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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 \(0.178 . \underline{2}\) 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 equation is y = -0.178x + 48.

Step by step solution

01

Understand the Problem

The question provides a relationship where for each additional person taking up running, the number of cigarettes smoked decreases. It also provides the value when no one runs. Our task is to express this relationship in the form of a linear equation.
02

Identify Slope and Intercept

The slope ( m) of the line represents the rate of change, which is provided as -0.178. If the number of people running increases by 1 unit, cigarettes decrease by 0.178 units. The y-intercept ( b) is 48 million, which indicates the number of cigarettes smoked when no one runs.
03

Write the Linear Equation

The equation of a line can generally be written as y = mx + b. Here, x represents the number of runners, m is the slope, and b is the intercept. Substitute m = -0.178 and b = 48 into the equation.
04

Final Equation

Substitute the given values into the equation form: y = -0.178x + 48 where y is the number of cigarettes smoked in millions per day and x is the number of people running.

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Key Concepts

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

Linear Equation
A linear equation is a fundamental tool in regression analysis. It helps describe a straight-line relationship between two variables. In this context, the relationship is between the number of people running and the number of cigarettes smoked daily.
Understanding linear equations can provide a clear picture of how changes in one variable affect another. The general form of a linear equation is \( y = mx + b \). Here:
  • \( y \) represents the dependent variable (cigarettes smoked).
  • \( x \) is the independent variable (people running).
  • \( m \) is the slope, showing how \( y \) changes with \( x \).
  • \( b \) is the intercept, indicating \( y \)'s value when \( x \) is zero.
Substituting the provided values \( m = -0.178 \) and \( b = 48 \) gives us a understanding of the predicted number of cigarettes smoked based on the number of runners.
Slope and Intercept
The slope and intercept are crucial elements in the formulation of a linear equation. They define the characteristics and behavior of the linear relationship depicted by the equation.

**Slope**
The slope, denoted by \( m \), indicates the rate at which the dependent variable changes for a one-unit increase in the independent variable. In our example, the slope is \(-0.178\). This means for every additional person who starts running, the cigarette consumption decreases by 0.178 million per day.

It's important to note that the slope is negative, indicating an inverse relationship between running and smoking.

**Intercept**
The intercept, denoted by \( b \), tells us the value of the dependent variable when the independent variable is zero. For this problem, the intercept is \(48\) million cigarettes. This intercept represents the starting point of the line on the \( y \)-axis, showing the number of cigarettes smoked when nobody is running. Together, the slope and intercept form the backbone of our linear equation \( y = -0.178x + 48 \).
Statistical Prediction
Statistical prediction is the process of using known data to predict unknown outcomes. In regression analysis, this involves using a linear equation to estimate the dependent variable based on different values of the independent variable.
The linear equation \( y = -0.178x + 48 \) derived from the exercise allows us to predict the number of cigarettes smoked for any given number of runners.

To make a prediction:
  • Substitute the number of people running into \( x \).
  • Perform the calculations to find \( y \), the predicted number of cigarettes smoked in millions.
This method of prediction is powerful because it provides a clear, numerical understanding of the relationship between running and smoking habits. By using statistical prediction, we can plan and strategize how to effectively decrease cigarette consumption.

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Most popular questions from this chapter

SAT Scores. The correlation between mean \(2018 \mathrm{Math}\) SAT scores and mean 2018 ERW (Evidence-Based Reading and Writing) SAT scores for all 50 states and the District of Columbia is \(0.985\). Would you expect the correlation between the mean state SAT scores for these two tests to be lower, about the same, or higher than the correlation between the scores of individuals on these two tests? Explain your answer.

The Price of Diamond Rings. Online advertisements contained pictures of diamond rings and listed their prices, diamond weights (in carats), and gold purity. Based on data for only the 18-carat gold ladies' rings in the advertisements, the least-squares regression line for predicting price (in dollars) from the weight of the diamond (in carats) is \(\frac{18}{18}\) $$ \text { price }=-6047.75+11975.14 \text { carats } $$ a. What does the slope of this line say about the relationship between price and number of carats? b. What is the predicted price when number of carats \(=0\) ? How would you interpret this price?

How Useful Is Regression? Figure 4.9 (page \(\underline{116}\) ) displays the relationship between golfers' scores on the first and second rounds of the 2019 Masters Tournament. The correlation is \(r=0.283\). Eigure \(4.3\) (page 105 ) gives data on number of boats registered in Florida and the number of manatees killed by boats for the years 1977 to 2018. The correlation is \(r=0.919\). Explain in simple language why knowing only these correlations enables you to say that prediction of manatee deaths from number of boats registered by a regression line will be much more accurate than prediction of a golfer's second-round score based on that golfer's first-round score.

What's My Grade? In Professor Krugman's economics course, the correlation between the students' total scores prior to the final examination and their final-examination scores is \(r=0.5\). The pre-exam totals for all students in the course have mean 280 and standard deviation 40 . The finalexam scores have mean 75 and standard deviation 8 . Professor Krugman has lost Julie's final exam but knows that her total before the exam was 300 . He decides to predict her final-exam score from her pre-exam total. a. What is the slope of the least-squares regression line of final-exam scores on pre-exam total scores in this course? What is the intercept? Interpret the slope in the context of the problem. b. Use the regression line to predict Julie's final-exam score. c. Julie doesn't think this method accurately predicts how well she did on the final exam. Use \(r^{2}\) to argue that her actual score could have been much higher (or much lower) than the predicted value.

Are You Happy? Exercise 4.26 (page 117) discusses a study in which the mean BRFSS life-satisfaction score of individuals in each state was compared with the mean of an objective measure of well-being (based on the "compensating-differentials method") for each state. Suppose that instead of the means for the states, the BRFSS life-satisfaction scores for individuals were compared with the corresponding measure of well-being (based on the compensating-differentials method) for these individuals. Would you expect the correlation between the mean state scores on these two measures to be lower, about the same, or higher than the correlation between the scores of individuals on these two measures? Explain your answer.
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