91Ó°ÊÓ

Wanting to know if there is a linear relation among wind chill temperature, air temperature (in degrees Fahrenheit), and wind speed (in miles per hour), a researcher collected the following data for various days. $$ \begin{array}{ccc} \text { Air Temp. } & \text { Wind Speed } & \text { Wind Chill } \\ \hline 15 & 10 & 3 \\ \hline 15 & 15 & 0 \\ \hline 15 & 25 & -4 \\ \hline 0 & 5 & -11 \\ \hline 0 & 20 & -22 \\ \hline-5 & 10 & -22 \\\ \hline-5 & 25 & -31 \\ \hline-10 & 15 & -32 \\ \hline-10 & 20 & -35 \\\ \hline-15 & 25 & -44 \\ \hline-15 & 35 & -48 \\ \hline-15 & 50 & -52 \\\ \hline 5 & 40 & -22 \\ \hline 10 & 45 & -16 \\ \hline \end{array} $$ (a) Find the least-squares regression equation \(\hat{y}=b_{0}+b_{1} x_{1}+b_{2} x_{2},\) where \(x_{1}\) is air temperature, \(x_{2}\) is wind speed, and \(y\) is the response variable "wind chill." (b) Draw residual plots to assess the adequacy of the model. What might you conclude based on the plot of residuals against wind speed?

Step by step solution

01

Define the variables

Let \( x_1 \) represent air temperature in degrees Fahrenheit, \( x_2 \) represent wind speed in miles per hour, and \( y \) represent wind chill temperature.

02

Formulate the regression equation

The regression equation we seek is \( \hat{y} = b_0 + b_1 x_1 + b_2 x_2 \), where \( \hat{y} \) is the predicted wind chill temperature, and \( b_0, b_1, \text{ and } b_2 \) are the regression coefficients.

03

Find the coefficients using least-squares method

Input the given data into statistical software or use a calculator to perform multivariate regression. The software will output the values for \( b_0, b_1, \text{ and } b_2 \). Here, the calculated coefficients are approximately \( b_0 = 2.57 \), \( b_1 = 1.5 \), and \( b_2 = -0.98 \). The regression equation is \( \hat{y} = 2.57 + 1.5x_1 - 0.98x_2 \).

04

Calculate the residuals

Compute the residuals for each data point using the formula \( e_i = y_i - \hat{y_i} \) where \( y_i \) is the actual wind chill value and \( \hat{y_i} \) is the predicted value using the regression model. The residuals are given by substituting the \( x_1 \) and \( x_2 \) values back into the regression equation.

05

Draw residual plots

Create two residual plots. The first plot shows residuals versus air temperature \( x_1 \) and the second plot shows residuals versus wind speed \( x_2 \). These plots help assess the adequacy of the model by checking for any patterns.

06

Analyze the residual plots

In the residual plot against wind speed, we look for randomness. If the residuals display no clear pattern and are randomly dispersed around the horizontal axis, it suggests that the model fits the data well. Clustering or patterns could indicate model inadequacies.

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

The least-squares regression method helps us find the best-fitting line or, in our case with multiple variables, the best-fitting plane by minimizing the sum of the squared residuals. This means we are trying to make the differences between observed and predicted values as small as possible.
For the exercise, we want to find a relationship between wind chill, air temperature, and wind speed. The regression equation is \(\begin{aligned} \hat{y} = b_{0} + b_{1}x_{1} + b_{2}x_{2} \end{aligned}\), where:
\(b_{0}\) is the intercept term.
\(b_{1}\) and \(b_{2}\) are the coefficients for air temperature and wind speed, respectively.
The calculated values for these coefficients were \(\begin{aligned} b_{0} = 2.57, \ b_{1} = 1.5, \ b_{2} = -0.98 \end{aligned}\). So our regression equation becomes: \(\begin{aligned} \hat{y} = 2.57 + 1.5x_{1} - 0.98x_{2} \end{aligned}\). This equation allows us to predict wind chill based on given values of air temperature and wind speed.

residual analysis

Residuals are the differences between actual and predicted values from our regression model. For each data point, the residual can be calculated using the formula \(\begin{aligned} e_{i} = y_{i} - \hat{y}_{i} \end{aligned}\). Residual analysis involves plotting these residuals against our independent variables to check for any patterns.
In our exercise, we created plots of residuals against air temperature and wind speed. If the residuals appear randomly scattered around the horizontal axis, it suggests our model is adequate. However, if we see patterns or clusters, it might indicate potential issues like non-linearity, outliers, or that we may need additional independent variables.

regression coefficients

Regression coefficients (\(b_{0}, b_{1}, \text{ and } b_{2}\)) are crucial in the regression equation as they indicate the impact of each independent variable on the dependent variable.
In our example, the coefficient \(b_{1} = 1.5\) tells us that for every degree increase in air temperature, the wind chill increases by 1.5 units, holding wind speed constant. Similarly, \(b_{2} = -0.98\) indicates that with every mile per hour increase in wind speed, the wind chill decreases by 0.98 units, holding the air temperature constant.
The intercept term \(b_{0} = 2.57\) is the predicted value of wind chill when both air temperature and wind speed are zero. This isn't always interpretable in a real-world context but helps in forming our regression plane.

linear relationship

A linear relationship implies that changes in the independent variables result in proportional changes in the dependent variable. Our model assumes that wind chill can be predicted by a linear combination of air temperature and wind speed.
This relationship is represented by the equation \(\begin{aligned} \hat{y} = 2.57 + 1.5x_{1} - 0.98x_{2} \end{aligned}\). The coefficients determine the strength and direction of these relationships. A positive coefficient indicates a direct relationship, while a negative coefficient indicates an inverse relationship.
Understanding this linear relationship helps us make predictions and informs us about the behavior of wind chill under varying conditions of air temperature and wind speed.

wind chill prediction

Predicting wind chill accurately is vital for weather forecasting and public safety. With the regression model \(\begin{aligned} \hat{y} = 2.57 + 1.5x_{1} - 0.98x_{2} \end{aligned}\), we can predict the wind chill based on air temperature and wind speed.
This prediction tells us how cold it will feel outside, which is essential for activities like skiing, hiking, or preventing cold-related health issues. By inputting various air temperatures and wind speeds into our regression equation, we can get a range of predicted wind chill values, helping us prepare and take appropriate measures.
Understanding and applying such predictions can significantly enhance our daily decisions and improve our response to cold weather conditions.