a. The regression equation is: Y' = -0.2419X + 17.4963
b. The predicted number of crimes for a city with 25 police officers is approximately 13.02 crimes.
c. The regression equation tells us that for each additional police officer added, the number of crimes goes down by 0.2419.
Here's the solution to the given problem:
Part (a): Regression equation
To find the regression equation, we first need to calculate the mean (average) of the number of police officers (X) and the mean of the number of crimes (Y). Then, we can calculate the slope (b) and y-intercept (a) of the regression line using the following formulas:
b = Σ[(xi - X)(yi - Y)] / Σ(xi - X)²
a = Y - bX
where:
xi and yi are the values of the independent (number of police officers) and dependent (number of crimes) variables for each city, respectively
X and Y are the mean values of the independent and dependent variables, respectively
Plugging in the given data, we get:
X = 19.375
Y = 10.375
b = -0.2419
a = 17.4963
Therefore, the regression equation is:
Y' = -0.2419X + 17.4963
Part (b): Predicted number of crimes for a city with 25 police officers
To predict the number of crimes for a city with 25 police officers, we can simply plug X = 25 into the regression equation:
Y' = -0.2419(25) + 17.4963 ≈ 13.0183
Therefore, the predicted number of crimes for a city with 25 police officers is approximately 13.02 crimes.
Part (c): Interpretation of the regression equation
The regression equation tells us that for each additional police officer added, the number of crimes goes down by 0.2419. This means that there is a negative relationship between the number of police officers and the number of crimes. However, it's important to note that correlation does not imply causation. There may be other factors that affect the number of crimes, and we cannot conclude that adding more police officers will necessarily cause a decrease in crime.