Question

Fitting a Line to Data In Exercises \(55-60\), find the least squares regression line \(y=a x+b\) for the points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\) by solving the system for \(a\) and \(b\). (If you are unfamiliar with summation notation, look at the discussion in Section 7.1.) $$ \left\\{\begin{array}{c} n b+\left(\sum_{i=1}^{n} x_{i}\right) a=\sum_{i=1}^{n} y_{i} \\ \left(\sum_{i=1}^{n} x_{i}\right) b+\left(\sum_{i=1}^{n} x_{i}^{2}\right) a=\sum_{i=1}^{n} x_{i} y_{i} \end{array}\right. $$ $$ \left\\{\begin{array}{r} 5 b+10 a=20.2 \\ 10 b+30 a=50.1 \end{array}\right. $$

Short answer

The least squares regression line is: \(y = 0.97x + 2.1\).

Step-by-step solution

Understand the Given Equations

From the given problem, we know that we have a system of equations to solve for the coefficients 'a' and 'b'. The two equations are given as: \(5b + 10a = 20.2\) (equation 1) and \(10b + 30a = 50.1\) (equation 2). These equations represent the normal equations for the least squares problem.

Solve for 'b' from the first equation

From equation 1, isolate 'b'. This is accomplished by subtracting \(10a\) from both sides of equation 1, then divide by 5. Doing this yields: \(b = (20.2 - 10a)/ 5 = 4.04 - 2a\).

Substitute 'b' into the second equation

Substitute \( b = 4.04 - 2a \) from step 2 into equation 2. By doing this we have: \(10(4.04 - 2a) + 30a = 50.1\), simplify this equation to \(40.4 - 20a + 30a = 50.1\), then simplify further to \(10a = 9.7\).

Solve for 'a' from the simplified equation from step 3

Isolate 'a' from the equation \(10a = 9.7\) by dividing both sides by 10. Doing this yields: \(a = 0.97\).

Substitute 'a' into the first equation to solve for 'b'

Substitute a = 0.97 in equation 1: \(5b + 10(0.97) = 20.2\), which simplifies to \(5b + 9.7 = 20.2\). By isolating 'b', you get \(b = (20.2 - 9.7) /5 = 2.1\).

Key concepts

System of Equations

At the heart of finding the least squares regression line is a system of equations, which comprises two or more equations involving the same variables. The goal is to find a common solution to all the equations in the system — in the context of regression analysis, these variables are the coefficients of the regression line, often labeled 'a' for the slope and 'b' for the y-intercept. Our exercise presents a system composed of two linear equations, which can be solved using various algebraic methods such as substitution, elimination, or matrix operations. The key when dealing with a system of equations is to manipulate the equations to isolate one variable, which can then be substituted into the other equation(s) to find the solution.

• Understand the structure and components of the system of equations.
• Use algebraic techniques to isolate and solve for the variables.

Regression Analysis

Regression analysis is a statistical tool used to determine the relationship between a dependent variable and one or more independent variables. The least squares regression line is a specific type of regression analysis aimed at minimizing the sum of the squared differences (residuals) between observed values and the values predicted by the model. In simpler terms, it's about finding the 'best fit' line through a scatter plot of data points, which helps in predicting outcomes. Our exercise tasks us with finding this line, represented as the equation y = ax + b, where 'a' represents the slope and 'b' is the y-intercept.

• Familiarize yourself with the relationship between the dependent and independent variables.
• Grasp the importance of minimizing residuals for the best fit line.

Summation Notation

Summation notation, denoted by the Greek letter Sigma (\f\(\f\text{Why are you misusing LaTeX syntax?}\f\)\f$), is a concise way of expressing the sum of a series of terms. It's an essential tool in algebra, especially when dealing with sequences and series. In the context of linear regression, summation notation is used to simplify expressions involving the sum of values, like the sum of all x-coordinates and the sum of products of paired scores (x, y). By understanding summation notation, one can easily read and translate these expressions into algebraic terms required to solve the system of equations for the regression line coefficients.

• Learn to read and interpret summation notation expressions.
• Apply summation notation in summarizing data series for analysis.

Algebraic Solutions

Finding algebraic solutions involves manipulating and solving equations to find the value of unknown variables. The steps to solve our system of equations for the least squares regression line employ foundational algebraic concepts — from isolating variables to substituting values and simplifying expressions until the unknowns, 'a' and 'b', are determined. Algebraic solutions provide a systematic approach to reach precise answers, and understanding these methods ensures students can tackle various types of equations confidently.

• Practice the manipulation of algebraic equations to isolate variables.
• Understand how substitution helps in solving complex systems of equations.