91Ó°ÊÓ

The article "Characterization of Highway Runoff in Austin, Texas, Area" (J. of Envir: Engr., 1998: 131-137) gave a scatter plot, along with the least squares line, of \(x=\) rainfall volume \(\left(\mathrm{m}^{3}\right)\) and \(y=\) runoff volume \(\left(\mathrm{m}^{3}\right)\) for a particular location. The accompanying values were read from the plot. $$ \begin{aligned} &\begin{array}{l|llllllll} x & 5 & 12 & 14 & 17 & 23 & 30 & 40 & 47 \\ \hline y & 4 & 10 & 13 & 15 & 15 & 25 & 27 & 46 \end{array}\\\ &\begin{array}{l|lllllll} x & 55 & 67 & 72 & 81 & 96 & 112 & 127 \\ \hline y & 38 & 46 & 53 & 70 & 82 & 99 & 100 \end{array} \end{aligned} $$ a. Does a scatter plot of the data support the use of the simple linear regression model? b. Calculate point estimates of the slope and intercept of the population regression line. c. Calculate a point estimate of the true average runoff volume when rainfall volume is 50 . d. Calculate a point estimate of the standard deviation \(\sigma\). e. What proportion of the observed variation in runoff volume can be attributed to the simple linear regression relationship between runoff and rainfall?

Step by step solution

01

Plot the Data

First, we need to create a scatter plot of the given data points to visually assess if a linear relationship is plausible. Plot the given pairs of \(x\) (rainfall volume) and \(y\) (runoff volume) on a graph. Observing the trend, if the points roughly follow a straight line, a simple linear regression model is suitable.

02

Calculate the Slope and Intercept

To find the slope \(b_1\) and intercept \(b_0\) of the regression line, use the formulas: \[ b_1 = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \]\[ b_0 = \frac{\sum y - b_1 \sum x}{n} \]Compute each component using the provided data and perform the calculations to find the values of \(b_1\) (the slope) and \(b_0\) (the intercept).

03

Estimate Runoff Volume for Given Rainfall

To estimate the runoff volume when the rainfall is 50, substitute \(x = 50\) into the regression equation \( y = b_0 + b_1x \). Calculate the value using the slope and intercept obtained in the previous step to get the estimated runoff volume.

04

Calculate Standard Deviation of the Residuals

The standard deviation of the residuals \(\sigma\) can be estimated using:\[ s = \sqrt{\frac{\sum (y_i - \hat{y}_i)^2}{n-2}} \]Where \(y_i\) are the observed values and \(\hat{y}_i\) are the predicted values from the regression line. Calculate \(\hat{y}_i\) for each rainfall volume, find the residuals \(y_i - \hat{y}_i\), and then compute \(s\).

05

Determine Proportion of Variation Explained

Calculate the coefficient of determination, \(R^2\), which gives the proportion of total variation in the runoff volume explained by the model:\[ R^2 = 1 - \frac{\sum (y_i - \hat{y}_i)^2}{\sum (y_i - \bar{y})^2} \]Where \(\bar{y}\) is the mean of \(y\) values. Evaluate \(R^2\) to determine this proportion.

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.

Scatter Plot

A scatter plot is a useful way to visualize the relationship between two numerical variables. Imagine you have a graph with dots scattered around. Each dot represents a pair of values, one on the x-axis and one on the y-axis. In this case, the x-axis is rainfall volume and the y-axis is runoff volume.

By plotting these data points, you can visually assess if there's a pattern or trend. If the dots seem to gather around a straight line, it indicates a linear relationship. This means you can use simple linear regression to model this relationship.

Creating a scatter plot is the first critical step in determining if a linear regression model is appropriate. It's like drawing a rough sketch before painting the full picture of the relationship using mathematical formulas.

Slope Estimation

Slope estimation in linear regression helps us understand how much the y-variable (in this case, runoff volume) changes for each unit change in the x-variable (rainfall volume). The slope is denoted as \( b_1 \) and can be calculated using a specific formula:

• First, multiply each pair of x and y values, then sum them up to obtain \( \sum xy \).
• Next, square each x value, sum those up to calculate \( \sum x^2 \).
• Use the slope formula: \( b_1 = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \).

Here, \( n \) is the number of data points.

The slope tells us the angle of the regression line relative to the x-axis. If \( b_1 \) is 2, for example, it implies that for each cubic meter increase in rainfall, the runoff volume increases by 2 cubic meters.

Understanding the slope is crucial as it quantifies the nature and strength of the linear relationship.

Standard Deviation

The standard deviation in the context of regression analysis describes how much the data points deviate from the fitted regression line, specifically the average distance of the observed values from their predicted values. It's denoted as \( \sigma \) or often just \( s \).

To calculate this value:

• First, compute each residual by subtracting the predicted value \( \hat{y}_i \) from the observed value \( y_i \).
• Square each of those residuals and sum them to get \( \sum (y_i - \hat{y}_i)^2 \).
• Divide by \( n-2 \) (degree of freedom), where \( n \) is the total number of observations.
• Finally, take the square root to find \( s \).

This measure helps in assessing the accuracy of the regression model.

A smaller standard deviation indicates that the data points are closer to the regression line, suggesting a better model fit.

Coefficient of Determination

The coefficient of determination, often referred to as \( R^2 \), quantifies how well the regression line explains the variation in the dependent variable (runoff volume in this case). It's a crucial measure to evaluate the effectiveness of the regression model.

To compute \( R^2 \):

• Find the total variance in the y-values, which is represented by \( \sum (y_i - \bar{y})^2 \), where \( \bar{y} \) is the mean of y.
• Subtract the unexplained variance, which is \( \sum (y_i - \hat{y}_i)^2 \), from the total variance.
• The formula is \( R^2 = 1 - \frac{\sum (y_i - \hat{y}_i)^2}{\sum (y_i - \bar{y})^2} \).

Expressed as a percentage, \( R^2 \) indicates the proportion of variance in the dependent variable that's predictable from the independent variable.

For instance, an \( R^2 \) of 0.85 means that 85% of the total variation in runoff volume can be explained by the linear relationship with rainfall volume. This metric helps assess how well the regression line captures the data dynamics.