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Chapter 13: Problem 15

Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the given points. $$ (0,0),(1,1),(3,4),(4,2),(5,5) $$

Short Answer

Expert verified
The least squares regression line for the given points is in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The exact values of \(m\) and \(b\) depend on the results from Step 3 and Step 4.

Step by step solution

01

Define the Variables

First, establish the given points as pairs of x and y values: \[x = [0, 1, 3, 4, 5]\] and \[y = [0, 1, 4, 2, 5]\]
02

Compute the Mean

Determine the mean (average) of x-values and y-values. The formulas to calculate the mean are \(x_{mean} = \frac{\sum x}{n}\) and \(y_{mean} = \frac{\sum y}{n}\), where \(n\) is the number of points. For the x-values, sum up all the x-values and divide by the number of points. Do the same for the y-values.
03

Compute the Slope

Use the formula for the regression line (slope) \(m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}\) To do this, find the sum of the product of x and y pairs (\(\sum xy\)). Then, subtract the product of total x and total y divided by \(n\) from this, resulting in the numerator. For the denominator, find the sum of square x (\(\sum x^2\)) minus square of the sum of x (\((\sum x)^2\)). Divide the numerator by the denominator to get the slope.
04

Compute the Y-Intercept

Calculate the y-intercept using the formula \(b = \frac{\sum y - m(\sum x)}{n}\). Here \(m\) is the slope determined in previous step. This will provide the y-intercept of the regression line.
05

Write the Regression Line Equation

The final step is to formulate the equation for the least squares regression line using the computed slope (m) and y-intercept (b). The equation will be \(y = mx + b\)

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Most popular questions from this chapter

After a change in marketing, the weekly profit of the firm in Exercise 35 is given by \(P=200 x_{1}+580 x_{2}-x_{1}^{2}-5 x_{2}^{2}-2 x_{1} x_{2}-7500\) Estimate the average weekly profit if \(x_{1}\) varies between 55 and 65 units and \(x_{2}\) varies between 50 and 60 units.

Use a symbolic integration utility to evaluate the double integral. $$ \int_{0}^{1} \int_{x}^{1} \sqrt{1-x^{2}} d y d x $$

Set up the integral for both orders of integration and use the more convenient order to evaluate the integral over the region \(R\). $$ \begin{aligned} &\int_{R} \int x y d A\\\ &R \text { : rectangle with vertices at }(0,0),(0,5),(3,5),(3,0) \end{aligned} $$

Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression quadratic for the given points. Then plot the points and graph the least squares regression quadratic. $$ (0,10),(1,9),(2,6),(3,0) $$

Evaluate the double integral. $$ \int_{0}^{\infty} \int_{0}^{\infty} x y e^{-\left(x^{2}+y^{2}\right)} d x d y $$
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