91Ó°ÊÓ

The wind chill factor depends on wind speed and air temperature. The following data represent the wind speed (in mph) and wind chill factor at an air temperature of \(15^{\circ}\) Fahrenheit. $$ \begin{array}{cc|cc} \begin{array}{l} \text { Wind } \\ \text { Speed, } x \\ (\mathrm{mph}) \end{array} & \begin{array}{l} \text { Wind Chill } \\ \text { Factor, } y \end{array} & \begin{array}{l} \text { Wind } \\ \text { Speed, } x \\ (\mathrm{mph}) \end{array} & \begin{array}{l} \text { Wind Chill } \\ \text { Factor, } y \end{array} \\ \hline 5 & 12 & 25 & -22 \\ \hline 10 & -3 & 30 & -25 \\ \hline 15 & -11 & 35 & -27 \\ \hline 20 & -17 & & \\ \hline \end{array} $$ (a) Draw a scatter diagram of the data treating wind speed as the explanatory variable. (b) Determine the correlation between wind speed and wind chill factor. Does this imply a linear relation between wind speed and wind chill factor? (c) Compute the least-squares regression line. (d) Plot the residuals against the wind speed. (e) Do you think the least-squares regression line is a good model? Why?

Step by step solution

01

Construct a Scatter Diagram

Plot the given data points on a coordinate axis, treating wind speed (in mph) as the x-axis and the wind chill factor as the y-axis.

02

Calculate the Correlation Coefficient

Use the formula for the Pearson correlation coefficient: \[r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}} \] where n is the number of data points.

03

Interpret the Correlation Coefficient

Determine whether the calculated correlation implies a linear relationship by checking if the r-value is close to 1 or -1.

04

Compute the Least-Squares Regression Line

Use the formulas to find the slope (m) and y-intercept (b) of the line: \[m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}\] and \[b = \frac{\sum y - m \sum x}{n}\]. Then write the equation in the form y = mx + b.

05

Plot Residuals

Calculate the residuals for each data point using the formula: Residual = Observed y - Predicted y. Plot these residuals against the wind speed.

06

Evaluate the Model

Analyze the residual plot to see if there are any patterns. If the residuals are randomly dispersed, the regression line is a good fit. Otherwise, consider other models.

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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, or scatter plot, is a type of graph used to visually display the relationship between two numerical variables. In the context of wind chill factor analysis, this is done to explore how wind speed affects the wind chill factor.
You plot the points on a coordinate grid with:

• x-axis (horizontal): wind speed (in mph)
• y-axis (vertical): wind chill factor

The key purpose of a scatter diagram is to unveil any apparent patterns or correlations visually.
For example, by plotting the data: (5, 12), (10, -3), (15, -11), (20, -17), (25, -22), (30, -25), and (35, -27), you can start to identify if there's a trend or pattern of how the wind chill changes with varying wind speeds.

correlation coefficient

The correlation coefficient, often denoted as ‘r’, is a statistical measure that calculates the strength of the relationship between the relative movements of two variables. In this case, it's wind speed and wind chill factor.

• Range: The value of the correlation coefficient can range from -1 to 1.
  
• Interpretations:
  • r = 1: Perfect positive linear relationship
  • r = -1: Perfect negative linear relationship
  • r = 0: No linear relationship

The formula to calculate the Pearson correlation coefficient is:
\[ r = \frac{n(\text{Σxy}) - (\text{Σx})(\text{Σy})}{\text{sqrt}([n \text{Σx}^2 - (\text{Σx})^2][n \text{Σy}^2 - (\text{Σy})^2])} \]
Calculating this for our dataset can help us understand if there's a strong relationship between wind speed and wind chill factor.

least-squares regression

Least-squares regression is a method for finding the line that best fits a set of data points. This line minimizes the sum of the squared differences (the residuals) between the observed values and the values predicted by the line.
The equation of the least-squares regression line is: \[ y = mx + b \]

• Slope (m): It determines how much y (wind chill factor) changes when x (wind speed) increases by one unit. The formula is: \[ m = \frac{n(\text{Σxy}) - (\text{Σx})(\text{Σy})}{n(\text{Σx}^2) - (\text{Σx})^2} \]
  
• y-intercept (b): It is the value of y when x is zero. The formula is: \[ b = \frac{\text{Σy} - m \text{Σx}}{n} \]
  

Using these formulas, you can determine the line that best predicts the wind chill factor based on wind speed.

residual analysis

Residual analysis involves examining the deviations of the actual data points from the predictions made by your regression model. A residual (or error) is the difference between the observed value and the value predicted by the regression line.

• Formula: Residual = Observed y - Predicted y
• Plotting: Plot these residuals against the x-axis (wind speed in this case).

This helps you assess the goodness of fit of your model. If the residuals are randomly distributed around zero, your model is likely a good fit. However, if there's a systematic pattern in the residuals, this suggests that your model may be missing some key aspects of the data, and another type of model might be more appropriate.

linear relationship

A linear relationship between two variables means that the relationship can be represented by a straight line on a graph. This means that changes in one variable bring about consistent and proportional changes in another variable.
In the context of wind speed and wind chill factor:

• Positive Linear Relationship: As wind speed increases, wind chill factor increases.
• Negative Linear Relationship: As wind speed increases, wind chill factor decreases.
• No Linear Relationship: Wind speed and wind chill factor do not follow any predictable pattern.

The correlation coefficient (r) helps determine if such a linear relationship exists.
A strong linear relationship is implied if the absolute value of r is close to 1. By analyzing the scatter plot, correlation coefficient, and least-squares regression line, we can evaluate the linearity of the relationship between wind speed and wind chill factor.