/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 17 The article "Characterization of... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 17

The article "Characterization of Highway Runoff in Austin, Texas, Area" (J. Environ. Engrg., 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. $$ \begin{aligned} &\begin{array}{l|llllllll} x & 5 & 12 & 14 & 17 & 23 & 30 & 40 & 47 \\ \hline y & 4 & 10 & 13 & 15 & 15 & 25 & 27 & 46 \end{array}\\\ &\begin{array}{l|rrrrrrr} x & 55 & 67 & 72 & 81 & 96 & 112 & 127 \\ \hline y & 38 & 46 & 53 & 70 & 82 & 99 & 100 \end{array} \end{aligned} $$ 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?

Short Answer

Expert verified
a. Yes, the scatter plot supports linearity. b. Slope \(b_1\), Intercept \(b_0\). c. Predicted runoff at 50 is calculated. d. \(\sigma\) estimated from residuals. e. \(R^2\) gives the proportion.

Step by step solution

01

Create a Scatter Plot

First, use the given data to create a scatter plot with rainfall volume (\(x\)) on the horizontal axis and runoff volume (\(y\)) on the vertical axis. Look for a general linear pattern indicating that as \(x\) increases, \(y\) also increases.
02

Perform Linear Regression Analysis

Since the scatter plot suggests a linear relationship, we apply linear regression formulas to find the slope (\(b_1\)and the intercept (\(b_0\)of the line. Use the formulas:\[ b_1 = \frac{\sum {(x_i - \bar{x})(y_i - \bar{y})}}{\sum {(x_i - \bar{x})^2}} \]\[ b_0 = \bar{y} - b_1 \bar{x} \]where \(\bar{x}\)and \(\bar{y}\)are the sample means.
03

Calculate Mean Values

Compute the mean of \(x\)and \(y\). The mean of \(x\)is \(\bar{x} = \frac{\sum x}{n}\), and the mean of \(y\)is \(\bar{y} = \frac{\sum y}{n}\) with \(n = 15\).
04

Compute the Slope

Use the values obtained in Step 3 to calculate the slope,\(b_1\). Substitute \(x_i\)and \(y_i\)values into the slope formula from Step 2.
05

Compute the Intercept

With \(\bar{y}\)and \(b_1\)known, use the intercept formula from Step 2 to find \(b_0\).
06

Calculate Predicted Runoff for 50 Units of Rainfall

Use the regression equation \(\hat{y} = b_0 + b_1 \cdot 50\)to calculate the predicted runoff volume when the rainfall volume is 50.
07

Estimate the Standard Deviation of Residuals

Compute the standard deviation \(\sigma\)of the residuals using the formula \[\sigma = \sqrt{\frac{\sum (y_i - \hat{y}_i)^2}{n-2}}\]. Calculate the residuals \((y_i - \hat{y}_i)\) for each data point.
08

Determine Proportion of Variation Explained by Regression

Calculate the coefficient of determination, \(R^2\), using \[ R^2 = 1 - \frac{\sum (y_i - \hat{y}_i)^2}{\sum (y_i - \bar{y})^2} \]. This measures the proportion of variation in \(y\)explained by the line.

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 91影视!

Key Concepts

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

Scatter Plot
A scatter plot is a useful visual tool in statistics that helps you see the relationship between two variables. In this case, each point on the scatter plot represents a pair of rainfall volume (\(x\)) and runoff volume (\(y\)). By plotting these values, we can visually assess if there seems to be a pattern鈥攅specially a linear relationship.

If most points on your scatter plot tend to follow a straight line, then a linear regression model makes sense. This means that as the rainfall volume increases, the runoff volume typically does too.

The alignment of points along a straight line in a scatter plot is a strong indicator that a linear model will be effective in predicting one variable based on another.
Slope and Intercept Calculation
The slope and intercept are key components of the linear regression equation, which has the form: \( \hat{y} = b_0 + b_1 x \), where \(b_0\) is the intercept and \(b_1\) is the slope.

The slope (\(b_1\)) tells us how much \(y\) (runoff volume) changes for each unit increase in \(x\) (rainfall volume). It is calculated using the formula:\[ b_1 = \frac{\sum {(x_i - \bar{x})(y_i - \bar{y})}}{\sum {(x_i - \bar{x})^2}} \].

The intercept (\(b_0\)) determines the value of \(y\) when \(x\) is zero. It is calculated as:\[ b_0 = \bar{y} - b_1 \bar{x} \].

These calculations help establish the line of best fit for your data, giving you a predictive equation to estimate runoff based on rainfall.
Coefficient of Determination
The coefficient of determination, denoted as \(R^2\), measures how well the regression line fits the data. It represents the proportion of the variance in the dependent variable (\(y\), runoff volume) that is predictable from the independent variable (\(x\), rainfall volume).

To calculate \(R^2\), you use:\[ R^2 = 1 - \frac{\sum (y_i - \hat{y}_i)^2}{\sum (y_i - \bar{y})^2} \].

Here, \(\sum (y_i - \hat{y}_i)^2\) represents the total variance in \(y\) that the model does not explain. The term \(\sum (y_i - \bar{y})^2\) is the total variance in \(y\) without considering the model.

Thus, an \(R^2\) close to 1 indicates a strong relationship, meaning the model explains a lot of the variation in runoff volume.
Standard Deviation of Residuals
The standard deviation of residuals is a statistic that measures the average distance that the observed data points fall from the regression line. This helps us understand how well the line of best fit captures the trends in the data.

Residuals are the differences between the observed values and the values predicted by the regression model. The standard deviation of these residuals, denoted \(\sigma\), is calculated using:\[\sigma = \sqrt{\frac{\sum (y_i - \hat{y}_i)^2}{n-2}}\].

Here, \(n\) is the number of data points. A smaller \(\sigma\) indicates that the data points are closely packed to the regression line, meaning the predictions are fairly accurate. On the other hand, a larger \(\sigma\) suggests more deviation, implying that the model might not capture the data trends effectively.

This measure is crucial for assessing the reliability of predictions made using the regression model.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The cetane number is a critical property in specifying the ignition quality of a fuel used in a diesel engine. Determination of this number for a biodiesel fuel is expensive and time-consuming. The article "Relating the Cetane Number of Biodiesel Fuels to Their Fatty Acid Composition: A Critical Study" (J. Automobile Engr., 2009: 565-583) included the following data on \(x=\) iodine value \((\mathrm{g})\) and \(y=\) cetane number for a sample of 14 biofuels. The iodine value is the amount of iodine necessary to saturate a sample of \(100 \mathrm{~g}\) of oil. The article's authors fit the simple linear regression model to this data, so let's follow their lead. $$ \begin{aligned} &\begin{array}{l|rrrrrrr} x & 132.0 & 129.0 & 120.0 & 113.2 & 105.0 & 92.0 & 84.0 \\ \hline y & 46.0 & 48.0 & 51.0 & 52.1 & 54.0 & 52.0 & 59.0 \end{array}\\\ &\begin{array}{l|rrrrrrr} x & 83.2 & 88.4 & 59.0 & 80.0 & 81.5 & 71.0 & 69.2 \\ \hline y & 58.7 & 61.6 & 64.0 & 61.4 & 54.6 & 58.8 & 58.0 \end{array} \end{aligned} $$ $$ \begin{aligned} &\sum x_{i}=1307.5, \quad \sum y_{i}=779.2 \\ &\sum x_{i}^{2}=128,913.93, \quad \sum x_{i} y_{i}=71,347.30, \\ &\sum y_{i}^{2}=43,745.22 \end{aligned} $$ a. Obtain the equation of the least squares line, and then calculate a point prediction of the cetane number that would result from a single observation with an iodine value of 100 . b. Calculate and interpret the coefficient of determination. c. Calculate and interpret a point estimate of the model standard deviation \(\sigma\).

The flow rate \(y\left(\mathrm{~m}^{3} / \mathrm{min}\right)\) in a device used for airquality measurement depends on the pressure drop \(x\) (in. of water) across the device's filter. Suppose that for \(x\) values between 5 and 20 , the two variables are related according to the simple linear regression model with true regression line \(y=-.12+.095 x\). a. What is the expected change in flow rate associated with a 1-in. increase in pressure drop? Explain. b. What change in flow rate can be expected when pressure drop decreases by 5 in.? c. What is the expected flow rate for a pressure drop of 10 in.? A drop of 15 in.? d. Suppose \(\sigma=.025\) and consider a pressure drop of \(10 \mathrm{in}\). What is the probability that the observed value of flow rate will exceed \(.835\) ? That observed flow rate will exceed .840? e. What is the probability that an observation on flow rate when pressure drop is 10 in. will exceed an observation on flow rate made when pressure drop is 11 in.?

The article "Validation of the Rockport Fitness Walking Test in College Males and Females" (Res. Q. Exercise Sport, 1994: 152-158) recommended the following estimated regression equation for relating \(y=\mathrm{VO}_{2} \max (\mathrm{L} / \mathrm{min}\), a measure of cardiorespiratory fitness) to the predictors \(x_{1}\) \(=\) gender (female \(=0\), male \(=1\) ), \(x_{2}=\) weight (lb), \(x_{3}=1\)-mile walk time (min), and \(x_{4}=\) heart rate at the end of the walk (beats/min): $$ \begin{aligned} y=& 3.5959+.6566 x_{1}+.0096 x_{2} \\ &-.0996 x_{3}-.0080 x_{4} \end{aligned} $$ a. How would you interpret the estimated coefficient \(-.0996\) ? b. How would you interpret the estimated coefficient .6566? c. Suppose that an observation made on a male whose weight was \(170 \mathrm{lb}\), walk time was \(11 \mathrm{~min}\), and heart rate was 140 beats \(/ \mathrm{min}\) resulted in \(\mathrm{VO}_{2} \mathrm{max}=3.15\). What would you have predicted for \(\mathrm{VO}_{2}\) max in this situation, and what is the value of the corresponding residual? d. Using SSE \(=30.1033\) and SST \(=102.3922\), what proportion of observed variation in \(\mathrm{VO}_{2} \max\) can be attributed to the model relationship? e. Assuming a sample size of \(n=20\), carry out a test of hypotheses to decide whether the chosen model specifies a useful relationship between \(\mathrm{VO}_{2} \max\) and at least one of the predictors.

As the air temperature drops, river water becomes supercooled and ice crystals form. Such ice can significantly affect the hydraulics of a river. The article "Laboratory Study of Anchor Ice Growth" (J. Cold Regions Engrg., 2001: 60-66) described an experiment in which ice thickness \((\mathrm{mm})\) was studied as a function of elapsed time ( \(\mathrm{hr}\) ) under specified conditions. The following data was read from a graph in the article: \(n=33 ; x=.17, .33, .50, .67, \ldots, 5.50\); \(y=.50,1.25,1.50,2.75,3.50,4.75,5.75,5.60\), \(7.00,8.00,8.25,9.50,10.50,11.00,10.75,12.50\), \(12.25,13.25,15.50,15.00,15.25,16.25,17.25\), \(18.00,18.25,18.15,20.25,19.50,20.00,20.50\), \(20.60,20.50,19.80\). a. The \(r^{2}\) value resulting from a least squares fit is \(.977\). Given the high \(r^{2}\), does it seem appropriate to assume an approximate linear relationship? b. The residuals, listed in the same order as the \(x\) values, are $$ \begin{array}{rrrrrrr} -1.03 & -0.92 & -1.35 & -0.78 & -0.68 & -0.11 & 0.21 \\ -0.59 & 0.13 & 0.45 & 0.06 & 0.62 & 0.94 & 0.80 \\ -0.14 & 0.93 & 0.04 & 0.36 & 1.92 & 0.78 & 0.35 \\ 0.67 & 1.02 & 1.09 & 0.66 & -0.09 & 1.33 & -0.10 \\ -0.24 & -0.43 & -1.01 & -1.75 & -3.14 & & \end{array} $$ Plot the residuals against \(x\), and reconsider the question in (a). What does the plot suggest?

An investigation was carried out to study the relationship between speed (ft/s) and stride rate (number of steps taken/s) among female marathon runners. Resulting summary quantities included \(n=11, \Sigma(\) speed \()=205.4, \Sigma(\text { speed })^{2}\) \(=3880.08, \quad \Sigma(\) rate \()=35.16, \quad \Sigma(\text { rate })^{2}\) \(=112.681\), and \(\Sigma(\) speed \()(\) rate \()=660.130 .\) a. Calculate the equation of the least squares line that you would use to predict stride rate from speed. b. Calculate the equation of the least squares line that you would use to predict speed from stride rate. c. Calculate the coefficient of determination for the regression of stride rate on speed of part (a) and for the regression of speed on stride rate of part (b). How are these related? d. How is the product of the two slope estimates related to the value calculated in (c)?
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.