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Chapter 2: Problem 36

find the least squares regression line. $$(1,0),(3,3),(5,6)$$

Short Answer

Expert verified
The line of least squares regression equation is y = 3 + 0x, or simply y = 3.

Step by step solution

01

Calculate the necessary averages

We need to calculate the averages of x-values, y-values, xy-product, and squares of x-values. For our given points (1,0),(3,3),(5,6), let's use the formulas\[\bar{x} = \frac{1+3+5}{3} = 3 \],\[\bar{y} = \frac{0+3+6}{3} = 3 \], \[ \bar{xy} = \frac{(1*0)+(3*3)+(5*6)}{3} = 7 \], and \[\bar{x^2} = \frac{(1^2)+(3^2)+(5^2)}{3} = 11 \]
02

Determine the slope of the line

We'll use these averages to find the slope (b1), using the formula \[b1= \frac{\bar{xy}-\bar{x}\bar{y}}{\bar{x^2}-\bar{x}^2} \]The slope b1 would be \[ \frac{7 - 3*3}{11-3^2} = 0 \]
03

Determine the Y-intercept

Get the Y-intercept (b0) using the formula\[b0= \bar{y}-b1\bar{x}\]So the Y-intercept would be \[ 3 - 0*3 = 3 \]

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Key Concepts

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

Slope Calculation
The slope of a regression line is a crucial element in understanding the relationship between two variables. In least squares regression, the slope tells us how much the dependent variable (y) is expected to increase (or decrease) with a one-unit increase in the independent variable (x).

To calculate the slope (\(b_1\)), you follow this formula:
  • Find the average of the products of x and y, denoted as \(\bar{xy}\).
  • Find the product of the averages of x and y, given by \(\bar{x}\bar{y}\).
  • Identify the average of the squares of x, \(\bar{x^2}\), and the square of the average of x, \(\bar{x}^2\).
  • Plug these into the formula: \[b_1 = \frac{\bar{xy} - \bar{x}\bar{y}}{\bar{x^2} - \bar{x}^2}\].
For our data points (1,0), (3,3), and (5,6), the slope is calculated to be 0. This indicates there's no linear relationship that can be drawn between x and y in this specific set of data, meaning changes in x do not lead to changes in y.
Regression Line
A regression line is a straight line that best fits the data points on a scatterplot. Its purpose is to convey the trend of the data and help predict values of the dependent variable based on the independent variable.

The line is described mathematically by the equation:
  • \( y = b_0 + b_1x \),
where \(b_0\) is the y-intercept and \(b_1\) is the slope of the line. In our example, since the slope (\(b_1\)) was calculated to be 0, the equation of the line simplifies to \( y = 3 \).

This simplified regression line conveys that all y-values are predicted to be 3 regardless of the x-values, reflecting the average y-value of the dataset.
Y-Intercept Determination
Determining the y-intercept of a regression line is essential for completing its equation. The y-intercept, \(b_0\), signifies the expected value of y when x is 0. This provides insight into the starting point or baseline level of your dependent variable.

To find the y-intercept, use the formula:
  • \[ b_0 = \bar{y} - b_1\bar{x} \]
  • This formula considers the average value of the dependent variable, subtracts the effect of the slope multiplied by the average independent variable.
In the provided data set, since the slope \(b_1\) is 0, the y-intercept \(b_0\) becomes simply \(\bar{y}\), which is 3. This constant y-intercept means, regardless of the value of x, the predicted y-value remains 3, indicating no variation in y related to x in this dataset.

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Most popular questions from this chapter

Let \(A, D,\) and \(P\) be \(n \times n\) matrices satisfying \(P^{-1} A P=D .\) Solve this equation for \(A .\) Must it be true that \(A=D ?\)

An industrial system has three industries and the input-output matrix \(D\) and external demand matrix \(E\) shown below. $$D=\left[\begin{array}{ccc} 0.2 & 0.4 & 0.4 \\ 0.4 & 0.2 & 0.2 \\ 0.0 & 0.2 & 0.2 \end{array}\right]$$ and $$E=\left[\begin{array}{c} 5000 \\ 2000 \\ 8000 \end{array}\right]$$ Solve for the output matrix \(X\) in the equation \(X=D X+E\)

Determine whether the matrix is idempotent. A square matrix \(A\) is idempotent if \(A^{2}=A\) $$\left[\begin{array}{rr} 2 & 3 \\ -1 & -2 \end{array}\right]$$

find the inverse of the elementary matrix. $$\left[\begin{array}{ll} 5 & 0 \\ 0 & 1 \end{array}\right]$$

Find an example of a singular \(2 \times 2\) matrix satisfying \(A^{2}=A\)
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