The Number Of Newly Reported Crime Cases In A County In New York State Is Shown In The Accompanying Table, Where $x$ Represents The Number Of Years Since 2003, And $y$ Represents The Number Of New Cases. Write The Linear
Introduction
Crime rates have been a significant concern for law enforcement agencies and policymakers in the United States. The state of New York, in particular, has seen a fluctuating trend in crime rates over the years. In this article, we will analyze the number of newly reported crime cases in a county in New York State using linear regression. We will examine the relationship between the number of years since 2003 and the number of new cases.
The Data
The accompanying table shows the number of newly reported crime cases in a county in New York State for the years 2003 to 2018.
| Year (x) | Number of New Cases (y) |
|---|---|
| 0 | 1500 |
| 1 | 1550 |
| 2 | 1600 |
| 3 | 1650 |
| 4 | 1700 |
| 5 | 1750 |
| 6 | 1800 |
| 7 | 1850 |
| 8 | 1900 |
| 9 | 1950 |
| 10 | 2000 |
| 11 | 2050 |
| 12 | 2100 |
| 13 | 2150 |
| 14 | 2200 |
| 15 | 2250 |
| 16 | 2300 |
| 17 | 2350 |
| 18 | 2400 |
Linear Regression
Linear regression is a statistical method used to model the relationship between a dependent variable (y) and one or more independent variables (x). In this case, we will use linear regression to model the relationship between the number of years since 2003 (x) and the number of new cases (y).
The linear regression equation is given by:
y = β0 + β1x + ε
where β0 is the intercept, β1 is the slope, and ε is the error term.
Calculating the Linear Regression Equation
To calculate the linear regression equation, we need to estimate the values of β0 and β1. We can use the ordinary least squares (OLS) method to estimate these values.
The OLS method minimizes the sum of the squared errors between the observed values of y and the predicted values of y.
The estimated values of β0 and β1 are given by:
β1 = Σ[(xi - x̄)(yi - ȳ)] / Σ(xi - x̄)²
β0 = ȳ - β1x̄
where x̄ is the mean of the x values, ȳ is the mean of the y values, and xi and yi are the individual x and y values.
Calculating the Slope (β1)
To calculate the slope (β1), we need to calculate the numerator and denominator of the equation.
The numerator is given by:
Σ[(xi - x̄)(yi - ȳ)]
= (0 - 10)(1500 - 2000) + (1 - 10)(1550 - 2000) + ... + (18 - 10)(2400 - 2000)
= -5000 - 4500 - 4000 - 3500 - 3000 - 2500 - 2000 - 1500 - 1000 - 500 + 500 + 1000 + 1500 + 2000 + 2500 + 3000 + 3500 + 4000 + 4500 + 5000
= -30000
The denominator is given by:
Σ(xi - x̄)²
= (0 - 10)² + (1 - 10)² + ... + (18 - 10)²
= 100 + 81 + 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100
= 585
Therefore, the slope (β1) is given by:
β1 = -30000 / 585
= -51.08
Calculating the Intercept (β0)
To calculate the intercept (β0), we need to use the estimated value of β1 and the mean of the y values (ȳ).
The mean of the y values (ȳ) is given by:
ȳ = (1500 + 1550 + ... + 2400) / 19
= 2000
Therefore, the intercept (β0) is given by:
β0 = ȳ - β1x̄
= 2000 - (-51.08)(10)
= 2000 + 510.8
= 2510.8
The Linear Regression Equation
The linear regression equation is given by:
y = β0 + β1x + ε
= 2510.8 - 51.08x + ε
Interpretation of the Results
The linear regression equation shows a negative relationship between the number of years since 2003 and the number of new cases. This means that as the number of years since 2003 increases, the number of new cases decreases.
The slope (β1) of -51.08 indicates that for every additional year since 2003, the number of new cases decreases by approximately 51.08.
The intercept (β0) of 2510.8 indicates that when x = 0 (i.e., in the year 2003), the number of new cases is approximately 2510.8.
Conclusion
In this article, we analyzed the number of newly reported crime cases in a county in New York State using linear regression. We found a negative relationship between the number of years since 2003 and the number of new cases. The slope of the linear regression equation indicates that for every additional year since 2003, the number of new cases decreases by approximately 51.08. The intercept of the linear regression equation indicates that when x = 0 (i.e., in the year 2003), the number of new cases is approximately 2510.8.
Limitations of the Study
This study has several limitations. Firstly, the data used in this study is limited to a single county in New York State. Secondly, the data is based on a single year (2003) as the starting point. Thirdly, the linear regression model assumes a linear relationship between the number of years since 2003 and the number of new cases. However, this relationship may not be linear in reality.
Future Research Directions
Future research directions include:
- Using a larger dataset: Using a larger dataset that includes multiple counties in New York State would provide a more comprehensive understanding of the relationship between the number of years since 2003 and the number of new cases.
- Using a different model: Using a different model, such as a non-linear model, may provide a more accurate representation of the relationship between the number of years since 2003 and the number of new cases.
- Including additional variables: Including additional variables, such as the number of police officers or the number of community programs, may provide a more comprehensive understanding of the relationship between the number of years since 2003 and the number of new cases.
Q&A: Linear Regression Analysis of Crime Rates in New York State ===========================================================
Introduction
In our previous article, we analyzed the number of newly reported crime cases in a county in New York State using linear regression. We found a negative relationship between the number of years since 2003 and the number of new cases. In this article, we will answer some frequently asked questions (FAQs) related to the linear regression analysis.
Q: What is the purpose of linear regression analysis in this study?
A: The purpose of linear regression analysis in this study is to examine the relationship between the number of years since 2003 and the number of new cases. We want to determine if there is a significant relationship between these two variables and if so, what the nature of the relationship is.
Q: What is the independent variable (x) in this study?
A: The independent variable (x) in this study is the number of years since 2003.
Q: What is the dependent variable (y) in this study?
A: The dependent variable (y) in this study is the number of new cases.
Q: What is the slope (β1) of the linear regression equation?
A: The slope (β1) of the linear regression equation is -51.08. This means that for every additional year since 2003, the number of new cases decreases by approximately 51.08.
Q: What is the intercept (β0) of the linear regression equation?
A: The intercept (β0) of the linear regression equation is 2510.8. This means that when x = 0 (i.e., in the year 2003), the number of new cases is approximately 2510.8.
Q: What is the significance of the linear regression equation?
A: The linear regression equation shows a negative relationship between the number of years since 2003 and the number of new cases. This means that as the number of years since 2003 increases, the number of new cases decreases.
Q: What are the limitations of this study?
A: This study has several limitations. Firstly, the data used in this study is limited to a single county in New York State. Secondly, the data is based on a single year (2003) as the starting point. Thirdly, the linear regression model assumes a linear relationship between the number of years since 2003 and the number of new cases. However, this relationship may not be linear in reality.
Q: What are the future research directions?
A: Future research directions include:
- Using a larger dataset: Using a larger dataset that includes multiple counties in New York State would provide a more comprehensive understanding of the relationship between the number of years since 2003 and the number of new cases.
- Using a different model: Using a different model, such as a non-linear model, may provide a more accurate representation of the relationship between the number of years since 2003 and the number of new cases.
- Including additional variables: Including additional variables, such as the number of police officers or the number of community programs, may provide a more comprehensive understanding of the relationship between the number of years since 2003 and the number of new cases.
Conclusion
In this article, we answered some frequently asked questions (FAQs) related to the linear regression analysis of crime rates in New York State. We hope that this article provides a better understanding of the relationship between the number of years since 2003 and the number of new cases.
References
- [1] Linear Regression Analysis. (n.d.). Retrieved from https://en.wikipedia.org/wiki/Linear_regression
- [2] Crime Rates in New York State. (n.d.). Retrieved from https://www.bjs.gov/index.cfm?ty=tp&tid=1044
Appendix
The data used in this study is available in the following table:
| Year (x) | Number of New Cases (y) |
|---|---|
| 0 | 1500 |
| 1 | 1550 |
| 2 | 1600 |
| 3 | 1650 |
| 4 | 1700 |
| 5 | 1750 |
| 6 | 1800 |
| 7 | 1850 |
| 8 | 1900 |
| 9 | 1950 |
| 10 | 2000 |
| 11 | 2050 |
| 12 | 2100 |
| 13 | 2150 |
| 14 | 2200 |
| 15 | 2250 |
| 16 | 2300 |
| 17 | 2350 |
| 18 | 2400 |
The linear regression equation is:
y = 2510.8 - 51.08x
This equation shows a negative relationship between the number of years since 2003 and the number of new cases.