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Chapter 5: Problem 2

What's the Line? An online article suggested that for each additional person who took up regular running for exercise, the number of cigarettes smoked daily would decrease by \(0.178 .{ }^{2}\) If we assume that 48 million cigarettes would be smoked per day if nobody ran, what is the equation of the regression line for predicting number of cigarettes smoked per day from the number of people who regularly run for exercise?

Short Answer

Expert verified
The regression line equation is \(y = 48 - 0.178x\).

Step by step solution

01

Understanding the Variables

Identify the variables. Let the number of regular runners be represented by \(x\), and the number of cigarettes smoked daily be represented by \(y\). According to the problem, the y-intercept is when \(x = 0\), so the y-intercept (\(b_0\)) is 48 million cigarettes.
02

Slope Calculation

The problem states that for each additional runner, the number of cigarettes smoked decreases by 0.178. Therefore, the slope (\(b_1\)) of the line is \(-0.178\). This represents the rate of decrease in the number of cigarettes smokes per additional runner.
03

Equation Formation

Use the slope and y-intercept to form the regression line equation: \(y = b_0 + b_1x\). Substitute \(b_0 = 48\) (in million cigarettes) and \(b_1 = -0.178\) into the equation.
04

Writing the Final Equation

Substitute the slope and y-intercept values to get the final equation: \(y = 48 - 0.178x\). This represents the regression line for predicting the number of cigarettes smoked per day based on the number of people who regularly run.

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Key Concepts

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

Slope Calculation
In linear regression analysis, calculating the slope is essential. The slope represents the rate of change between the two variables involved. To put it simply, it tells us how much one variable changes as the other variable changes.
In the equation of a line in the form of \( y = mx + c \), the slope is represented by \( m \). For our exercise, the slope \( b_1 \) is \(-0.178\).
This indicates that for every additional person who takes up regular running, the number of cigarettes smoked daily decreases by 0.178. In the context of a regression line, the slope helps us understand the strength and direction of the relationship between the variables.
Y-intercept
The y-intercept is a critical value in any linear equation which gives us the starting point of the line on the graph. It is the value of \( y \) when \( x \) is zero. This means it shows what happens when there are no runners in this context.
Given in the exercise, the y-intercept \( b_0 \) is 48 million. This means if nobody ran, there would be 48 million cigarettes smoked per day.
  • The y-intercept symbol is \( b_0 \)
  • It provides a baseline from which the effects of additional runners are measured.
The y-intercept serves as a crucial part in forming the complete linear equation.
Linear Equation
A linear equation is a representation of a straight line on a graph. It is defined using two key components: the slope and the y-intercept. Together, they form the complete equation \( y = mx + c \) which in context of our exercise becomes \( y = 48 - 0.178x \).
The formula clearly indicates:
  • \( y \): Predicted number of cigarettes smoked daily
  • \( x \): Number of regular runners
By substituting different values of \( x \) (number of runners) into the equation, we compute the predicted number of cigarettes smoked, helping us visualize the relationship as a linear model.
Predictive Modeling
Predictive modeling is the process of creating a mathematical model to predict future outcomes based on available data. In regression analysis, our linear equation serves as a predictive model.
Predictive models use patterns found in historical data to predict future behavior. For example, by using the equation \( y = 48 - 0.178x \), we can estimate how cigarette consumption decreases as more people take up running regularly.
  • The model provides insights into decision-making processes by estimating future results based on current trends.
  • It is useful for forming strategies to improve health outcomes by understanding factors that influence behavior.
Predictive modeling offers valuable foresight by effectively utilizing regression equations.

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Most popular questions from this chapter

More exercise, more weight loss. In the study described in Example 5.5, the researchers found that, in general, subjects who engaged in more physical activity had higher total energy expenditures. In particular, they found that physical activity explained \(3.3 \%\) of the variation in total energy expenditure. What is the numerical value of the correlation between physical activity and total energy expenditure?

Grade inflation and the SAT. The effect of a lurking variable can be surprising when individuals are divided into groups. In recent years, the mean SAT score of all high school seniors has increased. But the mean SAT score has decreased for students at each level of high school grades ( \(A, B, C\), and so on). Explain how grade inflation in high school (the lurking variable) can account for this pattem.

City Mileage, Highway Mileage. We expect a car's highway gas mileage to be related to its city gas mileage (in mpg). Data for all 1209 vehicles in the government's 2016 Fuel Economy Guide give the regression line $$ \text { highway mpg }=7.903+(0.993 \times \text { city mpg }) $$ for predicting highway mileage from city mileage. (a) What is the slope of this line? Say in words what the numerical value of the slope tells you. (b) What is the intercept? Explain why the value of the intercept is not statistically meaningful. (c) Find the predicted highway mileage for a car that gets \(16 \mathrm{mpg}\) in the city. Do the same for a car with city mileage of \(28 \mathrm{mpg}\). (d) Draw a graph of the regression line for city mileages between 10 and 50 mpg. (Be sure to show the scales for the \(x\) and \(y\) axes.)

Regression to the mean. Figure \(4.8\) (page 119) displays the relationship between golfers' scores on the first and second rounds of the 2016 Masters Tournament. The least-squares line for predicting second-round scores \((y)\) from first-round scores \((x)\) has equation \(y^{-} \hat{y}=62.91+0.164 x\). Find the predicted second-round scores for a player who shot 80 in the first round and for a player who shot 70 . The mean second-round score for all players was \(75.02\). So, a player who does well in the first round is predicted to do less well, but still better than average, in the second round. In addition, a player who does poorly in the first is predicted to do berter, but still worse than average, in the second. (Comment: This is regression to the mean. If you select individuals with extreme scores on some measure, they tend to have less extreme scores when measured again. That's because their extreme position is partly merit and partly luck, and the luck will be different next time. Regression to the mean contributes to lots of "effects." The rookie of the year often doesn't do as well the next year; the best player in an orchestral audition may play less well once hired than the runners-up; a student who feels she needs coaching after taking the SAT often does better on the next try without coaching.)

To Earn More, Get Married? Data show that men who are married, and also divorced or widowed men, earn quite a bit more than men the same age who have never been married. This does not mean that a man can raise his income by getting married because men who have never been married are different from married men in many ways other than marital status. Suggest several lurking variables that might help explain the association between marital status and income.
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