The Number Of Newly Reported Crime Cases In A County In New York State Is Shown In The Accompanying Table. Here, $x$ Represents The Number Of Years Since 2002, 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 model the relationship between the number of years since 2002 and the number of new crime cases in a county in New York State. The data is presented in the accompanying table.
Table: Number of New Crime Cases in a County in New York State
| Year (x) | Number of New Cases (y) |
|---|---|
| 0 | 150 |
| 1 | 180 |
| 2 | 220 |
| 3 | 250 |
| 4 | 280 |
| 5 | 310 |
| 6 | 340 |
| 7 | 370 |
| 8 | 400 |
| 9 | 430 |
Linear Regression Model
A linear regression model is a mathematical equation that describes the relationship between the dependent variable (y) and the independent variable (x). The general form of a linear regression model is:
y = β0 + β1x + ε
where:
- y is the dependent variable (number of new crime cases)
- x is the independent variable (number of years since 2002)
- β0 is the intercept or constant term
- β1 is the slope coefficient
- ε is the error term
Calculating the Linear Regression Coefficients
To calculate the linear regression coefficients, we need to use the following formulas:
β1 = Σ[(xi - x̄)(yi - ȳ)] / Σ(xi - x̄)²
β0 = ȳ - β1x̄
where:
- xi is the value of the independent variable (x) for each data point
- yi is the value of the dependent variable (y) for each data point
- x̄ is the mean of the independent variable (x)
- ȳ is the mean of the dependent variable (y)
Calculating the Mean of the Independent Variable (x)
To calculate the mean of the independent variable (x), we need to sum up all the values of x and divide by the number of data points.
x̄ = (0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9) / 10 x̄ = 5
Calculating the Mean of the Dependent Variable (y)
To calculate the mean of the dependent variable (y), we need to sum up all the values of y and divide by the number of data points.
ȳ = (150 + 180 + 220 + 250 + 280 + 310 + 340 + 370 + 400 + 430) / 10 ȳ = 275
Calculating the Slope Coefficient (β1)
To calculate the slope coefficient (β1), we need to use the formula:
β1 = Σ[(xi - x̄)(yi - ȳ)] / Σ(xi - x̄)²
First, we need to calculate the deviations from the mean for both x and y.
| Year (x) | Number of New Cases (y) | Deviation from Mean (x) | Deviation from Mean (y) |
|---|---|---|---|
| 0 | 150 | -5 | -125 |
| 1 | 180 | -4 | -95 |
| 2 | 220 | -3 | -55 |
| 3 | 250 | -2 | -25 |
| 4 | 280 | -1 | 5 |
| 5 | 310 | 0 | 35 |
| 6 | 340 | 1 | 65 |
| 7 | 370 | 2 | 95 |
| 8 | 400 | 3 | 125 |
| 9 | 430 | 4 | 155 |
Next, we need to calculate the products of the deviations.
| Year (x) | Number of New Cases (y) | Deviation from Mean (x) | Deviation from Mean (y) | Product of Deviations |
|---|---|---|---|---|
| 0 | 150 | -5 | -125 | 625 |
| 1 | 180 | -4 | -95 | 380 |
| 2 | 220 | -3 | -55 | 165 |
| 3 | 250 | -2 | -25 | 50 |
| 4 | 280 | -1 | 5 | -5 |
| 5 | 310 | 0 | 35 | 0 |
| 6 | 340 | 1 | 65 | 65 |
| 7 | 370 | 2 | 95 | 190 |
| 8 | 400 | 3 | 125 | 375 |
| 9 | 430 | 4 | 155 | 620 |
Finally, we can calculate the slope coefficient (β1) using the formula:
β1 = Σ[(xi - x̄)(yi - ȳ)] / Σ(xi - x̄)² β1 = (625 + 380 + 165 + 50 - 5 + 0 + 65 + 190 + 375 + 620) / (25 + 16 + 9 + 4 + 1 + 0 + 1 + 4 + 9 + 16) β1 = 2170 / 85 β1 = 25.53
Calculating the Intercept (β0)
To calculate the intercept (β0), we need to use the formula:
β0 = ȳ - β1x̄ β0 = 275 - 25.53(5) β0 = 275 - 127.65 β0 = 147.35
Linear Regression Equation
The linear regression equation is:
y = 147.35 + 25.53x
Interpretation of the Results
The linear regression equation shows a positive relationship between the number of years since 2002 and the number of new crime cases in a county in New York State. This means that as the number of years since 2002 increases, the number of new crime cases also increases. The slope coefficient (β1) of 25.53 indicates that for every additional year since 2002, the number of new crime cases increases by approximately 25.53.
Conclusion
Q: What is linear regression analysis?
A: Linear regression analysis is a statistical method used to model the relationship between a dependent variable and one or more independent variables. It is a widely used technique in data analysis and is commonly used to predict the value of a continuous outcome variable based on one or more predictor variables.
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(s) should be linear.
- Independence: Each observation should be independent of the others.
- Homoscedasticity: The variance of the residuals should be constant across all levels of the independent variable(s).
- Normality: The residuals should be normally distributed.
- No multicollinearity: The independent variables should not be highly correlated with each other.
Q: What is the difference between simple and multiple linear regression?
A: Simple linear regression involves modeling the relationship between a single dependent variable and a single independent variable. Multiple linear regression, on the other hand, involves modeling the relationship between a single dependent variable and multiple independent variables.
Q: How do I interpret the coefficients in a linear regression model?
A: The coefficients in a linear regression model represent the change in the dependent variable for a one-unit change in the independent variable, while holding all other independent variables constant. For example, if the coefficient for a particular independent variable is 2, it means that for every one-unit increase in that independent variable, the dependent variable is expected to increase by 2 units.
Q: What is the difference between a positive and negative coefficient in a linear regression model?
A: A positive coefficient indicates a positive relationship between the independent variable and the dependent variable, meaning that as the independent variable increases, the dependent variable also increases. A negative coefficient, on the other hand, indicates a negative relationship between the independent variable and the dependent variable, meaning that as the independent variable increases, the dependent variable decreases.
Q: How do I determine the significance of the coefficients in a linear regression model?
A: The significance of the coefficients in a linear regression model can be determined using a t-test or an F-test. The t-test is used to determine the significance of individual coefficients, while the F-test is used to determine the significance of the overall model.
Q: What is the difference between a p-value and a confidence interval?
A: A p-value represents the probability of observing a result at least as extreme as the one observed, assuming that the null hypothesis is true. A confidence interval, on the other hand, represents a range of values within which the true population parameter is likely to lie.
Q: How do I interpret the R-squared value in a linear regression model?
A: The R-squared value in a linear regression model represents the proportion of the variance in the dependent variable that is explained by the independent variable(s). For example, if the R-squared value is 0.7, it means that 70% of the variance in the dependent variable is explained by the independent variable(s).
Q: What is the difference between a linear regression model and a logistic regression model?
A: A linear regression model is used to model the relationship between a continuous dependent variable and one or more independent variables. A logistic regression model, on the other hand, is used to model the relationship between a binary dependent variable and one or more independent variables.
Q: How do I choose between a linear regression model and a logistic regression model?
A: The choice between a linear regression model and a logistic regression model depends on the nature of the dependent variable. If the dependent variable is continuous, a linear regression model is typically used. If the dependent variable is binary, a logistic regression model is typically used.