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Magazine Chapter 2: Problem 3
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. $$ (0,0),(1,1),(2,3),(3,3) $$
Short Answer
Expert verified
Line: \(y = 0.7x + 0.7\); Correlation Coefficient: \(r \approx 0.917\).
Step by step solution
01
Organize the Data
First, list the given data points clearly in a tabular format to facilitate calculations. Here, we have four data points as follows:\[(x_1, y_1) = (0, 0), (x_2, y_2) = (1, 1), (x_3, y_3) = (2, 3), (x_4, y_4) = (3, 3)\]
02
Calculate Sums
Calculate the necessary sums for the least squares formulas:Calculate \(\sum x_i = 0 + 1 + 2 + 3 = 6\), \(\sum y_i = 0 + 1 + 3 + 3 = 7\), \(\sum x_i^2 = 0^2 + 1^2 + 2^2 + 3^2 = 14\), \(\sum x_iy_i = 0\times0 + 1\times1 + 2\times3 + 3\times3 = 14\).
03
Use the Least Squares Formula to Find Slope (m)
The formula to find the slope \(m\) of the best-fit line is given by:\[ m = \frac{n(\sum x_iy_i) - (\sum x_i)(\sum y_i)}{n(\sum x^2_i) - (\sum x_i)^2} \] where \(n\) is the number of data points. Substituting the values, we get:\[ m = \frac{4(14) - (6)(7)}{4(14) - 6^2} = \frac{56 - 42}{56 - 36} = \frac{14}{20} = 0.7 \]
04
Use the Least Squares Formula to Find Y-Intercept (c)
The formula to find the intercept \(c\) of the best-fit line is:\[ c = \frac{(\sum y_i) - m (\sum x_i)}{n} \]Substitute the values calculated to find \(c\):\[ c = \frac{7 - 0.7 \times 6}{4} = \frac{7 - 4.2}{4} = \frac{2.8}{4} = 0.7 \]
05
Write the Equation of the Line
With \(m = 0.7\) and \(c = 0.7\), the equation of the line is:\[ y = 0.7x + 0.7 \]
06
Graph the Line on a Scatter Diagram
Plot the original points on a graph and draw the line \(y = 0.7x + 0.7\). This line should roughly pass through the middle of the data points standing as the best-fit line relative to the weight of each point.
07
Calculate the Correlation Coefficient (r)
The formula for the correlation coefficient \(r\) is:\[ r = \frac{n(\sum x_iy_i) - (\sum x_i)(\sum y_i)}{\sqrt{[n(\sum x^2_i) - (\sum x_i)^2][n(\sum y^2_i) - (\sum y_i)^2]}} \]Calculate \(\sum y_i^2 = 0^2 + 1^2 + 3^2 + 3^2 = 19\). Substituting in:\[ r = \frac{4(14) - (6)(7)}{\sqrt{[4(14) - 36][4(19) - 7^2]}} = \frac{14}{\sqrt{[56-36][76-49]}} = \frac{14}{\sqrt{20 \times 27}} \approx 0.917 \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Best-Fit Line
Using the least squares method to find a best-fit line is a robust way to analyze data points. This statistical technique helps to derive a straight line through a set of data points that minimizes the distance of all points from the line itself.
When we say 'best-fit line', we're referring to this specific straight line that has the least possible difference from the actual data points, summed across all the points. This line represents the most accurate prediction we can make for the given data.
When we say 'best-fit line', we're referring to this specific straight line that has the least possible difference from the actual data points, summed across all the points. This line represents the most accurate prediction we can make for the given data.
- **Slope (\( m \)):** Determines how steep the line is. It's calculated using the sum of the products of the x and y values (\( \sum x_iy_i \)) and other sum equations as detailed in the step-by-step solution.
- **Y-Intercept (\( c \)):** The point where the line crosses the y-axis. This illustrates the value of y when x is zero. The formula involves the sum of y-values and the slope previously calculated.
Correlation Coefficient
The correlation coefficient (\( r \)) is a numerical measure that illustrates how strongly two variables, x and y in our case, are related. This coefficient ranges from -1 to 1. \( r = 1 \) or \( r = -1 \) signifies a perfect linear relationship, while \( r = 0 \) indicates no linear relation.
In our example, the calculated correlation coefficient is approximately 0.917, which shows a strong positive relationship between x and y. The closer this number is to 1, the better the data points fit the calculated line, implying that x and y tend to increase together.
In our example, the calculated correlation coefficient is approximately 0.917, which shows a strong positive relationship between x and y. The closer this number is to 1, the better the data points fit the calculated line, implying that x and y tend to increase together.
- It helps us understand the direction of the relationship. A positive \( r \) means that as x increases, y also tends to increase.
- It also reveals the strength of this linear relationship. As in this case, a number near 1 indicates a strong linear link.
Scatter Diagram
A scatter diagram is a type of plot or mathematical diagram that uses Cartesian coordinates to display values typically for two variables (x and y) on a graph. It's a powerful visual tool to understand the relationship between these variables.
On this plot, each data point is represented by a dot. This allows us to visualize how closely related the variables are and if there's an observable pattern. In our exercise:
On this plot, each data point is represented by a dot. This allows us to visualize how closely related the variables are and if there's an observable pattern. In our exercise:
- Data points are plotted individually on the graph.
- The best-fit line, such as \( y = 0.7x + 0.7 \), is drawn through them. Ideally, the line will balance the points around it.
