Question

The data are the ages of criminals and their victims.

Criminal Age	Victim Age
32	24
24	20
28	25
17	22
12	16
34	25
25	21
29	26
18	23
13	17
37	28
28	23
32	29
20	25
14	18
39	29
29	24
34	30
21	27
14	19

1. Is there a significant correlation between the ages of the victims and the criminals? Provide support for your answer, and indicate the strength and direction of the correlation.
2. Conducting a simple linear regression, do they data support that a prediction of victim age can be obtained given the data provided in the file? Cite the elements of the output you used to draw your conclusion.
3. What is the equation of the regression, and what does that tell us?
4. How do correlation and regression differ?

Answer 1

Here in this scenario they provide the data of criminal age and victim age . We have to investigate the relationship between these two variable.

Here first we have to calculate the correlation between them. Correlation tells us the strength between the two variables and there relationship. After that if there is a correlation between them then we go fir Regression.

The correlation coefficient is as below,

the scatter plot is given below,

The Calculation of r :

Based on the given data,

X Values
∑ = 500
Mean = 25
∑(X - Mx)2 = SSx = 1360

Y Values
∑ = 471
Mean = 23.55
∑(Y - My)2 = SSy = 318.95

X and Y Combined
N = 20
∑(X - Mx)(Y - My) = 543

R Calculation
r = ∑((X - My)(Y - Mx)) / √((SSx)(SSy))
r = 543 / √((1360)(318.95)) = 0.8245

r = 0.8245

The value of R is 0.8245.

Yes, There is significant correlation between victim age and criminal age. We can also test this using hypothesis.

This is a strong positive correlation, which means that high X variable (Criminal age) scores go with high Y variable(victim age) scores.

The value of R2, the coefficient of determination, is 0.6798.

Here there is a correlation between them so here we can find the regression equation to predict the age of victims at given age of tge criminals.

b) Based on givenndata the regression equation is as below, here note that the victim age is considered as dependent variable y and criminal age is independent variable x.

Here the data support to do regression analysis, also here correlation between them is indicates that there is relationship between variables.

So the regression equation for victims age is given below,

ŷ = 13.568 + 0.399X

This is the prediction equation.

Calculation summary :

Sum of X = 500
Sum of Y = 471
Mean X = 25
Mean Y = 23.55
Sum of squares (SSX) = 1360
Sum of products (SP) = 543

Regression Equation = ŷ = bX + a

b = SP/SSX = 543/1360 = 0.39926

a = MY - bMX = 23.55 - (0.4*25) = 13.56838

ŷ = 13.568 + 0.3992X here the data support to do regression analysis.

c) The regression equation is ŷ = 13.568 + 0.399X.

Here the regression model is used often to predict the value based on the past and present data. Here based on regression model we can predict the victim age based on the criminal age.

d) The main difference between Correlation and regression is that,

Correlation coefficient is measures the strength between two variables and it is used to find only association between two variables. While regression analysis is used to predict the value of dependent variable based upon the past and present data.

Remember that, The correlation coefficient is relative measure on other hand regression is absolute measure.

This is the basic difference between correlation coefficient and regression coefficients.

Hope you understood. Thank you ?.