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

Calculate the necessary sums for part (a)

First, calculate the sums needed for the least-squares equation. For the given data pairs \((1, 2), (3, 1), (4, 6)\), we need:- Sum of x-values: \( \sum x = 1 + 3 + 4 = 8 \)- Sum of y-values: \( \sum y = 2 + 1 + 6 = 9 \)- Sum of x*y: \( \sum xy = 1\cdot2 + 3\cdot1 + 4\cdot6 = 2 + 3 + 24 = 29 \)- Sum of x^2: \( \sum x^2 = 1^2 + 3^2 + 4^2 = 1 + 9 + 16 = 26 \)The number of data points, \(n\), is 3.

02

Apply the least-squares formulas for part (a)

The least squares equation is given by \( y = ax + b \), where:\[ b = \frac{ \sum y \cdot \sum x^2 - \sum x \cdot \sum xy }{ n \cdot \sum x^2 - (\sum x)^2 } \]\[ a = \frac{ n \cdot \sum xy - \sum x \cdot \sum y }{ n \cdot \sum x^2 - (\sum x)^2 } \]Substitute the values calculated in Step 1:- \( b = \frac{ 9 \cdot 26 - 8 \cdot 29 }{ 3 \cdot 26 - 8^2 } = \frac{ 234 - 232 }{ 78 - 64 } = \frac{2}{14} \approx 0.143 \)- \( a = \frac{ 3 \cdot 29 - 8 \cdot 9 }{ 3 \cdot 26 - 8^2 } = \frac{ 87 - 72 }{ 78 - 64 } = \frac{15}{14} \approx 1.071 \)Thus, the equation is \( y = 1.071x + 0.143 \).

03

Calculate the sums for part (b)

For part (b), the data pairs are \((2, 1), (1, 3), (6, 4)\). Calculate:- Sum of x-values: \( \sum x = 2 + 1 + 6 = 9 \)- Sum of y-values: \( \sum y = 1 + 3 + 4 = 8 \)- Sum of x*y: \( \sum xy = 2\cdot1 + 1\cdot3 + 6\cdot4 = 2 + 3 + 24 = 29 \)- Sum of x^2: \( \sum x^2 = 2^2 + 1^2 + 6^2 = 4 + 1 + 36 = 41 \) and \(n = 3\).

04

Apply least-squares formulas for part (b)

Utilize the formulas for \(a\) and \(b\) as previously described:- \( b = \frac{ 8 \cdot 41 - 9 \cdot 29 }{ 3 \cdot 41 - 9^2 } = \frac{ 328 - 261 }{ 123 - 81 } = \frac{67}{42} \approx 1.595 \)- \( a = \frac{ 3 \cdot 29 - 9 \cdot 8 }{ 3 \cdot 41 - 9^2 } = \frac{ 87 - 72 }{ 123 - 81 } = \frac{15}{42} \approx 0.357 \)Thus, the equation is \( y = 0.357x + 1.595 \).

05

Compare data swap effect and solve equation

Compare results from part (a) and (b). Initially, x and y pairs appear swapped, but result differs in slopes and intercepts when directly swapped.To verify solving for x, take equation from part (a): \[ y = 1.071x + 0.143 \]To solve for \(x\): \[ x = \frac{y - 0.143}{1.071} \approx 0.934y - 0.134 \]This result demonstrates that solving for \(x\) in terms of \(y\) does not give the least-squares regression from data swap.

06

General observation from parts (a) through (d)

Even if swapping \((x, y)\), solving for the opposite variable in regression doesn't equate to swapping regression results.The least-squares equation \(y = ax + b\) solved for \(x\) doesn't necessarily yield the least-squares line for swapped data \((y, x)\).Parts (a) and (b) illustrate swapping affects only data position, not regression results.

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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 powerful statistical method used to model the relationship between a dependent variable and one or more independent variables. It helps us understand how the dependent variable, often denoted as \( y \), changes when any one of the independent variables, usually denoted as \( x \), is varied.

Linear regression works by drawing the best-fit straight line through a set of data points. It aims to minimize the distance between the points and the line, so it captures the trend of the data effectively. The resulting line, known as the regression line, allows us to make predictions and infer relationships between variables.

If there is only one independent variable, as in the exercises provided, the method is known as "simple linear regression." When multiple explanatory variables exist, it is referred to as "multiple linear regression." This helps in analyzing complex situations where several factors may affect the outcome.

Data Pairs

Data pairs are vital in applying linear regression as they represent the real-world observations we are trying to model. Each pair consists of an \( (x, y) \) value, where \( x \) is the independent variable, and \( y \) is the dependent variable. In this exercise, data pairs like \((1, 2), (3, 1), (4, 6)\) and \((2, 1), (1, 3), (6, 4)\) are provided for analysis.

Observation points from data pairs help define the direction and steepness of the regression line. To draw a meaningful line, the algorithm calculates various sums such as the sum of \( x \) values, \( y \) values, and crucially, \( x \cdot y \) products. By analyzing these sums, we can determine the line that best represents the trend of the data.

• Accurate data collection is crucial as it impacts the regression analysis.
• The more data points, the more reliable the regression results tend to be.
• Data pairs form the basis for calculating the regression slope and intercept, which describe the line's position and angle.

Slope and Intercept

The slope and intercept are key components of the linear regression equation, expressed generally as \( y = ax + b \). Here, \( a \) denotes the slope, while \( b \) represents the intercept.

The slope \( a \) indicates how much \( y \) is expected to increase when \( x \) increases by one unit. It defines the angle of the line drawn through the data points and reflects the direction of the relationship:

• A positive slope indicates that \( y \) increases as \( x \) increases.
• A negative slope shows that \( y \) decreases as \( x \) increases.

The intercept \( b \) is the value of \( y \) when \( x \) is zero. It places the line on the graph and gives the point where the line crosses the \( y \)-axis.

These parameters not only define the line but also help interpret the relationship between variables in context. For example, in part (a), the slope is approximately 1.071, showing a steep positive relationship, whereas, in part (b), the slope is a much gentler 0.357, suggesting a weaker positive trend.