91Ó°ÊÓ

Name the Relation, Part I For each of the following statements, explain whether you think the variables will have positive correlation, negative correlation, or no correlation. Support your opinion. (a) Number of children in the household under the age of 3 and expenditures on diapers (b) Interest rates on car loans and number of cars sold (c) Number of hours per week on the treadmill and cholesterol level (d) Price of a Big Mac and number of McDonald's French fries sold in a week (e) Shoe size and IQ

Name the Relation, Part II For each of the following statements, explain whether you think the variables will have positive correlation, negative correlation, or no correlation. Support your opinion. (a) Number of cigarettes smoked by a pregnant woman each week and birth weight of her baby (b) Years of education and annual salary (c) Number of doctors on staff at a hospital and number of administrators on staff (d) Head circumference and IQ (e) Number of movie goers and movie ticket price

Attending Class The following data represent the number of days absent, \(x\), and the final grade, \(y,\) for a sample of college students in a general education course at a large state university. $$ \begin{array}{lllllllllll} \hline \begin{array}{l} \text { No. of } \\ \text { absences, } x \end{array} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline \begin{array}{l} \text { Final } \\ \text { grade, } y \end{array} & 89.2 & 86.4 & 83.5 & 81.1 & 78.2 & 73.9 & 64.3 & 71.8 & 65.5 & 66.2 \\ \hline \end{array} $$ (a) Find the least-squares regression line treating number of absences as the explanatory variable and final grade as the response variable. (b) Interpret the slope and \(y\) -intercept, if appropriate. (c) Predict the final grade for a student who misses five class periods and compute the residual. Is the final grade above or below average for this number of absences? (d) Draw the least-squares regression line on the scatter diagram of the data. (e) Would it be reasonable to use the least-squares regression line to predict the final grade for a student who has missed 15 class periods? Why or why not?

The time it takes for a planet to complete its orbit around the sun is called the planet's sidereal year. In 1618 , Johannes Kepler discovered that the sidereal year of a planet is related to the distance the planet is from the sun. The following data show the distances of the planets, and the dwarf planet Pluto, from the sun and their sidereal years. $$ \begin{array}{lcc} \text { Planet } & \begin{array}{l} \text { Distance from Sun, } x \\ \text { (millions of miles) } \end{array} & \text { Sidereal Year, } \boldsymbol{y} \\ \hline \text { Mercury } & 36 & 0.24 \\ \hline \text { Venus } & 67 & 0.62 \\ \hline \text { Earth } & 93 & 1.00 \\ \hline \text { Mars } & 142 & 1.88 \\ \hline \text { Jupiter } & 483 & 11.9 \\ \hline \text { Saturn } & 887 & 29.5 \\ \hline \text { Uranus } & 1785 & 84.0 \\ \hline \text { Neptune } & 2797 & 165.0 \\ \hline \text { Pluto } & 3675 & 248.0 \\ \hline \end{array} $$ (a) Draw a scatter diagram of the data treating distance from the sun as the explanatory variable. (b) Determine the correlation between distance and sidereal year. Does this imply a linear relation between distance and sidereal year? (c) Compute the least-squares regression line. (d) Plot the residuals against the distance from the sun. (e) Do you think the least-squares regression line is a good model? Why?

The wind chill factor depends on wind speed and air temperature. The following data represent the wind speed (in mph) and wind chill factor at an air temperature of \(15^{\circ}\) Fahrenheit. $$ \begin{array}{cc|cc} \begin{array}{l} \text { Wind } \\ \text { Speed, } x \\ (\mathrm{mph}) \end{array} & \begin{array}{l} \text { Wind Chill } \\ \text { Factor, } y \end{array} & \begin{array}{l} \text { Wind } \\ \text { Speed, } x \\ (\mathrm{mph}) \end{array} & \begin{array}{l} \text { Wind Chill } \\ \text { Factor, } y \end{array} \\ \hline 5 & 12 & 25 & -22 \\ \hline 10 & -3 & 30 & -25 \\ \hline 15 & -11 & 35 & -27 \\ \hline 20 & -17 & & \\ \hline \end{array} $$ (a) Draw a scatter diagram of the data treating wind speed as the explanatory variable. (b) Determine the correlation between wind speed and wind chill factor. Does this imply a linear relation between wind speed and wind chill factor? (c) Compute the least-squares regression line. (d) Plot the residuals against the wind speed. (e) Do you think the least-squares regression line is a good model? Why?

American Black Bears The American black bear (Ursus americanus) is one of eight bear species in the world. It is the smallest North American bear and the most common bear species on the planet. In 1969 , Dr. Michael R. Pelton of the University of Tennessee initiated a long-term study of the population in the Great Smoky Mountains National Park. One aspect of the study was to develop a model that could be used to predict a bear's weight (since it is not practical to weigh bears in the field). One variable thought to be related to weight is the length of the bear. The following data represent the lengths and weights of 12 . American black bears. (a) Which variable is the explanatory variable based on the goals of the research? (b) Draw a scatter diagram of the data. (c) Determine the linear correlation coefficient between weight and length. (d) Does a linear relation exist between the weight of the bear and its length?

Professor Katula feels that there is a relation between the number of hours a statistics student studies each week and the student's age. She conducts a survey in which 26 statistics students are asked their age and the number of hours they study statistics each week. She obtains the following results: $$ \begin{array}{ll|ll|ll} \text { Age, } & \text { Hours } & \text { Age, } & \text { Hours } & \text { Age, } & \text { Hours } \\ \boldsymbol{x} & \text { Studying, } \boldsymbol{y} & \boldsymbol{x} & \text { Studying, } \boldsymbol{y} & \boldsymbol{x} & \text { Studying, } \boldsymbol{y} \\ \hline 18 & 4.2 & 19 & 5.1 & 22 & 2.1 \\ \hline 18 & 1.1 & 19 & 2.3 & 22 & 3.6 \\ \hline 18 & 4.6 & 20 & 1.7 & 24 & 5.4 \\ \hline 18 & 3.1 & 20 & 6.1 & 25 & 4.8 \\ \hline 18 & 5.3 & 20 & 3.2 & 25 & 3.9 \\ \hline 18 & 3.2 & 20 & 5.3 & 26 & 5.2 \\ \hline 19 & 2.8 & 21 & 2.5 & 26 & 4.2 \\ \hline 19 & 2.3 & 21 & 6.4 & 35 & 8.1 \\ \hline 19 & 3.2 & 21 & 4.2 & & \\ \hline \end{array} $$ (a) Draw a scatter diagram of the data. Comment on any potential influential observations. (b) Find the least-squares regression line using all the data points. (c) Find the least-squares regression line with the data point (35,8.1) removed. (d) Draw each least-squares regression line on the scatter diagram obtained in part (a). (e) Comment on the influence that the point (35,8.1) has on the regression line.

31\. Putting It Together: A Tornado Model Is the width of a tornado related to the amount of distance for which the tornado is on the ground? Go to www.pearsonhighered.com/sullivanstats to obtain the data file \(4_{-} 3_{-} 31\) using the file format of your choice for the version of the text you are using. The data represent the width (yards) and length (miles) of tornadoes in the state of Oklahoma in \(2013 .\) (a) What is the explanatory variable? (b) Explain why this data should be analyzed as bivariate quantitative data. (c) Draw a scatter diagram of the data. What type of relation appears to exist between the width and length of a tornado? (d) Determine the correlation coefficient between width and length. (e) Is there a linear relation between a tornado's width and its length on the ground? (f) Find the least-squares regression line. (g) Predict the length of a tornado whose width is 500 yards. (h) Was the tornado whose width was 180 yards and length was 1.9 miles on the ground longer than would be expected? (i) Interpret the slope. (j) Explain why it does not make sense to interpret the intercept. (k) What proportion of the variability in tornado length is explained by the width of the tornado? (I) Plot residuals against the width. Does the residual plot suggest the two variables are linearly related? (m) Draw a boxplot of the residuals. Are there any outliers? (n) A major tornado was 4576 yards wide that had a length of 16.2 miles. Is this an influential tornado? Explain.

Consider the following set of data: $$ \begin{array}{lllllllll} \hline x & 2.2 & 3.7 & 3.9 & 4.1 & 2.6 & 4.1 & 2.9 & 4.7 \\ \hline y & 3.9 & 4.0 & 1.4 & 2.8 & 1.5 & 3.3 & 3.6 & 4.9 \\ \hline \end{array} $$ (a) Draw a scatter diagram of the data and compute the linear correlation coefficient (b) Draw a scatter diagram of the data and compute the linear correlation coefficient with the additional data point \((10.4,9.3) .\) Comment on the effect the additional data point has on the linear correlation coefficient. Explain why correlations should always be reported with scatter diagrams.

Consider the following data set: $$ \begin{array}{lllllllll} \hline x & 5 & 6 & 7 & 7 & 8 & 8 & 8 & 8 \\\ \hline y & 4.2 & 5 & 5.2 & 5.9 & 6 & 6.2 & 6.1 & 6.9 \\ \hline x & 9 & 9 & 10 & 10 & 11 & 11 & 12 & 12 \\ \hline y & 7.2 & 8 & 8.3 & 7.4 & 8.4 & 7.8 & 8.5 & 9.5 \\ \hline \end{array} $$ (a) Draw a scatter diagram with the \(x\) -axis starting at 0 and ending at 30 and with the \(y\) -axis starting at 0 and ending at 20 . (b) Compute the linear correlation coefficient. (c) Now multiply both \(x\) and \(y\) by 2 . (d) Draw a scatter diagram of the new data with the \(x\) -axis starting at 0 and ending at 30 and with the \(y\) -axis starting at 0 and ending at 20. Compare the scatter diagrams. (e) Compute the linear correlation coefficient. (f) Conclude that multiplying each value in the data set by a nonzero constant does not affect the correlation between the variables.