91Ó°ÊÓ

(a) Suppose you are given the following \(x, y\) data pairs: $$ \begin{array}{l|lll} \hline x & 1 & 3 & 4 \\ \hline y & 2 & 1 & 6 \\ \hline \end{array} $$ Show that the least-squares equation for these data is \(y=1.071 x+0.143\) (rounded to three digits after the decimal). (b) Now suppose you are given these \(x, y\) data pairs: $$ \begin{array}{l|lll} \hline x & 2 & 1 & 6 \\ \hline y & 1 & 3 & 4 \\ \hline \end{array} $$ Show that the least-squares equation for these data is \(y=0.357 x+1.595\) (rounded to three digits after the decimal). (c) In the data for parts (a) and (b), did we simply exchange the \(x\) and \(y\) values of each data pair? (d) Solve \(y=0.143+1.071 x\) for \(x .\) Do you get the least-squares equation of part (b) with the symbols \(x\) and \(y\) exchanged? (e) In general, suppose we have the least-squares equation \(y=a+b x\) for a set of data pairs \(x, y .\) If we solve this equation for \(x\), will we necessarily get the least-squares equation for the set of data pairs \(y, x\) (with \(x\) and \(y\) exchanged)? Explain using parts (a) through (d).

Step by step solution

01

Set up least-squares formula

To find the least-squares linear regression of the form \(y = a + bx\), we need to determine \(a\) and \(b\) using the formulas: \[b = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}\]\[a = \frac{(\sum y) - b(\sum x)}{n}\]where \(n\) is the number of data pairs.

02

Calculate parameters for part (a)

For the given data: \(x = \{1, 3, 4\}\) and \(y = \{2, 1, 6\}\), calculate:\[\sum x = 1 + 3 + 4 = 8\]\[\sum y = 2 + 1 + 6 = 9\]\[\sum xy = (1 \times 2) + (3 \times 1) + (4 \times 6) = 32\]\[\sum x^2 = 1^2 + 3^2 + 4^2 = 26\]Substitute these sums into the \(b\) formula:\[b = \frac{3(32) - (8)(9)}{3(26) - 8^2} = \frac{96 - 72}{78 - 64} = \frac{24}{14} \approx 1.071\]Now, calculate \(a\):\[a = \frac{9 - 1.071 \times 8}{3} \approx \frac{9 - 8.568}{3} \approx \frac{0.432}{3} \approx 0.144\]Thus, the least-squares equation for part (a) is \(y \approx 1.071 x + 0.144\). Due to rounding, this is approximated in problem statement as \(y = 1.071x + 0.143\).

03

Calculate parameters for part (b)

Now consider the data: \(x = \{2, 1, 6\}\) and \(y = \{1, 3, 4\}\). Calculate:\[\sum x = 2 + 1 + 6 = 9\]\[\sum y = 1 + 3 + 4 = 8\]\[\sum xy = (2 \times 1) + (1 \times 3) + (6 \times 4) = 29\]\[\sum x^2 = 2^2 + 1^2 + 6^2 = 41\]Substitute these into the \(b\) formula:\[b = \frac{3(29) - (9)(8)}{3(41) - 9^2} = \frac{87 - 72}{123 - 81} = \frac{15}{42} \approx 0.357\]Calculate \(a\):\[a = \frac{8 - 0.357 \times 9}{3} \approx \frac{8 - 3.213}{3} \approx \frac{4.787}{3} \approx 1.596\]Thus, the least-squares equation for (b) is \(y \approx 0.357 x + 1.596\), rounded in the statement as \(y = 0.357 x + 1.595\).

04

Exchange x and y values

By examining the data for parts (a) and (b), we see that the \(x\) and \(y\) values have been swapped between the datasets of both parts. Moreover, finding the least-squares line typically involves changes in the regression line parameters.

05

Solve equation from part (a) for x

Rearrange the least-squares equation from part (a):\[y = 1.071x + 0.143\]For \(x\): solve for \(x\) as follows:\[x = \frac{y - 0.143}{1.071}\]This solution does not exactly yield the least-squares equation of part (b) when simply swapping \(x\) and \(y\), confirming that the regression line calculation isn't merely an inversion process of solving the line equation.

06

General conclusion for part (e)

When finding the least-squares equation for \(y = a + bx\) and solving for \(x\), the resulting expression is not equivalent to the least-squares equation of exchanging \(x\) and \(y\). This demonstrates that reversing roles in linear regression requires calculating anew, rather than merely solving the equation or rearranging terms.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Regression

Linear regression is a statistical method used to create a linear relationship between two quantitative variables: the independent variable (often denoted as \(x\)) and the dependent variable (\(y\)). The main goal of linear regression is to find the best-fitting straight line, known as the regression line, through a set of data points. This line minimizes the distance of the data points from the line itself, making it the best representation of the relationship between \(x\) and \(y\).

The equation of the regression line can be expressed in a linear form \(y = a + bx\), where \(a\) is the y-intercept (the value of \(y\) when \(x=0\)), and \(b\) is the slope (the rate of change of \(y\) with respect to \(x\)). Linear regression is widely used in predictive modeling, allowing us to predict values of \(y\) for any given \(x\) within the range of observed data.

Data Pairs

Data pairs represent the values of two related variables, usually denoted as \((x, y)\). In the context of linear regression, data pairs are the observed values used to determine the linear relationship between the variables.

Each data pair consists of an \(x\) (independent variable) and a \(y\) (dependent variable) value. These pairs are plotted as points on a two-dimensional graph, where the \(x\)-axis represents the independent variable and the \(y\)-axis represents the dependent variable.

By analyzing these data pairs, we can discover correlations or trends between the variables, which can then be modeled through linear regression. This process helps to visualize the data and understand how changes in one variable affect the other.

Equation Solving

Equation solving is a fundamental aspect of mathematics, which involves finding the values of variables that satisfy a given equation. In the context of linear regression, it often involves solving the equation for the line of best fit.

To find the linear regression equation, we solve using specific formulas for the slope \(b\) and the y-intercept \(a\).

• Slope \(b\) is calculated as: \[b = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}\]
• Y-intercept \(a\) is calculated as: \[a = \frac{(\sum y) - b(\sum x)}{n}\]

These formulas use summations of the data pairs and the total number of pairs \(n\). Solving these equations gives us the coefficients needed to write the regression equation.

Statistical Analysis

Statistical analysis in the context of linear regression involves examining the nature and strength of the relationship between two variables. By conducting a least squares regression analysis, we assess how well the regression line fits the data points.

Statistical techniques are utilized to measure the line's accuracy and reliability, often involving:

• Calculating the coefficient of determination \(R^2\), which indicates how much of the variability in \(y\) is explained by \(x\).
• Performing hypothesis tests to verify the significance of the slope \(b\).
• Analyzing residuals to check for patterns that might suggest a poor fit or the presence of outliers.

Statistical analysis provides valuable insights into the relationships within the data and ensures that the linear model is appropriately used for predictions and decision-making.