Question

You are part of a team investigating the identifying motor vehicle accidents. A multiple regression model is to be constructed to predict the number of motor vehicle accidents in a town per year based upon the population of the town, the number of recorded traffic offenses per year and the average annual temperature in the town.

Data has been collected on 30 randomly selected towns:

Number of motor vehicle accidents per year	Population (× 1000)	No. of recorded traffic offences (× 100)	Average temperature °F
355	181	29	78
490	257	56	82
597	441	34	81
475	50	95	81
922	495	102	82
736	38	165	81
305	167	25	84
1,128	378	191	78
745	369	86	76
476	237	63	84
143	100	4	84
203	118	21	79
909	489	106	78
410	210	39	77
642	138	131	81
847	308	138	82
604	418	40	77
719	194	132	78
350	319	8	84
327	70	61	76
1,038	259	192	78
756	299	115	81
635	440	40	79
796	283	131	85
301	64	56	81
135	26	26	79
639	31	150	81
325	210	13	77
441	43	98	79
522	370	26	82

a)Find the multiple regression equation using all three explanatory variables. Assume that X₁ is population, X₂ is number of recorded traffic offenses per year and X₃ is average annual temperature. Give your answers to 3 decimal places.

y^ =  + population + no. traffic offences + average temp

b)At a level of significance of 0.05, the result of the F test for this model is that the null hypothesis isis not rejected.

For parts c) and d), using the data, separately calculate the correlations between the response variable and each of the three explanatory variables.

c)The explanatory variable that is most correlated with number of motor vehicle accidents per year is:

population
number of traffic offenses
average annual temperature

d)The explanatory variable that is least correlated with number of motor vehicle accidents per year is:

population
number of traffic offenses
average annual temperature

e)The value of R² for this model, to 2 decimal places, is equal to

f)The value of s_e for this model, to 3 decimal places, is equal to

g)Construct a new multiple regression model by removing the variable average annual temperature. Give your answers to 3 decimal places.

The new regression model equation is:

y^ =  + population + no. traffic offences

h)In the new model compared to the previous one, the value of R² (to 2 decimal places) is:

increased
decreased
unchanged

i)In the new model compared to the previous one, the value of s_e (to 3 decimal places) is:

increased
decreased
unchanged

Answer 1

All analysis is carried out in excel software.

a) The multiple regression equation for three explanatory variables is describe as

where y is number of motor vehicle accidents, x₁ is population, x₂ is number of recorded traffic offenses per year and x₃ is average annual temperature.

The resulted outcome is using OLS method under excel command,

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.997319
R Square	0.994646
Adjusted R Square	0.994028
Standard Error	20.08388
Observations	30
ANOVA
	df	SS	MS	F	Significance F
Regression	3	1948343	649447.6	1610.085	1.24E-29
Residual	26	10487.42	403.3623
Total	29	1958830
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	236.3423	116.8627	2.022394	0.053535	-3.87235	476.5569
x1	0.989568	0.025325	39.07436	1.27E-24	0.937512	1.041625
x2	3.800536	0.066973	56.74689	8.68E-29	3.66287	3.938202
x3	-2.52262	1.44612	-1.74441	0.092901	-5.49516	0.449922

b) At a level of significance of 0.05, the result of the F test for this model is 1610.085 and p-value is less than 0.05. We concluded that the null hypothesis is rejected.

c)

Corr(y,x1) = 0.5697

Corr(y,x2) = 0.8234

Corr(y,x3) = -0.1057

The explanatory variable that is most correlated with number of motor vehicle accidents per year is number of traffic offenses .

d) The explanatory variable that is least correlated with number of motor vehicle accidents per year is average annual temperature

e)

e)The value of R² for this model is equal to 0.9946

f)The value of s_e for this model is equal to 20.083

g)Construct a new multiple regression model by removing the variable average annual temperature.

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.997005
R Square	0.994019
Adjusted R Square	0.993576
Standard Error	20.82985
Observations	30
ANOVA
	df	SS	MS	F	Significance F
Regression	2	1947115	973557.7	2243.827	9.68E-31
Residual	27	11714.83	433.8827
Total	29	1958830
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	33.052	9.02213	3.663436	0.00107	14.54011	51.56388
x1	0.991293	0.026246	37.76948	6.56E-25	0.937441	1.045145
x2	3.808852	0.069285	54.97381	2.96E-29	3.666691	3.951013

The new regression model equation is:

i) In the new model compared to the previous one, the value of s_e increased.