Question

You are moving up in the Judicial system for Orange County, California.  You are seeking a reliable and rational way to predict workload so you can allocate resources appropriately.  You think you can use the number of arrests over the past month to predict the total number of trials that will occur three months later. In order to investigate this, you gather data for the last 48 months and develop a linear regression model where X is the total number of felony arrests for a month and Y is the total number of trials three months later.  You get the following results:

Y = 4.3 + 0.47X	Sy|x = 35.6	sb = 0 .07	r2 = 0.52
X-bar = 500	sx = 100	n = 48

State your null and alternative hypotheses.  Explain the meaning of the slope, intercept and the coefficient of determination, specific to your equation.  Next, test the slope to see if it is significant (significantly different from 0 that is).  Based on the regression equation, how many trials can the judicial system expect if there are 623 felony arrests?   Put a 95% confidence around that estimate. Based on your evaluation, what can you say about your null and alternative hypotheses?

Answer 1

You are moving up in the Judicial system for Orange County, California.  You are seeking a reliable and rational way to predict workload so you can allocate resources appropriately.  You think you can use the number of arrests over the past month to predict the total number of trials that will occur three months later. In order to investigate this, you gather data for the last 48 months and develop a linear regression model where X is the total number of felony arrests for a month and Y is the total number of trials three months later.  You get the following results:

Y = 4.3 + 0.47X	Sy|x = 35.6	sb = 0 .07	r2 = 0.52
X-bar = 500	sx = 100	n = 48

State your null and alternative hypotheses.

Ho: There is no significant linear relations between the total number of felony arrests for a month and total number of trials three months later.

H1: There is a significant linear relations between the total number of felony arrests for a month and total number of trials three months later.

Explain the meaning of the slope, intercept and the coefficient of determination, specific to your equation.  

The regression line is Y = 4.3 + 0.47X.

When the total number of felony arrests for a month increases by 1, total number of trials three months later increases by 0.47.

When the total number of felony arrests for a month is 0, total number of trials three months later is 4.3.

R square = 0.52.   52% of variation in total number of trials three months later is explained by the total number of felony arrests for a month.

Next, test the slope to see if it is significant (significantly different from 0 that is).  

t = 0.47/0.07 =6.714

DF = n-2 =46

Table value of t with 46 DF at 0.05 level =2.013

Calculated t =6.714 > the table value

The null hypothesis is rejected.

The slope is significantly different from 0.

Based on the regression equation, how many trials can the judicial system expect if there are 623 felony arrests?  

When x=623, predicted Y = 4.3 + 0.47*623

=297.11

Put a 95% confidence around that estimate. Based on your evaluation, what can you say about your null and alternative hypotheses?

95% confidence around the estimate = (-584.403, 1178.623)

There is a significant linear relations between the total number of felony arrests for a month and total number of trials three months later.