Concept explainers
Remainder Round all answers to two decimal places unless otherwise indicated.
The Effect of Sampling Error on Linear Regression A stream that feeds a lake is flooding, and during this flooding period, the depth of water in the lake is increasing. The actual depth of the water at a certain point in the lake is given by the linear function
|
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0 | 51.9 |
0.5 | 52.5 |
1 | 52.9 |
1.5 | 53.3 |
2 | 53.7 |
Plot the data points.
Find the equation of the regression line for D as a function of t, and explain in practical terms the meaning of the slope.
Add the graph of the regression line to the plot of the data points.
Add the graph of the depth function
What was the actual depth of the water at t = 3 hours?
What prediction would the hydrologist’s regression line give for the depth of the water at t = 3?
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Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
- Remainder Round all answers to two decimal places otherwise indicated. College Enrollment This is a continuation of Exercise 9. We use the data in the college enrolment table that appears in Exercise 9. a.Find the equation of the regression line model for college enrolment as a function of time, and add its graph to the data plot made in Exercise 9. b.Explain the meaning of the slope of the line you found in part a. c.Express, using functional notation, the enrolment in American private colleges in2010, and then estimate that value. d.Enrollment in American private colleges in 2013 was 5.74 million. Does it appear that the trend established in the mid-2000s was valid in 2013?arrow_forwardRemainder Round all answers to two decimal places unless otherwise indicated. Running Ants A scientist collected the following data on the speed, in centimeters per second, at which ants ran at the given ambient temperature, in degrees Celsius. Temperature Speed 25.6 2.62 27.5 3.03 30.3 3.57 30.4 3.56 32.2 4.03 33.0 4.17 33.8 4.32 a.Find the equation of the regression line, giving the speed as a function of the temperature. b.Explain in practical terms the meaning of the slope of the regression line. c.Express, using functional notation, the speed at which the ants run when the ambient temperature is 29 degrees Celsius, and then estimate that value. d.The scientist observes the ants running at a speed of 2.5 centimeters per second. What is the ambient temperature?arrow_forwardRemainder Round all answers to two decimal places unless otherwise indicated. Energy Cost of Running Physiologists have studied the steady-state oxygen consumption measured per unit of mass in a running animal as a function of its velocity i.e., its speed. They have determined that the relationship is approximately linear, at least over an appropriate range of velocities. The table below gives the velocity v, in kilometres per hour, and the oxygen consumption E, in milliliters of oxygen per gram per hour, for the rhea, a large flightless South American bird. For comparison, 10 kilometers per hour is about 6.2 miles per hour. Velocity v Oxygen Consumption E 2 1.0 5 2.1 10 4.0 12 4.3 a.Find the equation of the regression line for E in terms of v. b.The slope of the linear function giving oxygen consumption in terms of velocity is called the cost of transport for the animal, since it measures the energy required to move a unit mass by 1 unit distance. What is the cost of transport for the rhea? c.Physiologists have determined the general approximate formula C=8.5W-0.40 for the cost of transport C of an animal weighing W grams. If the rhea weighs 22, 000 grams, is its cost of transport from part b higher or lower than what the general formula would predict? Is the rhea a more or a less efficient runner than a typical animal its size? d.What would your equation from part a lead you to estimate for the oxygen consumption of a rhea at rest?arrow_forward
- Remainder Round all answers to two decimal places unless otherwise indicated. Long Jump The following table shows the length, in meters, of the winning long jump in the Olympic Games for the indicated year. One meter is 39.37 inches. Year 1900 1904 1908 1912 Length 7.19 7.34 7.48 7.60 a.Find the equation of the regression that gives the length as a function of time. Round the regression line parameters to three decimal places. b.Explain in practical terms the meaning of the slope of the regression line. c.Plot the data points and the regression line. d.Would you expect the regression line formula to be a good model of the winning length over a long period of time? Be sure to explain your reasoning. e.There were no Olympic Games in 1916 because of World War I, but the winning long jump in the 1920 Olympic Games was 7.15 meters. Compare this with the value that the regression line model gives. Is the result consistent with your answer to part d?arrow_forwardRemainder Round all answers to two decimal places unless otherwise indicated. Rice production in Asia Improved agricultural practices, including better utilization of fertilizers and use of improved plant varieties, have resulted in increased rice yields in Asia. The accompanying table shows the average yield Y, in tons per hectare, as a function of the number of years t since 1980. t= years since 1980 Y= average yield 5 3.32 10 3.61 15 3.73 20 3.95 25 4.11 a.One study used a linear function to approximate yield as a function of time. Find an approximate linear model for Y as a function of t. b.Explain what the slope of the linear model tells you. c.Use the model to calculate Y30, and explain in practical terms what your answer means. d.What yield would the model estimate for 2025? e.The 1993 report of the International Rice Research Institute predicted that rice requirements in Asis would grow exponentially. If the rice requirements in 2025 are 6.44 tons per hectare, will rice production in 2025 meet those requirements?arrow_forwardRemainder Round all answers to two decimal places unless otherwise indicated. Tourism The number, in millions, of international tourists who visited the United States is given in the following table. Date 2010 2011 2012 2013 Millions of tourists 59.74 62.33 66.66 69.77 a.Plot the data. b.Find the equation of the regression line and add its graph to your data plot. Round the regression line parameters to two decimal places. c.Explain in practical terms the meaning of the slope. d.Express, using functional notation, the number of tourists who visited the United States in 2014, and then estimate that value. The actual number was 74.73 million.arrow_forward
- Remainder Round all answers to two decimal places unless otherwise indicated. Is a Linear Model Appropriate? The number, in thousands, of bacteria in a petri dish is given by the following table. Time is measured in hours. Time in hours since experiment began Number of bacteria in thousands 0 1.2 1 2.4 2 4.8 3 9.6 4 19.2 5 38.4 6 76.8 The table below shows enrollment, in millions of people, in private colleges in the United States during the years from 2004 through 2008. Date Enrollment in millions 2004 4.29 2005 4.47 2006 4.58 2007 4.76 2008 5.13 a.Plot the data points for number of bacteria. Does it look reasonable to approximate these data with a straight line? b.Plot the data points for college enrollment. Does it look reasonable to approximate these data with a straight line?arrow_forwardRemainder Round all answers to two decimal places unless otherwise indicated. 2. Federal Methamphetamine Arrests The table below shows the number A, in thousands, of federal arrests for methamphetamine t years after 2006. t = years since 2006 A= thousands of arrests 0 5.85 1 5.54 2 4.72 3 4.70 Find the equation of the regression line for A as a function of t.arrow_forwardRemainder Round all answers to two decimal places unless otherwise indicated. Driving You are driving on a highway. The following table gives your speed S, in miles per hour, as a function of the time t, in seconds, since you started making your observations. Time t 0 15 30 45 60 Speed S 54 59 63 66 68 a.Find the equation of the regression that expresses S as a linear function of t. b.Explain in practical terms the meaning of the slope of the regression line. c.On the basis of the regression line model, when do you predict that your speed will reach 70 miles per hour? Round your answer to the nearest second. d.Plot the data points and the regression line. e.Use your plot in part d to answer the following: Is your prediction in part c likely to give a time earlier or later than the actual time when y our speed reaches 70 miles per hour?arrow_forward
- Remainder Round all answers to two decimal places unless otherwise indicated. 7. DirecTV Subscribers The table on the next page shows the number S, in millions, of subscribers to DirecTV t years after 1995. t = years since 1995 S = subscribers, in millions 0 1.20 4 6.68 7 11.18 9 13.00 16 19.89 19 20.35 a.Find the equation of the regression line for S as a function of t. b.What number does this equation give for DirecTV subscribers in 2013? The actual number was 20.25 million. c.Explain in practical terms the meaning the meaning of the slope of the line you found in part a. d.Plot the data points and the regression line.arrow_forwardRemainder Round all answers to two decimal places unless otherwise indicated. Motor Vehicle Fatalities The table below shows the traffic fatality rate R, in fatalities per 100 million vehicle miles travelled, t years after 2010. Find the equation of the regression line for R as a function of t. t = years since 2010 R = rate 0 1.11 1 1.10 2 1.14 3 1.09 4 1.07arrow_forwardAverage Rate of change A fucntion f is given. (a) Find the average rate of change of f between x=0 and x=0 ,and the average rate of change of f between x=15 and x=50 . (b) Were the two average rates of change that you found in part (a) the same? (c) Is the function linear? If so, what is its rate of change? f(x)=12x6arrow_forward
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