91Ó°ÊÓ

The article "Effects of Bike Lanes on Driver and Bicyclist Behavior" (ASCE Transportation Engrg. J., 1977: 243-256) reports the results of a regression analysis with \(x=\) available travel space in feet (a convenient measure of roadway width, defined as the distance between a cyclist and the roadway center line) and separation distance \(y\) between a bike and a passing car (determined by photography). The data, for ten streets with bike lanes, follows: $$ \begin{array}{r|rrrrr} x & 12.8 & 12.9 & 12.9 & 13.6 & 14.5 \\ \hline y & 5.5 & 6.2 & 6.3 & 7.0 & 7.8 \\ x & 14.6 & 15.1 & 17.5 & 19.5 & 20.8 \\ \hline y & 8.3 & 7.1 & 10.0 & 10.8 & 11.0 \end{array} $$ a. Verify that \(\sum x_{i}=154.20, \sum y_{i}=80\), \(\sum x_{i}^{2}=2452.18, \quad \sum x_{i} y_{i}=1282.74, \quad\) and \(\sum y_{i}^{2}=675.16 .\) b. Derive the equation of the estimated regression line. c. What separation distance would you predict for another street that has \(15.0\) as its available travel space value? d. What would be the estimate of expected separation distance for all streets having available travel space value \(15.0\) ?

Step by step solution

01

Calculate the Sums

Let's first verify the sums of the values provided. We have the datasets for available travel space \(x\) and separation distance \(y\). We must confirm:- \(\sum x_{i} = 154.2\)- \(\sum y_{i} = 80\)- \(\sum x_{i}^2 = 2452.18\)- \(\sum x_{i}y_{i} = 1282.74\)- \(\sum y_{i}^2 = 675.16\). Check each sum by calculating these: - \(\sum x = 12.8 + 12.9 + 12.9 + 13.6 + 14.5 + 14.6 + 15.1 + 17.5 + 19.5 + 20.8 = 154.2\)- \(\sum y = 5.5 + 6.2 + 6.3 + 7.0 + 7.8 + 8.3 + 7.1 + 10.0 + 10.8 + 11.0 = 80\)- \(\sum x^2 = (12.8^2 + 12.9^2 + ... + 20.8^2) = 2452.18\)- \(\sum xy = (12.8 \times 5.5 + 12.9 \times 6.2 + ... + 20.8 \times 11.0) = 1282.74\)- \(\sum y^2 = (5.5^2 + 6.2^2 + ... + 11.0^2) = 675.16\). Thus, these values are verified as correct.

02

Calculate Means

To derive the equation of the regression line, first calculate the means of \(x\) and \(y\). The mean of \(x\) is \(\bar{x} = \frac{\sum x_{i}}{n} = \frac{154.2}{10} = 15.42\). The mean of \(y\) is \(\bar{y} = \frac{\sum y_{i}}{n} = \frac{80}{10} = 8.0\).

03

Calculate the Slope

The slope \(b\) of the regression line is calculated using:\[ b = \frac{\sum (x_i y_i) - n \bar{x} \bar{y}}{\sum (x_i^2) - n \bar{x}^2} \]Substitute the given sums and calculated means:\[ b = \frac{1282.74 - 10 \times 15.42 \times 8.0}{2452.18 - 10 \times 15.42^2} = \frac{1282.74 - 1233.6}{2452.18 - 2380.71} = \frac{49.14}{71.47} = 0.6876 \]

04

Calculate the Intercept

The intercept \(a\) of the regression line is calculated using:\[ a = \bar{y} - b \bar{x} \] hus, substitute the values:\[ a = 8.0 - 0.6876 \times 15.42 = 8.0 - 10.600 = -2.600 \]

05

Write the Regression Equation

The equation of the estimated regression line is:\[ y = a + bx = -2.600 + 0.6876x \]

06

Predict Separation Distance

For a street with \(x = 15.0\), substitute into the regression equation to predict \(y\):\[ y = -2.600 + 0.6876 \times 15.0 = -2.600 + 10.314 = 7.714 \]

07

Estimate Expected Separation Distance

The predicted separation distance for streets with available travel space \(15.0\) is the same calculation as predicting it for one street; hence, the expected separation distance is also \(7.714\).

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Key Concepts

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

Statistical Modeling

Statistical modeling is a way to represent data and make predictions about it. Think of it as a recipe that helps us understand complex data by breaking it down into simpler parts. It's like using a map to plan a trip - it guides you by showing the relationship between different places.

In our exercise, we are talking about available travel space and separation distance. They are like two friends who often hang out together. Statistical models help us figure out how one might behave if we know something about the other.

A good model can tell us if these friends will likely meet at a particular place, like helping us predict separation distance using travel space.

Linear Regression

Linear regression is a type of statistical modeling. It helps us draw a straight line through data points on a graph, showing a relationship between two variables. Think of it like finding the shortest path between two towns.

The line is called the 'regression line.' It helps us see trends and make predictions. For example, in our bike lanes study, we are trying to find a line that best represents the relationship between available travel space and separation distance.

We use calculations to find two important numbers:

• The slope, which tells us how steep the line is. It shows how much y (separation distance) changes when x (available travel space) changes.
• The intercept, which tells us where the line crosses the y-axis. It's like the starting point on our map.

Together, these numbers form the equation of our regression line.

Data Analysis

Data analysis involves examining, cleaning, transforming, and modeling data to discover useful information. It's like being a detective who examines clues to solve a mystery.

In this exercise, we begin with raw data about bike lanes and several calculations to verify sums of values, such as total travel space and separation distances. We calculate these using simple arithmetic, like adding or multiplying numbers. These steps help ensure our data is ready for analysis.

By calculating averages and looking at how data points are spread, we get a clear picture of what the data tells us. It's like piecing together a puzzle until we can see the whole picture.

Predictive Modeling

Predictive modeling uses mathematical techniques to predict future outcomes based on historical data. It’s like looking at weather patterns to forecast tomorrow's weather.

In the case of our regression line from the bike lanes study, we use it to predict the separation distance on streets with specific travel space. For instance, if another street has a travel space of 15 feet, we can plug this number into our regression line equation to predict the distance a car might maintain while passing a bicycle.

This process helps make informed decisions. It means we can plan for future road projects by expecting how road layout influences driver behavior. Predictive models like these are valuable tools for planning and decision-making.