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Chapter 9: Problem 14

Prisons Does prison really deter violent crime? Let \(x\) represent percent change in the rate of violent crime and \(y\) represent percent change in the rate of imprisonment in the general U.S. population. For 7 recent years, the following data have been obtained (Source: The Crime Drop in America, edited by Blumstein and Wallman, Cambridge University Press). $$ \begin{array}{l|rrrrrrr} \hline x & 6.1 & 5.7 & 3.9 & 5.2 & 6.2 & 6.5 & 11.1 \\ \hline y & -1.4 & -4.1 & -7.0 & -4.0 & 3.6 & -0.1 & -4.4 \\ \hline \end{array} $$ Complete parts (a) through (e), given \(\Sigma x=44.7, \Sigma y=-17.4, \Sigma x^{2}=315.85\), \(\Sigma y^{2}=116.1, \Sigma x y=-107.18\), and \(r \approx 0.084 .\) (f) Critical Thinking Considering the values of \(r\) and \(r^{2}\), does it make sense to use the least-squares line for prediction? Explain.

Short Answer

Expert verified
With an \( r \approx 0.084 \) and \( r^2 \approx 0.007 \), the linear model is not suitable for prediction because it explains less than 1% of the variation.

Step by step solution

01

Calculate the Coefficient of Determination

First, calculate the coefficient of determination, denoted as \( r^2 \), which quantifies the proportion of the variance in the dependent variable that is predictable from the independent variable. Since the correlation coefficient \( r \) is given as approximately 0.084, we calculate \( r^2 \) as follows: \[ r^2 = (0.084)^2 = 0.007056. \] This means that about 0.7% of the variation in the rate of violent crime is explained by imprisonment rates.
02

Interpret the Values of r and r虏

Given that \( r = 0.084 \), we identify that there is a very weak positive linear relationship between percent change in violent crime and percent change in imprisonment rates. Additionally, an \( r^2 \) value of 0.007056 indicates that only approximately 0.7% of the variation in the violent crime rate can be explained by changes in imprisonment rates. This is an insufficient explanation indicating a lack of association between these variables.
03

Evaluate the Use of Least-Squares Line

Because both \( r \) and \( r^2 \) values are extremely low, using a least-squares regression line to predict the percent change in violent crime from the percent change in imprisonment rates would not be reliable. The small \( r^2 \) value suggests that the linear model does not fit the data well and is not an appropriate choice for prediction.

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Key Concepts

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

Coefficient of Determination
The Coefficient of Determination, represented as \( r^2 \), is a statistical measure that helps us understand the proportion of variance in the dependent variable that is predictable from the independent variable. In simpler terms, it tells us how well our data points fit a statistical model or a line of best fit.
To calculate this, we square the correlation coefficient \( r \). For instance, if \( r \) is 0.084, the coefficient of determination \( r^2 \) would be calculated as follows:
  • \( (0.084)^2 = 0.007056 \)
This indicates that only about 0.7% of the change in violent crime rates can be explained by changes in imprisonment rates. Such a low \( r^2 \) value suggests that our independent variable (imprisonment rates) does not have much explanatory power concerning changes in violent crime rates. Therefore, the relationship in the data is weak, making predictions based on this model quite unreliable.
Correlation Coefficient
The Correlation Coefficient, denoted as \( r \), is a statistical measure that expresses the extent to which two variables are linearly related. It ranges between -1 and 1.
- A value of 1 implies a perfect positive linear relationship - A value of -1 implies a perfect negative linear relationship- A value of 0 indicates no linear relationship at allIn our study of violent crime rates and imprisonment rates, the correlation coefficient is given as 0.084. But what does this number signify?Given that \( r = 0.084 \):
  • It indicates a very weak positive linear relationship.
  • This minimal correlation suggests that changes in imprisonment rates have almost no predictive power in explaining the changes in violent crime rates.
Thus, because \( r \) is so close to zero, it is an indication that the linear relationship is weak, and other factors likely influence the dependent variable more significantly than imprisonment rates.
Least-Squares Line
The Least-Squares Line is a method used in regression analysis to find the line that best fits a given set of data points, minimizing the sum of the squares of the vertical distances of the points from the line. It is often used for making predictions about one variable based on the known values of another variable.
Despite its utility, the least-squares method relies heavily on the value of \( r \) and \( r^2 \) to judge its effectiveness.

In our scenario:
  • Given the very low \( r \) of 0.084, and an \( r^2 \) of 0.007056, the least-squares line doesn鈥檛 provide a good fit for the data.
  • This is because the small \( r^2 \) value suggests that the line does not adequately represent the variation observed in the data.
Therefore, relying on the least-squares line for predictions between violent crime rates and imprisonment rates in this context would not be effective. Other methods or additional variables might be needed to build a more accurate predictive model.

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Most popular questions from this chapter

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