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Often we have a set of data from an experiment and we want to fit a straight line (or other curve) to that data. If the system has more equations than unknowns it is called over determined. Then we can solve the equation Ax = b approximately by trying to minimize || Ax - b ||. This approach involves solving a linear system called the normal equation, AtAx = ATb. A experiment was conducted with the following results We would like to determine if x and y have a linear or quadratic relationship. Use the normal equation to fit a straight line y = a0 + a1x to the data. Use the normal equation to fit a parabola y = a0 + a1x + a2x2 to the data. Graph the data using appropriate software and your fitted curves on the same axis. Which do you think fits the data best? Construct or research a mathematical method to determine which line fits best. Describe the method and use it to determine which of the curves gives the best fit.

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Related questions

Average price per square foot of floor area in new one-family houses sold. Price per square foot of floor area excludes value of improved lot Source: Year 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 Price per square foot $55.18 $57.41 $60.21 $62.45 $64.38 $66.81 $67.83 $67.70 $70.43 $71.93 $75.68 $79.21 $85.13 $90.63 $91.99 Instead of working on portions of the scatter plots, we want to approximate the data by a line we can use to predict prices in the future. The line that will best fit the given data is called the linear regression. The approximation will be acceptable whether the determina- tion coefficient R (r**2) is close to 1 or the correlation coefficient r is close to -1 or +1.

a) using your entered data to find the equation y = ax + b of the best line through the data points and give the values of coefficients a and b, and of the determination coefficient R or correlation coefficient r.

b. Use Excel to plot the scatter plot and the regression line on the same graph.

c. Is the linear regression a visually good approximation of the data? Sustain your argument by a mathematical proof.

d. use the linear regression to predict the price per square foot area in 2009 e. In what year will the price per square foot be greater than $199?

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Algebra

Use a transformation of the graph of y = x3 to graph the function. h(x)=4 - (x - 2)5 Choose the correct graph below. Find the vertical, horizontal, and oblique asymptotes, if any, for the given rational function. Select the correct choice below and fill in any answer boxes within your choice. The vertical asymptote(s) is / are x = (Use a comma to separate answers as needed. Use integers or fractions for any numbers in the expression.) There is no vertical asymptote. Select the correct choice below and fill in any answer boxes within your choice. The horizontal asymptote(s) is / are y = (Use a comma to separate answers as needed. Use integers or fractions for any numbers in the expression.) There is no horizontal asymptote. Select the correct choice below and fill in any answer boxes within your choice. The oblique asymptote{s)is / are y = (Use a comma to separate answers as needed. Use integers or fractions for any numbers in the expression.) There is no oblique asymptote. Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value. f(x) = 6x2 + 12x - 3 The quadratic function has a value. The value is For the data given below, answer parts (a) through (d). Find the equation of the line containing the first and the last data points. (Type your answer in slope-intercept form. Use integers or fractions for any numbers in the expression.) Draw a scatter diagram and the line found in part (a) on the same axes. Choose the correct graph below. Use a graphing utility to find the line of best fit. Which of the following is the equation of the line of best fit? Use a graphing utility to draw the scatter diagram and graph the line of best fit on it. Choose the correct graph below. Complete the square of the given quadratic expression. Then, graph the function using the technique of shifting. f(x) = x2 - 8x + 15 Complete the square by entering the correct numbers into the expression below. Choose the correct graph below. Analyze the graph of the function. What is the domain of R(x)? What is the equation of the vertical asymptote(s) of R(x)? Select the correct choice below and fill in any answer boxes within your choice. What is the equation of the horizontal or oblique asymptote of R(x)? Select the correct choice below and fill in any answer boxes within your choice. Choose the correct graph for R(x) below. Find k such that f(x) = x3 - kx2 + kx + 4 has the factor x - 2. k = (Simplify.) F(x) = 4x4 - 8X2 + 3 Determine whether F is even. odd. or neither. There is a local minimum of - 1 at 1. Determine a second local minimum, (a) Determine whether F is even. odd. or neither. Neither Even Odd There is a local minimum of - 1 at 1. Determine a second local minimum. The second local minimum is . F(x) = 4x4 - 8X2 + 3 Determine whether F is even. odd. or neither. There is a local minimum of - 1 at 1. Determine a second local minimum, (a) Determine whether F is even. odd. or neither. Neither Even Odd There is a local minimum of - 1 at 1. Determine a second local minimum. The second local minimum is .The function f is defined as follows. Find the domain of the function. Locate any intercepts. Graph the function. Based on the graph, find the range. Is f continuous on its domain? The domain of the function f is (Type your answer in interval notation.) Locate any intercepts. Select the correct choice below and. if necessary, fill in the answer box to complete your choice. The intercept(s) is/are (Type an ordered pair. Use a comma to separate answers as needed.) There are no intercepts. Choose the correct graph of f(x) below. Is f continuous on its domain? Select the correct choice below and. if necessary, fill in the answer box to complete your choice. Find the difference quotient of f; that is. find for the following function. Be sure to simplify. Determine whether the given function is linear or nonlinear. If it is linear, determine the slope. Is the function a linear function? Determine the slope. Select the correct choice below and fill in any answer boxes within your choice. The slope of the function is .The function is not linear. Analyze the graph of the function. What is the domain of f(x)? What is the equation of the vertical asymptote(s) of f(x)? Select the correct choice below and fill in any answer boxes within your choice. x = (Use a comma to separate answers as needed. Type an integer or a fraction.) There is no vertical asymptote. What is the equation of the horizontal or oblique asymptote of f(x)? Select the correct choice below and fill in any answer boxes within your choice. y = (Simplify your answer.) There is no horizontal or oblique asymptote. Choose the correct graph for f(x) below. Use a transformation of the graph of y = x3 to graph the function. f(x) = (x - 6)5 + 5 Select the graph of f(x) = (x - 6)3 + 5. Determine whether the graph below is that of a function by using the vertical-line test. If it is, use the graph to find its domain and range. the intercepts, if any. any symmetry with respect to the x-axis: y-axis: or the origin. Is the graph that of a function? If the graph is that of a function, what are the domain and range of the function? Select the correct choice below and fill in any answer boxes within your choice. What are the intercepts? Select the correct choice below and fill in any answer boxes within your choice. (Type an ordered pair. Use a comma to separate answers as needed.) There are no intercepts. The graph is not a function. Determine if the graph is symmetrical. It is symmetrical with respect to the origin. It is symmetrical with respect to the x-axis. It is symmetrical with respect to the y-axis. The graph is not symmetrical. The graph is not a function. Solve the following inequality. Choose the correct solution.

Show transcribed image text Use a transformation of the graph of y = x3 to graph the function. h(x)=4 - (x - 2)5 Choose the correct graph below. Find the vertical, horizontal, and oblique asymptotes, if any, for the given rational function. Select the correct choice below and fill in any answer boxes within your choice. The vertical asymptote(s) is / are x = (Use a comma to separate answers as needed. Use integers or fractions for any numbers in the expression.) There is no vertical asymptote. Select the correct choice below and fill in any answer boxes within your choice. The horizontal asymptote(s) is / are y = (Use a comma to separate answers as needed. Use integers or fractions for any numbers in the expression.) There is no horizontal asymptote. Select the correct choice below and fill in any answer boxes within your choice. The oblique asymptote{s)is / are y = (Use a comma to separate answers as needed. Use integers or fractions for any numbers in the expression.) There is no oblique asymptote. Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value. f(x) = 6x2 + 12x - 3 The quadratic function has a value. The value is For the data given below, answer parts (a) through (d). Find the equation of the line containing the first and the last data points. (Type your answer in slope-intercept form. Use integers or fractions for any numbers in the expression.) Draw a scatter diagram and the line found in part (a) on the same axes. Choose the correct graph below. Use a graphing utility to find the line of best fit. Which of the following is the equation of the line of best fit? Use a graphing utility to draw the scatter diagram and graph the line of best fit on it. Choose the correct graph below. Complete the square of the given quadratic expression. Then, graph the function using the technique of shifting. f(x) = x2 - 8x + 15 Complete the square by entering the correct numbers into the expression below. Choose the correct graph below. Analyze the graph of the function. What is the domain of R(x)? What is the equation of the vertical asymptote(s) of R(x)? Select the correct choice below and fill in any answer boxes within your choice. What is the equation of the horizontal or oblique asymptote of R(x)? Select the correct choice below and fill in any answer boxes within your choice. Choose the correct graph for R(x) below. Find k such that f(x) = x3 - kx2 + kx + 4 has the factor x - 2. k = (Simplify.) F(x) = 4x4 - 8X2 + 3 Determine whether F is even. odd. or neither. There is a local minimum of - 1 at 1. Determine a second local minimum, (a) Determine whether F is even. odd. or neither. Neither Even Odd There is a local minimum of - 1 at 1. Determine a second local minimum. The second local minimum is . F(x) = 4x4 - 8X2 + 3 Determine whether F is even. odd. or neither. There is a local minimum of - 1 at 1. Determine a second local minimum, (a) Determine whether F is even. odd. or neither. Neither Even Odd There is a local minimum of - 1 at 1. Determine a second local minimum. The second local minimum is .The function f is defined as follows. Find the domain of the function. Locate any intercepts. Graph the function. Based on the graph, find the range. Is f continuous on its domain? The domain of the function f is (Type your answer in interval notation.) Locate any intercepts. Select the correct choice below and. if necessary, fill in the answer box to complete your choice. The intercept(s) is/are (Type an ordered pair. Use a comma to separate answers as needed.) There are no intercepts. Choose the correct graph of f(x) below. Is f continuous on its domain? Select the correct choice below and. if necessary, fill in the answer box to complete your choice. Find the difference quotient of f; that is. find for the following function. Be sure to simplify. Determine whether the given function is linear or nonlinear. If it is linear, determine the slope. Is the function a linear function? Determine the slope. Select the correct choice below and fill in any answer boxes within your choice. The slope of the function is .The function is not linear. Analyze the graph of the function. What is the domain of f(x)? What is the equation of the vertical asymptote(s) of f(x)? Select the correct choice below and fill in any answer boxes within your choice. x = (Use a comma to separate answers as needed. Type an integer or a fraction.) There is no vertical asymptote. What is the equation of the horizontal or oblique asymptote of f(x)? Select the correct choice below and fill in any answer boxes within your choice. y = (Simplify your answer.) There is no horizontal or oblique asymptote. Choose the correct graph for f(x) below. Use a transformation of the graph of y = x3 to graph the function. f(x) = (x - 6)5 + 5 Select the graph of f(x) = (x - 6)3 + 5. Determine whether the graph below is that of a function by using the vertical-line test. If it is, use the graph to find its domain and range. the intercepts, if any. any symmetry with respect to the x-axis: y-axis: or the origin. Is the graph that of a function? If the graph is that of a function, what are the domain and range of the function? Select the correct choice below and fill in any answer boxes within your choice. What are the intercepts? Select the correct choice below and fill in any answer boxes within your choice. (Type an ordered pair. Use a comma to separate answers as needed.) There are no intercepts. The graph is not a function. Determine if the graph is symmetrical. It is symmetrical with respect to the origin. It is symmetrical with respect to the x-axis. It is symmetrical with respect to the y-axis. The graph is not symmetrical. The graph is not a function. Solve the following inequality. Choose the correct solution.

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