91Ó°ÊÓ

The following data are based on information from the book Life in America's Small Cities (by G. S. Thomas, Prometheus Books). Let \(x\) be the percentage of 16 - to 19 -year-olds not in school and not high school graduates. Let \(y\) be the reported violent crimes per 1000 residents. Six small cities in Arkansas (Blytheville, El Dorado, Hot Springs, Jonesboro, Rogers, and Russellville) reported the following information about \(x\) and \(y\) : $$ \begin{array}{r|rrrrrr} \hline x & 24.2 & 19.0 & 18.2 & 14.9 & 19.0 & 17.5 \\ \hline y & 13.0 & 4.4 & 9.3 & 1.3 & 0.8 & 3.6 \\ \hline \end{array} $$ Complete parts (a) through (e), given \(\Sigma x=112.8, \Sigma y=32.4, \Sigma x^{2}=2167.14\), \(\Sigma y^{2}=290.14, \Sigma x y=665.03\), and \(r \approx 0.764\). (f) If the percentage of 16 - to 19 -year-olds not in school and not graduates reaches \(24 \%\) in a similar city, what is the predicted rate of violent crimes per 1000 residents?

Step by step solution

01

Understand the Linear Regression Formula

To find the predicted rate of violent crimes when the percentage of 16- to 19-year-olds not in school and not graduates is 24%, we use the linear regression equation: \[ y = mx + b \] where \( m \) is the slope and \( b \) is the y-intercept.

02

Calculate the Slope (m)

The slope \( m \) is given by the formula:\[ m = \frac{n \Sigma xy - \Sigma x \Sigma y}{n \Sigma x^2 - (\Sigma x)^2} \]Plugging in the values for \( n = 6 \), \( \Sigma xy = 665.03 \), \( \Sigma x = 112.8 \), \( \Sigma y = 32.4 \), and \( \Sigma x^2 = 2167.14 \), we calculate:\[ m = \frac{6 \times 665.03 - 112.8 \times 32.4}{6 \times 2167.14 - (112.8)^2} \]\[ m = \frac{3990.18 - 3650.72}{13002.84 - 12722.24} \] \[ m = \frac{339.46}{280.6} \approx 1.21 \]

03

Calculate the Y-Intercept (b)

The y-intercept \( b \) is given by the formula:\[ b = \frac{\Sigma y - m \Sigma x}{n} \]Substitute the calculated \( m \) and other given values:\[ b = \frac{32.4 - 1.21 \times 112.8}{6} \]\[ b = \frac{32.4 - 136.488}{6} \]\[ b = \frac{-104.088}{6} \approx -17.35 \]

04

Predict the Violent Crime Rate for 24%

Now that we have \( m \) and \( b \), use the linear regression equation to predict \( y \) when \( x = 24 \):\[ y = 1.21 \times 24 - 17.35 \]\[ y = 29.04 - 17.35 \]\[ y = 11.69 \]

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Key Concepts

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

Statistics Education

Understanding statistics is essential for analyzing and interpreting the world around us. One vital application of statistics is linear regression, a powerful method to establish a relationship between two variables.
In this exercise, we see a practical example of linear regression used to study the impact of education on crime rates in small cities within Arkansas. Different variables are represented in statistics: here, \(x\) denotes the percentage of young individuals not in school, and \(y\), the rate of violent crimes. These examples exemplify the importance of statistics education in solving real-world problems.
Grasping these statistical concepts allows students to comprehend how data can predict societal trends and behaviors. By understanding the relationships between different sets of data, students not only excel academically but are also better equipped to make informed decisions in their personal and professional lives.

Predictive Modeling

Predictive modeling is a statistical technique involving the use of mathematical formulas to predict future outcomes based on historical data.
A common form of predictive modeling is linear regression, as used in this exercise to forecast crime rates. Here, linear regression models the prediction that a city with 24% of teenagers not in school or not high school graduates will have a predicted crime rate of 11.69 per 1000 residents.
Predictive modeling requires selecting and interpreting variables correctly, ensuring the model is both reliable and valid. This involves calculating the model's slope and y-intercept, which tells us how much change in the independent variable \(x\) affects the dependent variable \(y\).
Through these methods, predictive modeling greatly aids in decision-making processes across multiple fields such as economics, engineering, and public policy.

Statistical Analysis

Statistical analysis is the process of collecting, reviewing, and interpreting data to uncover patterns and trends.
This exercise integrated comprehensive statistical analysis by calculating several key metrics, including sums of squares and products \(\Sigma x\), \(\Sigma y\), and \(\Sigma xy\), to determine the slope \(m\) and y-intercept \(b\) of the linear equation.
By examining these statistics, we gain insights into the direct relationship between education and crime rates. The analysis shows us the strength and direction of this relationship. Statistical analysis empowers learners to derive conclusions from supplied data.
With proper analysis, data reveals not just past records, but also enables accurate predictions and strategic planning.

Correlation Coefficient

The correlation coefficient \(r\) is a statistical measure that expresses the extent to which two variables are linearly related; it shows the strength and direction of this relationship.
In this exercise, the calculated correlation coefficient is approximately 0.764, signifying a strong positive relationship between the percentage of teenagers not in school and crime rates.
Values of \(r\) range from -1 to 1. An \(r\) close to 1 implies a strong positive correlation; conversely, an \(r\) near -1 suggests a strong negative correlation. An \(r\) around 0 indicates no correlation.
Understanding the correlation coefficient helps students grasp how much one variable is likely to affect another. It explains why certain variables in a linear regression model, like education and crime in this example, can be reliable for predictive modeling and strategic decision-making.