91Ó°ÊÓ

(a) By hand, draw a scatter diagram treating \(x\) as the explanatory variable and y as the response variable. (b) Select two points from the scatter diagram and find the equation of the line containing the points selected. (c) Graph the line found in part (b) on the scatter diagram. (d) By hand, determine the least-squares regression line. (e) Graph the least-squares regression line on the scatter diagram. (f) Compute the sum of the squared residuals for the line found in part (b). (g) Compute the sum of the squared residuals for the leastsquares regression line found in part (d). (h) Comment on the fit of the line found in part (b) versus the least-squares regression line found in part ( \(d\) ). $$ \begin{array}{rrrrrr} \hline x & -2 & -1 & 0 & 1 & 2 \\ \hline y & 7 & 6 & 3 & 2 & 0 \\ \hline \end{array} $$

Step by step solution

01

- Draw the Scatter Diagram

Plot the points \((-2,7)\), \((-1,6)\), \(0,3)\), \(1,2)\), and \(2,0)\) on a graph with x as the explanatory variable and y as the response variable.

02

- Select Two Points

Choose the points \((-2,7)\) and \(2,0)\) to find the equation of the line.

03

- Find the Equation of the Line

Using the formula \((y_2 - y_1) / (x_2 - x_1)\) to find the slope: \((-2-(-7)) / (2-(-2)) = -7/4\). The equation of the line is \(y - y_1 = m(x - x_1)\) => \(y - 7 = -7/4(x + 2)\). This simplifies to \(y = -7/4x + 3.25\).

04

- Graph the Line

Plot the line \(y = -7/4x + 3.25\) on the scatter diagram.

05

- Determine the Least-Squares Regression Line

Calculate the slope (b1) and intercept (b0) for the least-squares regression line using the formulas: \(b_1 = \frac{\text{SS}_{xy}}{\text{SS}_x}\) and \(b_0 = ȳ - b_1x̄\), where \(\text{SS}_{xy} = Σ(xi - x̄)(yi - ȳ)\) and \(\text{SS}_x = Σ(xi - x̄)^2\). After calculations, we get \(b_1 = -1.7\) and \(b_0 = 3\). The equation is \(y = -1.7x + 3\).

06

- Graph the Least-Squares Regression Line

Plot the least-squares regression line \(y = -1.7x + 3\) on the scatter diagram.

07

- Compute Sum of Squared Residuals (Part b)

Calculate the residuals for each point using the line \(y = -7/4x + 3.25\), square them and sum: \(Σ(\text{residuals}^2) = 0.0625\).

08

- Compute Sum of Squared Residuals (Part d)

Calculate the residuals for each point using the line \(y = -1.7x + 3\), square them and sum: \(Σ(\text{residuals}^2) = 0.4\).

09

- Comment on the Fit

The sum of the squared residuals (0.0625) for the line found in part (b) is lower than that for the least-squares regression line (0.4). Therefore, the line from part (b) fits the data better.

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Key Concepts

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

Scatter Diagram

A scatter diagram is a graph that shows the relationship between two variables. Each point on the graph represents a pair of values. In our exercise, we plotted points using the given data.

The x-values are the explanatory variable, and the y-values are the response variable. The explanatory variable helps explain changes in the response variable.

To create a scatter diagram, we plot the pairs time data:

• (-2, 7)
• (-1, 6)
• (0, 3)
• (1, 2)
• (2, 0)

The scatter diagram helps us visually see if there is any correlation or pattern.

Sum of Squared Residuals

Residuals are the differences between the observed values and the values predicted by a model. In regression analysis, we often calculate residuals to assess the model's accuracy. The smaller the residuals, the better the model fits the data.

When squared, residuals highlight larger deviations more sharply. We sum these squared residuals to measure a model's total prediction error. This sum is key in determining the goodness of fit for a regression line.

In our exercise, we computed the sum of squared residuals for two lines:

• The manually drawn line: y = -7/4x + 3.25 (for this line, the sum of squared residuals is 0.0625)


• The least-squares regression line: y = -1.7x + 3 (for this line, the sum of squared residuals is 0.4)

### Why is this important? Summing squared residuals enables us to compare different models. The model with a smaller sum generally offers a better fit. Here, we see the line from part (b) fits better than the least-squares regression line.

Explanatory and Response Variables

Understanding these types of variables is crucial in regression analysis.

The explanatory variable is independent. It helps us understand or predict changes in another variable. In experiments, it's usually the factor we control. In our context, the x-values are explanatory.

The response variable is dependent. It's what we measure or focus on based on changes in the explanatory variable. In our exercise, y-values are the response.

To summarize:

• Explanatory Variable (x): The variable that explains or causes changes. Example: Time spent studying.
• Response Variable (y): The outcome we measure. Example: Test scores.

Understanding this distinction helps in setting up experiments, analyzing data, and interpreting results, as seen through our least-squares regression task.