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Chapter 3: Problem 32

In 2013, data was collected from the U.S. Department of Transportation and the Insurance Institute for Highway Safety. According to the collected data, the number of deaths per 100,000 individuals in the U.S would decrease by 24.45 for every 1 percentage point gain in seat belt usage. Let \(\hat{y}=\) predicted number of deaths per 100,000 individuals in 2013 and \(x=\) seat belt use rate in a given state. a. Report the slope \(b\) for the equation \(\hat{y}=a+b x\). b. If the \(y\) intercept equals \(32.42,\) then predict the number of deaths per 100,000 people in a state if (i) no one wears seat belts, (ii) \(74 \%\) of people wear seat belts (the value for Montana), (iii) \(100 \%\) of people wear seat belts.

Short Answer

Expert verified
a. The slope \( b \) is \(-24.45\). b. (i) 32.42, (ii) 14.327, (iii) 7.97.

Step by step solution

01

Determine the Slope

The problem states that the number of deaths per 100,000 decreases by 24.45 for every 1 percentage point increase in seat belt usage. This change per 1 percentage point increase is the slope. Therefore, the slope \( b \) is \(-24.45\).
02

Understand the Linear Equation

Given that \( \hat{y} = a + b x \), and given \( a = 32.42 \) (the y-intercept) and \( b = -24.45 \) (the slope), the equation becomes: \[ \hat{y} = 32.42 - 24.45x \]
03

Predict Deaths with 0% Seat Belt Usage

Substitute \( x = 0 \) into the equation: \[ \hat{y} = 32.42 - 24.45(0) \] which simplifies to \( \hat{y} = 32.42 \). This means that if no one wears seat belts, the predicted number of deaths per 100,000 is 32.42.
04

Predict Deaths with 74% Seat Belt Usage

Substitute \( x = 0.74 \) into the equation: \[ \hat{y} = 32.42 - 24.45(0.74) \] Calculate \( 24.45 \times 0.74 = 18.093 \), thus the equation becomes \( \hat{y} = 32.42 - 18.093 = 14.327 \). The predicted number of deaths per 100,000 with 74% seat belt usage is 14.327.
05

Predict Deaths with 100% Seat Belt Usage

Substitute \( x = 1 \) into the equation: \[ \hat{y} = 32.42 - 24.45(1) \] which simplifies to \( \hat{y} = 32.42 - 24.45 = 7.97 \). Thus, if everyone wears seat belts, the predicted number of deaths per 100,000 is 7.97.

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Key Concepts

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

Understanding the Slope in Linear Regression
In any linear regression model, the slope is a crucial component as it quantifies the relationship between the independent variable and the dependent variable. In our context of analyzing seat belt usage and fatalities, the slope represents how changes in seat belt usage affect the number of deaths.

The slope, symbolized by \( b \) in the equation \( \hat{y} = a + bx \), tells us how much the predicted value \( \hat{y} \) changes for each unit increase in \( x \), the seat belt usage rate. Here, our slope is \(-24.45\), which means for every 1 percentage point increase in seat belt usage, the number of deaths per 100,000 is expected to decrease by 24.45.

A negative slope indicates that the relationship between seat belt usage and fatalities is inversely proportional. It's a clear mathematical representation that emphasizes the importance of seat belt usage in reducing fatalities.
Decoding the Y-intercept
The y-intercept in a linear equation provides the starting value of the dependent variable when the independent variable is zero. In our equation \( \hat{y} = 32.42 - 24.45x \), the y-intercept is \( 32.42 \).

This value represents the predicted number of deaths per 100,000 individuals when the seat belt usage rate is 0%. Essentially, it's the baseline prediction in the absence of any seat belt usage.

The y-intercept offers a reference point against which changes in the predicted value are measured. In practical terms, it suggests what could be the state of fatalities if no preventive measures like seat belt usage were in place.
Making Predictions with Linear Regression
Predicting outcomes using a linear regression equation involves substituting values for the independent variable into the equation. This allows us to estimate the dependent variable or the value we are trying to predict.

In this scenario, we use the equation \( \hat{y} = 32.42 - 24.45x \) to estimate the number of deaths at different seat belt usage rates:
  • For 0% seat belt usage \((x = 0)\), substituting into the equation gives \( \hat{y} = 32.42 \). This means the predicted fatalities are 32.42 per 100,000 individuals.
  • For 74% seat belt usage \((x = 0.74)\), the equation becomes \( \hat{y} = 32.42 - 24.45 \times 0.74 = 14.327 \). This shows a significant reduction in fatalities.
  • For 100% seat belt usage \((x = 1)\), the equation simplifies to \( \hat{y} = 32.42 - 24.45 = 7.97 \). This is the scenario with the lowest predicted fatalities.
These predictions help illustrate the critical impact that increased seat belt usage has on reducing fatalities on the roads.

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Most popular questions from this chapter

The figure shows recent data on \(x=\) the number of televisions per 100 people and \(y=\) the birth rate (number of births per 1000 people ) for six African and Asian nations. The regression line, \(\hat{y}=29.8-0.024 x\), applies to the data for these six countries. For illustration, another point is added at (81,15.2) , which is the observation for the United States. The regression line for all seven points is \(\hat{y}=31.2-0.195 x\). The figure shows this line and the one without the U.S. observation. a. Does the U.S. observation appear to be (i) an outlier on \(x\), (ii) an outlier on \(y\), or (iii) a regression outlier relative to the regression line for the other six observations? b. State the two conditions under which a single point can have a dramatic effect on the slope and show that they apply here. c. This one point also drastically affects the correlation, which is \(r=-0.051\) without the United States but \(r=-0.935\) with the United States. Explain why you would conclude that the association between birth rate and number of televisions is (i) very weak without the U.S. point and (ii) very strong with the U.S. point. d. Explain why the U.S. residual for the line fitted using that point is very small. This shows that a point can be influential even if its residual is not large.

Most cars are fuel efficient when running at a steady speed of around 40 to \(50 \mathrm{mph}\). A scatterplot relating fuel consumption (measured in mpg) and steady driving speed (measured in mph) for a mid-sized car is shown below. The data are available in the Fuel file on the book's Web site. (Source: Berry, I. M. (2010). The Effects of Driving Style and Vehicle Performance on the Real-World Fuel Consumption of U.S. Light-Duty Vehicles. Masters thesis, Massachusetts Institute of Technology, Cambridge, MA.) a. The correlation equals \(0.106 .\) Comment on the use of the correlation coefficient as a measure for the association between fuel consumption and steady driving speed. b. Comment on the use of the regression equation as a tool for predicting fuel consumption from the velocity of the car. c. Over what subrange of steady driving speed might fitting a regression equation be appropriate? Why?

Is there a relationship between how many sit-ups you can do and how fast you can run 40 yards? The EXCEL output shows the relationship between these variables for a study of female athletes to be discussed in Chapter 12 .a. The regression equation is \(\hat{y}=6.71-0.024 x .\) Find the predicted time in the 40 -yard dash for a subject who can do (i) 10 sit-ups, (ii) 40 sit-ups. Based on these times, explain how to sketch the regression line over this scatterplot. b. Interpret the \(y\) -intercept and slope of the equation in part a, in the context of the number of sit-ups and time for the 40 -yard dash. c. Based on the slope in part a, is the correlation positive or negative? Explain.

A study shows that there is a positive correlation between \(x=\) size of a hospital (measured by its numbers of beds) and \(y=\) median number of days that patients remain in that hospital. a. Does this mean that you can shorten a hospital stay by choosing a small hospital? b. Identify a third variable that could be a common cause of \(x\) and \(y .\) Construct a hypothetical scatterplot (like Figure 3.22 for crime and education), identifying points according to their value on the third variable, to illustrate your argument.

According to data selected from GSS in \(2014,\) the correlation between \(y=\) email hours per week and \(x=\) Internet hours per week is \(0.33 .\) The regression equation is predicted email hours \(=3.54+\) 0.25 Internet hours a. Based on the correlation value, the slope had to be positive. Why? b. Your friend says she spends 60 hours on the Internet and 10 hours on email in a week. Find her predicted email use based on the regression equation. c. Find her residual. Interpret.
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