. The published findings are summarized exactly as follows: y = 1.08x – 4.36 (r = 0.97; n = 48) For a PTH level of x = 1000 ng/l with the standard intact test, the predicted PTH level with the turbo test is 1075.64 ng/l. is 1036.4 ng/l. cannot be predicted accurately because it would be extrapolating. is 970 ng/l. 17. Babies typically learn to crawl approximately six months after birth. However, it may take longer for babies to learn to crawl in the winter, when they are often bundled in clothes that restrict their movement. Thus, there may be an association between a baby’s crawling age and the average temperature during the month they first try to crawl. Below are the average ages (in weeks) at which babies began to crawl for a sample of babies born in each of the 12 months of the year. In addition, the average temperature (in °F) for the month that is six months after the birth month is also listed. Birth month Average crawling age Average temperature January 29.84 66 February 30.52 73 March 29.70 72 April 31.84 63 May 28.58 52 June 31.44 39 July 33.64 33 August 32.82 30 September 33.83 33 October 33.35 37 November 33.38 48 December 32.32 57 We want to investigate if the average age at which infants begin to crawl (y) can be predicted from the average outdoor temperature (x) six months after birth, when they are likely to begin crawling. We decide to fit a least-squares regression line to the data with x as the explanatory variable and y as the response variable. We compute the following quantities. r = correlation between x and y = –0.7 = mean of the values of x = 50.25 = mean of the values of y = 31.77 sx = standard deviation of the values of x = 15.85 sy = standard deviation of the values of y = 1.76 Which of the following is closest to the slope of the least-squares line? 0.08 –0.08 6.30 –6.30 18. Babies typically learn to crawl approximately six months after birth. However, it may take longer for babies to learn to crawl in the winter, when they are often bundled in clothes that restrict their movement. Thus, there may be an association between a baby’s crawling age and the average temperature during the month they first try to crawl. Below are the average ages (in weeks) at which babies began to crawl for a sample of babies born in each of the 12 months of the year. In addition, the average temperature (in °F) for the month that is six months after the birth month is also listed. Birth month Average crawling age Average temperature January 29.84 66 February 30.52 73 March 29.70 72 April 31.84 63 May 28.58 52 June 31.44 39 July 33.64 33 August 32.82 30 September 33.83 33 October 33.35 37 November 33.38 48 December 32.32 57 We want to investigate if the average age at which infants begin to crawl (y) can be predicted from the average outdoor temperature (x) six months after birth, when they are likely to begin crawling. We decide to fit a least-squares regression line to the data with x as the explanatory variable and y as the response variable. We compute the following quantities. r = correlation between x and y = –0.7 = mean of the values of x = 50.25 = mean of the values of y = 31.77 sx = standard deviation of the values of x = 15.85 sy = standard deviation of the values of y = 1.76 The fraction of the variation in the values of a response y that is explained by the least-squares regression of y on x is the intercept of the least-squares regression line. correlation coefficient. square of the correlation coefficient. slope of the least-squares regression line.