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 1995, And $y$ Represents The Number Of New Cases. Write The Linear Regression
Introduction
Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables. In this analysis, we will use linear regression to examine the relationship between the number of years since 1995 and the number of newly reported crime cases in a county in New York State. The data is presented in the accompanying table.
Table: Number of Newly Reported Crime Cases in New York State
| Year (x) | Number of New Cases (y) |
|---|---|
| 0 | 1000 |
| 1 | 1200 |
| 2 | 1100 |
| 3 | 1300 |
| 4 | 1400 |
| 5 | 1500 |
| 6 | 1600 |
| 7 | 1700 |
| 8 | 1800 |
| 9 | 1900 |
Linear Regression Model
A linear regression model can be written in the form:
y = β0 + β1x + ε
where y is the dependent variable (number of new cases), x is the independent variable (number of years since 1995), β0 is the intercept or constant term, β1 is the slope coefficient, and ε is the error term.
Calculating the Linear Regression Coefficients
To calculate the linear regression coefficients, we need to minimize the sum of the squared errors (SSE) between the observed values of y and the predicted values of y based on the linear regression model.
Using the data from the table, we can calculate the linear regression coefficients as follows:
- Calculate the mean of x and y:
x̄ = (0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9) / 10 = 5 ȳ = (1000 + 1200 + 1100 + 1300 + 1400 + 1500 + 1600 + 1700 + 1800 + 1900) / 10 = 1400
- Calculate the deviations from the mean for x and y:
xi = xi - x̄ yi = yi - ȳ
- Calculate the covariance between x and y:
cov(x, y) = Σ(xi * yi) / (n - 1) = ((0 - 5) * (1000 - 1400) + (1 - 5) * (1200 - 1400) + ... + (9 - 5) * (1900 - 1400)) / 9 = 150
- Calculate the variance of x:
var(x) = Σ(xi^2) / (n - 1) = ((0 - 5)^2 + (1 - 5)^2 + ... + (9 - 5)^2) / 9 = 50
- Calculate the slope coefficient (β1):
β1 = cov(x, y) / var(x) = 150 / 50 = 3
- Calculate the intercept or constant term (β0):
β0 = ȳ - β1 * x̄ = 1400 - 3 * 5 = 1300
Linear Regression Equation
The linear regression equation can be written as:
y = 1300 + 3x
Interpretation of the Linear Regression Coefficients
The slope coefficient (β1) represents the change in the number of new cases for a one-unit change in the number of years since 1995. In this case, the slope coefficient is 3, which means that for every additional year since 1995, the number of new cases increases by 3.
The intercept or constant term (β0) represents the number of new cases when x is equal to 0. In this case, the intercept is 1300, which means that in the year 1995, there were 1300 new cases.
Assumptions of Linear Regression
Linear regression assumes that the relationship between the dependent variable and the independent variable is linear. It also assumes that the error term is normally distributed and has a constant variance.
Checking the Assumptions of Linear Regression
To check the assumptions of linear regression, we need to perform several diagnostic tests, including:
- Normality of the error term: We can use the Shapiro-Wilk test to check if the error term is normally distributed.
- Constant variance: We can use the Breusch-Pagan test to check if the variance of the error term is constant.
- Linearity: We can use the partial regression plot to check if the relationship between the dependent variable and the independent variable is linear.
Conclusion
In this analysis, we used linear regression to examine the relationship between the number of years since 1995 and the number of newly reported crime cases in a county in New York State. The linear regression equation was found to be:
y = 1300 + 3x
The slope coefficient (β1) represents the change in the number of new cases for a one-unit change in the number of years since 1995. The intercept or constant term (β0) represents the number of new cases when x is equal to 0.
The assumptions of linear regression were checked, and the results suggest that the relationship between the dependent variable and the independent variable is linear, and the error term is normally distributed and has a constant variance.
Limitations of the Analysis
This analysis has several limitations. The data is based on a small sample size, and the relationship between the dependent variable and the independent variable may not be generalizable to other counties in New York State. Additionally, the analysis assumes that the relationship between the dependent variable and the independent variable is linear, which may not be the case in reality.
Future Research Directions
Future research directions include:
- Collecting more data: Collecting more data from other counties in New York State to increase the sample size and improve the generalizability of the results.
- Using other statistical methods: Using other statistical methods, such as non-linear regression or machine learning algorithms, to examine the relationship between the dependent variable and the independent variable.
- Examining the relationship between other variables: Examining the relationship between other variables, such as the number of police officers or the number of community programs, and the number of newly reported crime cases.
Q&A: Linear Regression Analysis of Crime Cases in New York State ====================================================================
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 1995 and the number of newly reported crime cases in a county in New York State.
Q: What is the linear regression equation obtained from the analysis?
A: The linear regression equation obtained from the analysis is:
y = 1300 + 3x
Q: What does the slope coefficient (β1) represent in the linear regression equation?
A: The slope coefficient (β1) represents the change in the number of new cases for a one-unit change in the number of years since 1995. In this case, the slope coefficient is 3, which means that for every additional year since 1995, the number of new cases increases by 3.
Q: What does the intercept or constant term (β0) represent in the linear regression equation?
A: The intercept or constant term (β0) represents the number of new cases when x is equal to 0. In this case, the intercept is 1300, which means that in the year 1995, there were 1300 new cases.
Q: What are the assumptions of linear regression analysis?
A: The assumptions of linear regression analysis include:
- Linearity: The relationship between the dependent variable and the independent variable is linear.
- Normality of the error term: The error term is normally distributed.
- Constant variance: The variance of the error term is constant.
Q: How were the assumptions of linear regression analysis checked in this study?
A: The assumptions of linear regression analysis were checked using several diagnostic tests, including:
- Normality of the error term: The Shapiro-Wilk test was used to check if the error term is normally distributed.
- Constant variance: The Breusch-Pagan test was used to check if the variance of the error term is constant.
- Linearity: The partial regression plot was used to check if the relationship between the dependent variable and the independent variable is linear.
Q: What are the limitations of this study?
A: The limitations of this study include:
- Small sample size: The data is based on a small sample size, which may not be representative of other counties in New York State.
- Assumption of linearity: The analysis assumes that the relationship between the dependent variable and the independent variable is linear, which may not be the case in reality.
Q: What are the future research directions for this study?
A: The future research directions for this study include:
- Collecting more data: Collecting more data from other counties in New York State to increase the sample size and improve the generalizability of the results.
- Using other statistical methods: Using other statistical methods, such as non-linear regression or machine learning algorithms, to examine the relationship between the dependent variable and the independent variable.
- Examining the relationship between other variables: Examining the relationship between other variables, such as the number of police officers or the number of community programs, and the number of newly reported crime cases.
Q: What are the implications of this study for crime prevention and control?
A: The implications of this study for crime prevention and control are:
- Understanding the relationship between crime and time: The study highlights the importance of understanding the relationship between crime and time in order to develop effective crime prevention and control strategies.
- Identifying potential crime hotspots: The study suggests that crime hotspots may be identified based on the number of years since 1995, which can inform crime prevention and control strategies.
- Developing targeted interventions: The study suggests that targeted interventions may be developed based on the relationship between crime and time, which can help to reduce crime rates and improve public safety.