Write down the method of linear least squares fit and derive expressions for thecorrelation coefficient and covariance and explain their significance.

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Answered: Write down the method of linear least…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.CM: Cumulative Review

Problem 21CM

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Apply the linear least-squares method to each set of data to obtain a mathematical model showing the relationships between variables.
An engineer is testing a new car model to determine how its fuel efficiency, measured in L/(100 km), is related to its speed, which is measured in km/hour. The engineer calculates the average speed for 30 trials. The average speed is an example of a (statistic or parameter) The engineer would like to find the least squares regression line predicting fuel used (y) from speed (x) for the 30 cars he observed. He collected the data below. Speed 62 65 80 82 85 87 90 96 98 100 Fuel 12 13 14 13 14 14 15 15 16 15 Speed 100 102 104 107 112 114 114 117 121 122 Fuel 16 17 16 17 18 17 18 17 18 19 Speed 124 127 127 130 132 137 138 142 144 150 Fuel 18 19 20 19 21 23 22 23 24 26 The regression line equation is Round each number to four decimal places.
What is the least-squares line approximation to a set of datapoints? How is the line determined?
An engineer wants to determine how the weight of a gas-powered car, x, affects gas mileage, y. The accompanying data represent the weights of various domestic cars and their miles per gallon in the city for the most recent model year. Complete parts (a) Find the least-squares regression line treating weight as the explanatory variable and miles per gallon as the response variable.
An engineer found that by including small amounts of a compound in rechargeable batteries for portable computers, she could extend their lifetimes. She experimented with different amounts of the additive and the data are Amount of additive Life (hours) 1 4 2. 3 3 4 a) Obtain the least squares fit of a straight line to the amount of additive. b) Test whether or not the slope B-0. Take a = 0.10 as your level of significance.
Assume that the table below displays the number of freight tonnes carried (in millions) in Australia between 1990 and 2015.Year199019952000200520102015Number of freight tonnes (millions)6684104126160186a) Draw a scatter diagram of the data.b) Find the least squares regression line of number of freight tonnes (in millions) carried on a year. (Note: Use only the last two digits of the year).c) Draw the line in (b) on the scatter diagram in (a). Comment on whether you feel the line is a good fit of the data.d) Use the line in (b) to predict the number of freight tonnes that might be carried in Australia in both 2020 and 2025.e) Do you think your predictions in (d) will be relatively accurate? Why or why not?
A researcher wishes to determine whether the rate of water flow (in liters per second) over an experiment soil bed can be used to predict the amount of soil washed away (in kilograms). The researcher measures the amount of soil washed away for various flow rates, and from these data calculates the least squares regression line to be: amount of eroded soil = 1.3(flow rate) + 0.4 One of the flow rates used by the researcher was 0.3 liters per second and for this flow rate the amount of eroded soil was 0.8 kilograms. These values were used in determining the equation of the regression line. Determine the residual corresponding to these values.

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Write down the method of linear least squares fit and derive expressions for the correlation coefficient and covariance and explain their significance.