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Chapter 3: Problem 47

Example 13 found the regression line \(\hat{y}=-3.1+0.33 x\) for all 51 observations on \(y=\) murder rate and \(x=\) percent with a college education. a. Show that the predicted murder rates increase from 1.85 to 10.1 as percent with a college education increases from \(x=15 \%\) to \(x=40 \%\), roughly the range of observed \(x\) values. b. When the regression line is fitted only to the 50 states, \(\hat{y}=8.0-0.14 x\). Show that the predicted murder rate decreases from 5.9 to 2.4 as percent with a college education increases from \(15 \%\) to \(40 \%\). c. D.C. has the highest value for \(x\) (38.3) and is an extreme outlier on \(y\) (41.8). Is it a regression outlier? Why? d. What causes results to differ numerically according to whether \(\mathrm{D} . \mathrm{C}\). is in the data set? Which line is more appropriate as a summary of the relationship? Why?

Short Answer

Expert verified
The predicted rates increase from 1.85 to 10.1 in the first model and decrease from 5.9 to 2.4 in the second, indicating D.C.'s significant outlier effect. The second model is more appropriate without D.C.

Step by step solution

01

Calculate Predicted Rates with First Regression

Using the first regression equation \( \hat{y} = -3.1 + 0.33x \), we will calculate the predicted murder rates. - For \( x = 15\% \):\[\hat{y} = -3.1 + 0.33 \times 15 = -3.1 + 4.95 = 1.85\]- For \( x = 40\% \):\[\hat{y} = -3.1 + 0.33 \times 40 = -3.1 + 13.2 = 10.1\]Thus, the predicted murder rates increase from 1.85 to 10.1 as \( x \) increases from 15\% to 40\%.
02

Calculate Predicted Rates with Second Regression

Using the second regression equation \( \hat{y} = 8.0 - 0.14x \), we calculate the predicted murder rates.- For \( x = 15\% \):\[\hat{y} = 8.0 - 0.14 \times 15 = 8.0 - 2.1 = 5.9\]- For \( x = 40\% \):\[\hat{y} = 8.0 - 0.14 \times 40 = 8.0 - 5.6 = 2.4\]Thus, the predicted murder rates decrease from 5.9 to 2.4 as \( x \) increases from 15\% to 40\%.
03

Analyze Potential Outlier

D.C. has the highest \( x \) value (38.3) and a murder rate \( y \) of 41.8. Since the predicted rate at \( x = 38.3 \) from the first model is 9.54, the discrepancy is large (32.26). It indicates that D.C. has a murder rate significantly higher than predicted, making it a potential regression outlier.
04

Discuss Differences and Appropriateness of Models

The inclusion of D.C. makes the predicted murder rate increase with education, although D.C. is an educational outlier with a high murder rate, skewing results. Without D.C., a negative relationship reflects the data better due to eliminating the outlier effect. The second model better summarizes the relationship for the states, given D.C.'s disproportional impact.

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Key Concepts

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

Linear Regression
Linear regression is a statistical technique used to model the relationship between a dependent variable and one or more independent variables using a linear function. It helps to predict or explain outcomes and assess the strength of relationships. In simple linear regression, such as the one used in the original problem, we have one independent variable, in this case, the percentage of people with a college education, and one dependent variable, the murder rate. The linear regression equation can be written as:\[y = a + bx\]where \(y\) is the predicted value (murder rate), \(x\) is the independent variable (college education percentage), \(a\) is the y-intercept, and \(b\) is the slope of the line. The slope \(b\) indicates the change in the dependent variable for a one-unit change in the independent variable. In our example, the two different regression equations provided different insights, one indicating an increase in murder rates with higher education rates, and the other a decrease. These differing results illustrate how influential data points like outliers can alter the perceived relationship between variables.
Outliers in Data
Outliers are data points that differ significantly from other observations in the dataset. They can occur due to variability in the measurement or it could indicate experimental errors. In the context of linear regression, outliers can significantly affect the results of the analysis by skewing the trend line.

D.C. was identified as an outlier in the original problem because it had a particularly high murder rate not consistent with the pattern seen in other states. To determine the influence of an outlier, you can compare models with and without the outlier. In this case, including D.C. shifted the relationship from negative to positive, illustrating its disruptive impact. By distorting the linear regression outcome, outliers can mask the true relationship between variables. Therefore, it's essential to identify and examine outliers during analysis to make informed decisions about data inclusion.
Predictive Modeling
Predictive modeling involves using statistical techniques like regression analysis to predict future outcomes based on historical data. It's a crucial tool in data analysis to make educated predictions about unknown future events.

In this context, we used the regression equations as predictive models to estimate murder rates based on varying percentages of the population with a college education. The process involves applying the regression equations to predict values within your dataset; this allows for understanding potential trends or patterns.

For effective predictive modeling:
  • Build a model with the relevant variables, ensuring they are meaningfully linked to the outcome.
  • Examine your dataset for outliers, which can distort predictions.
  • Validate the accuracy of your model with various subsets of data.
The exercise provided two distinct predictive models. The practical choice depends on whether D.C. is included in the dataset, highlighting the importance of context understanding in predictive modeling decisions.

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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.

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.

For the 100 cars on the lot of a used-car dealership, would you expect a positive association, negative association, or no association between each of the following pairs of variables? Explain why. a. The age of the car and the number of miles on the odometer b. The age of the car and the resale value c. The age of the car and the total amount that has been spent on repairs d. The weight of the car and the number of miles it travels on a gallon of gas e. The weight of the car and the number of liters it uses per \(100 \mathrm{~km}\).

The following table shows data on gender \((\) coded as \(1=\) female \(, 2=\) male \()\) and preferred type of chocolate \((\) coded as \(1=\) white, \(2=\) milk, \(3=\) dark \()\) for a sample of 10 students. The students' teacher enters the data into software and reports a correlation of 0.640 between gender and type of preferred chocolate. He concludes that there is a moderately strong positive correlation between someone's gender and chocolate preference. What's wrong with this analysis?

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?
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