91Ó°ÊÓ

Attending Class The following data represent the number of days absent, \(x\), and the final grade, \(y,\) for a sample of college students in a general education course at a large state university. $$ \begin{array}{lllllllllll} \hline \begin{array}{l} \text { No. of } \\ \text { absences, } x \end{array} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline \begin{array}{l} \text { Final } \\ \text { grade, } y \end{array} & 89.2 & 86.4 & 83.5 & 81.1 & 78.2 & 73.9 & 64.3 & 71.8 & 65.5 & 66.2 \\ \hline \end{array} $$ (a) Find the least-squares regression line treating number of absences as the explanatory variable and final grade as the response variable. (b) Interpret the slope and \(y\) -intercept, if appropriate. (c) Predict the final grade for a student who misses five class periods and compute the residual. Is the final grade above or below average for this number of absences? (d) Draw the least-squares regression line on the scatter diagram of the data. (e) Would it be reasonable to use the least-squares regression line to predict the final grade for a student who has missed 15 class periods? Why or why not?

Step by step solution

01

- Calculate the Means

First, find the mean of the number of absences, denoted as \( \bar{x} \), and the mean of the final grades, denoted as \( \bar{y} \). Use the formulas \( \bar{x} = \frac{\sum x}{n} \) and \( \bar{y} = \frac{\sum y}{n} \), where \( \sum x \) and \( \sum y \) are the sums of the absences and the grades respectively, and \( n \) is the number of data points.

02

- Calculate the Slope

Next, calculate the slope \( m \) of the least-squares regression line using the formula \[ m = \frac{ \sum (x - \bar{x})(y - \bar{y}) }{ \sum (x - \bar{x})^2 } \].

03

- Calculate the Intercept

Now, calculate the y-intercept \( b \) using the formula \[ b = \bar{y} - m \bar{x} \].

04

- Formulate the Regression Equation

Use the slope \( m \) and y-intercept \( b \) to write the least-squares regression line equation: \( y = mx + b \).

05

- Interpret the Slope and Intercept

Interpret the slope and intercept in the context of the problem. The slope represents the change in the final grade for each additional class missed. The y-intercept represents the final grade when no classes are missed.

06

- Make a Prediction

Use the regression equation to predict the final grade for a student who misses 5 classes. Substitute \( x = 5 \) into the equation.

07

- Calculate the Residual

The residual is the difference between the actual final grade and the predicted final grade. Use the formula \( \text{Residual} = y_{\text{actual}} - y_{\text{predicted}} \) to compute it.

08

- Determine if the Grade is Above or Below Average

Determine if the computed residual is positive or negative. A positive residual indicates that the actual final grade is above the predicted grade, while a negative residual indicates it is below.

09

- Draw the Regression Line

Plot the data points on a scatter diagram and draw the regression line using the equation from Step 4. Ensure the line passes through the predicted grades at the given values of \( x \).

10

- Assess the Reasonableness

Assess if it would be reasonable to use the regression line to predict the final grade for a student who misses 15 classes. Consider the data range and whether extrapolation is acceptable in this context.

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Key Concepts

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

slope interpretation

The slope in a least-squares regression line represents the rate of change in the response variable (final grade) for each unit increase in the explanatory variable (number of absences). For instance, in this case, a slope of -2.83 indicates that for each additional day a student is absent, their final grade is expected to decrease by 2.83 points. Understanding the slope helps to quantify the relationship between the variables and to make informed predictions about the impact of changes in the explanatory variable.

y-intercept

The y-intercept is the value of the response variable (final grade) when the explanatory variable (number of absences) is zero. Essentially, it represents the expected final grade for a student who has not missed any classes. In our example, if the y-intercept is 89.2, it indicates that a student with zero absences is predicted to have a final grade of 89.2. It provides a baseline value from which the effect of absences can be measured.

predictions using regression

Predictions using the regression line are made by substituting the value of the explanatory variable into the regression equation. For instance, to predict the final grade for a student who misses 5 classes, substitute 5 for x in the equation. If our equation is y = -2.83x + 89.2, then the predicted grade would be y = -2.83(5) + 89.2 = 75.05. This method provides a straightforward way to estimate outcomes based on observed relationships.

residual calculation

A residual is the difference between the observed actual value and the predicted value from the regression equation. It is calculated as Residual = y_actual - y_predicted. Residuals help assess the accuracy of the regression model. For instance, if the actual grade for a student with 5 absences is 73.9, the residual would be 73.9 - 75.05 = -1.15. A negative residual indicates that the actual grade is below the predicted grade, while a positive residual indicates it is above.

scatter diagram

A scatter diagram (or scatter plot) visually represents the relationship between the explanatory and response variables. Each point on the scatter plot represents a pair of values. For example, one point might represent 2 absences and a grade of 83.5. Plotting these pairs helps identify patterns or trends in the data. By adding the least-squares regression line to the scatter plot, we can see the general direction and strength of the relationship, as well as how well the line fits the data.