Question

The State of Florida is considering raising the fine for red light violations this session. The theory, of course, is that higher fines deter violations. You decide to test this by taking a sample of ten states and record their 2003 state fine for running a red light and the number of cited violations per 1,000 vehicle miles (standard measure in transportation to control for highway volume). Assume enforcement is relatively equal across the states:

State Fine                                                        Violations Measure

                                      50                                                                              10

                                    100                                                                                8

                                    120                                                                                9

                                      40                                                                                7

                                      80                                                                              10

                                      60                                                                              12

                                      90                                                                                7

                                    100                                                                                8

                                      50                                                                                9

                                    110                                                                                2                    

1.State your null formally and in lay terms for a simple regression

2. Calculate r and the regression line (y = a + bx) and reject/accept at a=.05.

3. Explain your findings in lay terms using b (if relevant), r, r-square, and p-value.

4.Calculate a 95% confidence interval for the slope and explain in layterms.

5. Calculate a 95% confidence interval for Y when X is 80, explain in layterms.

Answer 1

1) given regression line y = a + bx, slope = b

Hypothesis

H₀: b = 0, There is no linear relationship between State Fine and Violations Measure

H₁: b 0, There is a linear relationship between State Fine and Violations Measure.

2)

Sl.No.	State Fine (x)	Violations measure (y)	xi^2	yi^2	xi*yi
1	50	10	2500	100	500
2	100	8	10000	64	800
3	120	9	14400	81	1080
4	40	7	1600	49	280
5	80	10	6400	100	800
6	60	12	3600	144	720
7	90	7	8100	49	630
8	100	8	10000	64	800
9	50	9	2500	81	450
10	110	2	12100	4	220
Sum	800	82	71200	736	6280
Average	80	8.2

Sxx	7200
Syy	63.6
Sxy	-280

correlation coefficient r = = = -0.414

y = a + bx

b = Sxy/Sxx = -280/7200 = -0.039

a = = 8.2 - (-0.039*80) = 11.32

y = 11.32 - 0.039*x

ANOVA Table

Source	df	SS	MS =(SS/df)	F	critical F	p-value
Regression	1	=b1*Sxy =10.89	10.89	1.653	5.32	0.235
Error	= n-2 = 8	= SStotal - SSregression =52.71	6.59
Total	n -1 = 10 - 1 =9	Syy = 63.6

Since critical value is more than test statistic F we fail to reject null hypothesis and there is no significant evidence to conclude that There is a linear relationship between State Fine and Violations Measure.

3) Since there is significant linear relationship, the interpretation of b is not relevant.

r = -0.414, which says that there is moderate negative relationship between State Fine and Violations Measure. In lay terms we increase in state fine decreases the violations measure.

r-square = 0.1713, Which means model only explains 17.13% of the variability between Violations Measure with state fines as independent variable.

4) 95% CI for slope =

t_{0.025,8 =} 2.306

by substituting all values we get

95% CI for slope = (-0.109, 0.031)

we are 95% confident that the slope of the regression equation will lie in between (-0.109, 0.031)

p-value = 0.235,

5) 95% confidence interval for Y when X is 80

when x₀ = 80, y₀ = 11.32 - 0.039*80 = 8.2

we get 95% confidence interval = ( 6.33 , 10.07)

we are 95% confident that the predicted value of Y when X = 80 using the regression equation will lie in between (6.33 , 10.07).