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Chapter 1: Chapter 1

Problem 1

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(a)$ find the equation of the leastsquares line for the data and (b) draw a scatter diagram for the data and graph the least-squares line. $$ \begin{array}{lllll} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 \\ \hline \boldsymbol{y} & 4 & 6 & 8 & 11 \\ \hline \end{array} $$

Short Answer

Expert verified
The least-squares line for the given data points is \(y = 2.3x + 2.5\). When graphing the scatter plot of the data points (1, 4), (2, 6), (3, 8), and (4, 11), and the least-squares line, the linear regression line shows the best fit relationship among the given data points.
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TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Calculate mean values of x and y

To find the mean of x values, sum up all the x values and divide by the number of data points; do the same for y. In this case: Mean of x values: \(\bar{x} = \frac{1+2+3+4}{4} = 2.5\) Mean of y values: \(\bar{y} = \frac{4+6+8+11}{4} = 7.25\) ###Step 2: Calculate elements for slope (m) and y-intercept (b) ###

Step 2: Calculate elements for slope (m) and y-intercept (b)

We will be using the following formulas for m and b: \(m = \frac{\sum(x_i-\bar{x})(y_i-\bar{y})}{\sum(x_i-\bar{x})^2}\) \(b = \bar{y} - m\bar{x}\) Firstly, calculate the numerators and denominators of the slope formula: \((x_1-\bar{x})(y_1-\bar{y}) = (1-2.5)(4-7.25) = 1.5*3.25 = 4.875\) \((x_2-\bar{x})(y_2-\bar{y}) = (2-2.5)(6-7.25) = 0.5*1.25 = 0.625\) \((x_3-\bar{x})(y_3-\bar{y}) = (3-2.5)(8-7.25) = 0.5*0.75= 0.375\) \((x_4-\bar{x})(y_4-\bar{y}) = (4-2.5)(11-7.25) = 1.5*3.75= 5.625\) Now calculate the sum of these values: \(\sum(x_i-\bar{x})(y_i-\bar{y}) = 4.875 + 0.625 + 0.375 + 5.625 = 11.5\) Next, calculate denominators: \((x_1-\bar{x})^2 = (1-2.5)^2 = 1.5^2 = 2.25\) \((x_2-\bar{x})^2 = (2-2.5)^2 = 0.5^2 = 0.25\) \((x_3-\bar{x})^2 = (3-2.5)^2 = 0.5^2 = 0.25\) \((x_4-\bar{x})^2 = (4-2.5)^2 = 1.5^2 = 2.25\) Now calculate the sum of these values: \(\sum(x_i-\bar{x})^2 = 2.25 + 0.25 + 0.25 + 2.25 = 5\) ###Step 3: Calculate the slope (m) and y-intercept (b) ###

Step 3: Calculate the slope (m) and y-intercept (b)

Now, we have all the elements needed to calculate the slope (m): \(m = \frac{11.5}{5} = 2.3\) Next, use the slope to calculate the y-intercept (b): \(b = \bar{y} - m\bar{x} = 7.25 - 2.3*2.5 = 2.5\) Thus, the equation of the least-squares line is: \(y = 2.3x + 2.5\) ###Step 4: Draw a scatter plot and graph the least-squares line###

Step 4: Draw a scatter plot and graph the least-squares line

To create a scatter plot, plot the given data points on a graph: (1, 4), (2, 6), (3, 8), and (4, 11) Now, graph the least-squares line using the equation: \(y = 2.3x + 2.5\) The graph should display all four data points and the linear regression line that best fits the given data, showing their relationship.

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