91Ó°ÊÓ

Exercises 33 and 34 refer to the following setting. Thirty randomly selected seniors at Council High School were asked to report the age (in years) and mileage of their main vehicles. Here is a scatterplot of the data: We used Minitab to perform a least-squares regression analysis for these data. Part of the computer output from this regression is shown below. Predictor \(\quad\) coef \(\quad\) stdev \(\quad\) t-ratio \(\quad \mathrm{P}\) Constant \(-13832 \qquad 8773 \qquad-1.58 \qquad 0.126\) Age \(\quad 14954 \qquad 1546 \qquad 9.67 \quad 0.000\) \(s=22723 \qquad R-s q=77.08 \qquad R-s q(a d j)=76.18\) Drive my car (3.2, 4.3) (a) Explain what the value of r2 tells you about how well the least-squares line fits the data. (b) The mean age of the students’ cars in the sample was x 8 years. Find the mean mileage of the cars in the sample. Show your work. (c) Interpret the value of s in the context of this setting. (d) Would it be reasonable to use the least-squares line to predict a car’s mileage from its age for a Council High School teacher? Justify your answer.

Step by step solution

01

Interpretation of R-Squared

The R-squared ( R-sq ) value is approximately 77.08%. R-squared represents the proportion of the variance in the dependent variable (mileage) that is predictable from the independent variable (age of cars). In this case, an R-squared value of 77.08% means that 77.08% of the variation in the car mileage can be explained by the age of the car. This indicates a strong relationship since a substantial portion of the variability in mileage is accounted for by the regression model.

02

Calculating Mean Mileage of Cars

To find the mean mileage of cars, we make use of the regression equation. The regression equation is derived from: \[\text{Mileage} = \text{Constant} + (\text{Slope} \times \text{Mean Age})\]Using the given coefficients, the equation becomes:\[\text{Mileage} = -13832 + 14954 \times 8\]Calculate the predicted mean mileage:\[\text{Mileage} = -13832 + 119632 = 105800 \]Thus, the mean mileage of the cars is 105,800 miles.

03

Interpretation of Standard Error s

The standard error (s) provided is 22723. This measure indicates the average distance that the observed values fall from the regression line. In this context, on average, each car's actual mileage is 22,723 miles away from the mileage predicted by the regression model. It gives us an idea of the expected variability of actual mileage around the predicted value based on their ages.

04

Application of the Regression Model to Teachers

The regression model was created based on data from students' cars at Council High School. Since we're considering applying this to teachers, whose cars may have different characteristics or usage patterns, it would be inappropriate without further validation. Teachers' cars might differ in terms of types, usage frequency, or maintenance, which could affect their mileage independently of age. Therefore, using this model for teachers without additional data could lead to erroneous predictions.

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 91Ó°ÊÓ!

Key Concepts

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

Least-Squares Regression

Least-squares regression is a common statistical method used to model the relationship between two variables. In our specific context, we're looking at how the age of a car relates to its mileage. The least-squares approach aims to find the line that best fits the data points by minimizing the sum of the squares of the vertical distances (also known as residuals) between the observed values and the values predicted by the line.
The line is represented by a regression equation of the form: \( y = a + bx \), where:

• \( y \) is the dependent variable (mileage).
• \( x \) is the independent variable (age of the car).
• \( a \) is the y-intercept (constant term).
• \( b \) is the slope of the line.

In the provided exercise, the constant term is \(-13832\), and the slope is \(14954\), indicating that for every additional year of a car's age, its mileage typically increases by \(14954\) miles. This calculated line provides the best estimate of the mileage based on the age of the car. Understanding and interpreting the coefficients in the regression equation is key to making predictions and drawing conclusions from your data.

R-squared Interpretation

The R-squared value, or \( R^2 \), serves as a statistical measure of how well the regression line explains the variance in the dependent variable. In simple terms, it tells us how much of the variation in mileage is explained by the age of the cars in our model. The closer \( R^2 \) is to 1, the better the model explains the variability of the response data around its mean.
In this exercise, the R-squared value is found to be approximately \(77.08\%\). This indicates that around 77% of the variability in the mileage can be explained by the age of the vehicles. It signifies a strong relationship, suggesting that age is a good predictor of car mileage for these students. This insight is crucial as it helps us gauge the model's reliability and assess whether other factors might need to be considered for a more comprehensive understanding.

Standard Error

The standard error (\( s \)) measures the average distance that the data points lie from the regression line. It is essentially a gauge of the accuracy of predictions made by the regression model. A smaller standard error represents more precise predictions.
In the given scenario, the standard error is \(22723\) miles. This means that on average, the cars' actual mileages deviate by about \(22723\) miles from the predicted mileage according to the regression line. While the standard error gives us a sense of variability, it also provides a tangible context for the predictions' reliability.

• A higher standard error often indicates greater variability among data points not explained by the model.
• Having this in mind helps in assessing the confidence we can place on the predictive power of our regression line.

Understanding the standard error is key in practical applications, especially when predictions involve crucial decisions.

Data Variability

Data variability refers to the extent to which the data points spread around the center. In the context of regression analysis, it reveals how consistent the relationship between variables is throughout the dataset.
High variability suggests that while our model may provide an average trend, individual data points can deviate notably from this trend. Such deviations indicate that factors other than the primary independent variable (in this case, car age) might influence the dependent variable (mileage).
It is essential to recognize and account for this variability when evaluating a regression model because:

• It helps identify the limitations of the model — the more the variability, the lesser the model's accuracy in some situations.
• Understanding variability assists in identifying if additional variables should be considered to better explain the dependent variable.

By addressing data variability directly, we gain insights into the data's structure and ensure that our analytical conclusions are well-founded and actionable.