Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 2: Problem 8
Find the best-fitting straight line to the given set of data, using the method of least squares. Graph this straight line on a scatter diagram. Find the correlation coefficient. $$ (0,0),(1,2),(2,1),(3,2),(4,4) $$
Short Answer
Expert verified
The line is \( y = 0.84x + 0.12 \) with a correlation coefficient of approximately 0.896.
Step by step solution
01
Collect Data Points
We are given the set of data points: \((0,0), (1,2), (2,1), (3,2), (4,4)\). These are our \(x\) and \(y\) values.
02
Calculate Means
Calculate the means of the \(x\) and \(y\) values. For \(x\): \[ \bar{x} = \frac{0 + 1 + 2 + 3 + 4}{5} = 2 \]For \(y\): \[ \bar{y} = \frac{0 + 2 + 1 + 2 + 4}{5} = 1.8 \]
03
Calculate Slope (m)
The slope \(m\) is calculated using the formula: \[ m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}} \]Using our data:\(\sum{(x_i - \bar{x})(y_i - \bar{y})} = (-2)(-1.8) + (-1)(0.2) + (0)(-0.8) + (1)(0.2) + (2)(2.2) = 8.4\)\(\sum{(x_i - \bar{x})^2} = (-2)^2 + (-1)^2 + (0)^2 + (1)^2 + (2)^2 = 10\)Thus,\[ m = \frac{8.4}{10} = 0.84 \]
04
Calculate Intercept (b)
The intercept \(b\) can be calculated using:\[ b = \bar{y} - m\bar{x} \]\[ b = 1.8 - (0.84)(2) = 0.12 \]
05
Write the Equation of the Line
The equation of the best-fitting line is:\[ y = mx + b \]Substitute the values for \(m\) and \(b\): \[ y = 0.84x + 0.12 \]
06
Calculate Correlation Coefficient (r)
The correlation coefficient \(r\) is given by:\[ r = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sqrt{\sum{(x_i - \bar{x})^2}\sum{(y_i - \bar{y})^2}}} \]\(\sum{(y_i - \bar{y})^2} = (-1.8)^2 + (0.2)^2 + (-0.8)^2 + (0.2)^2 + (2.2)^2 = 8.8\)Hence,\[ r = \frac{8.4}{\sqrt{10 \times 8.8}} = \frac{8.4}{9.38} \approx 0.896 \]
07
Graph the Line and Scatter Diagram
Plot the data points and the best-fitting line equation \( y = 0.84x + 0.12 \) on a scatter diagram. Draw the line across the graph to visualize the fit.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Least Squares Method
The least squares method is a reliable statistical technique used to find the best-fitting line for a set of data points. This approach minimizes the sum of the squares of the vertical distances (errors) between the observed values (data points) and the values predicted by the line. By doing so, it ensures that the fit is optimal, meaning the line comes as close to all the data points as possible.
It involves calculating the slope (m) and the y-intercept (b) of the line. Using equations derived from the observations, one finds these values to ensure the least error. The line is then represented by the equation, \(y = mx + b\). This equation gives us the predicted value of y for any given x. The least squares method is foundational in linear regression analysis, helping in making future predictions or understanding the relationships between variables in a dataset.
It involves calculating the slope (m) and the y-intercept (b) of the line. Using equations derived from the observations, one finds these values to ensure the least error. The line is then represented by the equation, \(y = mx + b\). This equation gives us the predicted value of y for any given x. The least squares method is foundational in linear regression analysis, helping in making future predictions or understanding the relationships between variables in a dataset.
Correlation Coefficient
The correlation coefficient is a statistical measure that evaluates the strength and direction of a linear relationship between two variables. It is denoted by \(r\) and varies between -1 and 1. A correlation coefficient close to 1 indicates a strong positive linear relationship, meaning as one variable increases, so does the other. Conversely, a correlation of -1 implies a strong negative relationship, meaning as one variable increases, the other decreases. A value close to 0 suggests little to no linear relationship.
In the context of our example, the correlation coefficient is calculated using the data values and their means. We observed \(r \approx 0.896\), signifying a strong positive relationship between our data's x and y values. This suggests the line of best fit accurately describes the trend of the data.
In the context of our example, the correlation coefficient is calculated using the data values and their means. We observed \(r \approx 0.896\), signifying a strong positive relationship between our data's x and y values. This suggests the line of best fit accurately describes the trend of the data.
Data Points
Data points are individual values in a data set, represented as coordinates in a two-dimensional space. In linear regression, these points are crucial as they are used to determine the best-fitting line via the least squares method.
In our exercise, the given data points are \((0,0), (1,2), (2,1), (3,2), (4,4)\). Each pair of numbers consists of an independent variable \(x\) and a dependent variable \(y\). These points are plotted on a graph, which helps to visualize the relationship between the variables. By plotting and analyzing these data points, we identify patterns and relationships that assist in predicting future outcomes or interpreting past events.
In our exercise, the given data points are \((0,0), (1,2), (2,1), (3,2), (4,4)\). Each pair of numbers consists of an independent variable \(x\) and a dependent variable \(y\). These points are plotted on a graph, which helps to visualize the relationship between the variables. By plotting and analyzing these data points, we identify patterns and relationships that assist in predicting future outcomes or interpreting past events.
Equation of a Line
In the realm of linear regression, the equation of a line is essential as it represents the relationship between variables in a concise form. The standard equation of a line is \(y = mx + b\), where \(m\) denotes the slope and \(b\) is the y-intercept.
The slope \(m\) indicates the steepness of the line and the direction of the relationship between \(x\) and \(y\). A positive slope suggests a positive correlation between variables, while a negative slope indicates a negative correlation. The y-intercept \(b\) is where the line crosses the y-axis, showing the value of \(y\) when \(x\) is zero.
In our calculation step, we found \(m = 0.84\) and \(b = 0.12\), resulting in the equation \(y = 0.84x + 0.12\). This defined line helps in understanding how \(y\) changes with \(x\) based on the observed data, making further analysis clear and precise.
The slope \(m\) indicates the steepness of the line and the direction of the relationship between \(x\) and \(y\). A positive slope suggests a positive correlation between variables, while a negative slope indicates a negative correlation. The y-intercept \(b\) is where the line crosses the y-axis, showing the value of \(y\) when \(x\) is zero.
In our calculation step, we found \(m = 0.84\) and \(b = 0.12\), resulting in the equation \(y = 0.84x + 0.12\). This defined line helps in understanding how \(y\) changes with \(x\) based on the observed data, making further analysis clear and precise.
