Managers of an outdoor coffee stand in Coast City are examining the relationship between (hot) coffee sales and daily temperature, hoping to be able to predict a day's total coffee sales from the maximum temperature that day. The bivariate data values for the coffee sales (denoted by y , in dollars) and the maximum temperature (denoted by x , in degrees Fahrenheit) for each of fifteen randomly selected days during the past year are given below. These data are plotted in the scatter plot in Figure 1. Temperature, x (in degrees Fahrenheit) Coffee sales, y (in dollars) 53.1 2215.5 76.6 1982.7 70.1 1937.3 39.7 2251.1 72.5 1603.9 48.4 2024.2 46.7 2135.3 83.9 1536.5 58.4 1965.6 38.5 1944.3 68.2 1746.1 75.4 1472.1 65.6 1819.5 53.9 1627.4 45.1 1782.3 Send data to calculator Send data to Excel Coffee sales (in dollars) y 1200 1400 1600 1800 2000 2200 2400 x 40 50 60 70 80 90 Temperature (in degrees Fahrenheit) Figure 1 The least-squares regression line for these data has a slope of approximately −10.42 Answer the following. Carry your intermediate computations to at least four decimal places, and round your answers as specified below. (If necessary, consult a list of formulas.) (a) What is the value of the y-intercept of the least-squares regression line for these data? Round your answer to at least two decimal places. (b) What is the value of the sample correlation coefficient for these data? Round your answer to at least three decimal places.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.3: Least Squares Approximation

Problem 31EQ

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Managers of an outdoor coffee stand in Coast City are examining the relationship between (hot) coffee sales and daily temperature, hoping to be able to predict a day's total coffee sales from the maximum temperature that day. The bivariate data values for the coffee sales (denoted by y , in dollars) and the maximum temperature (denoted by x , in degrees Fahrenheit) for each of fifteen randomly selected days during the past year are given below. These data are plotted in the scatter plot in Figure 1.

Temperature, x (in degrees Fahrenheit)	Coffee sales, y (in dollars)
53.1	2215.5
76.6	1982.7
70.1	1937.3
39.7	2251.1
72.5	1603.9
48.4	2024.2
46.7	2135.3
83.9	1536.5
58.4	1965.6
38.5	1944.3
68.2	1746.1
75.4	1472.1
65.6	1819.5
53.9	1627.4
45.1	1782.3

Send data to calculator

Send data to Excel

Coffee sales
(in dollars)

1200

1400

1600

1800

2000

2200

2400

Temperature
(in degrees Fahrenheit)

Figure 1

The least-squares regression line for these data has a slope of approximately −10.42

Answer the following. Carry your intermediate computations to at least four decimal places, and round your answers as specified below. (If necessary, consult a list of formulas.)

(a)	What is the value of the y-intercept of the least-squares regression line for these data? Round your answer to at least two decimal places.

(b)	What is the value of the sample correlation coefficient for these data? Round your answer to at least three decimal places.