Question

A highway department is studying the relationship between traffic flow and speed. The following model has been hypothesized:

y = β₀ + β₁x + ε where

• y = traffic flow in vehicles per hour
• x = vehicle speed in miles per hour.

The following data were collected during rush hour for six highways leading out of the city.

Traffic Flow (y)	Vehicle Speed (x)
1,256	35
1,329	40
1,226	30
1,333	45
1,347	50
1,122	25

In working further with this problem, statisticians suggested the use of the following curvilinear estimated regression equation.

ŷ = b₀ + b₁x + b₂x²

(a) Develop an estimated regression equation for the data of the form ŷ = b₀ + b₁x + b₂x² (Round b₀ to the nearest integer and b₁ to two decimal places and b₂ to three decimal places.)

ŷ =

Find the value of the test statistic. (Round your answer to two decimal places.)

Base on the model predict the traffic flow in vehicles per hour at a speed of 38 miles per hour. (Round your answer to two decimal places.)

vehicles per hour

A study of emergency service facilities investigated the relationship between the number of facilities and the average distance traveled to provide the emergency service. The following table gives the data collected.

Number of Facilities	Average Distance (miles)
9	1.67
11	1.12
16	0.82
21	0.63
27	0.50
30	0.47

Develop an estimated regression equation for the data corresponding to a second-order model with one predictor variable. (Round your numerical values to four decimal places.)

Answer 1

Problem 1:

SPSS 22.0V Program Code * Curve Estimation. TSET NEWVAR=NONE. CURVEFIT /VARIABLES=Y WITH X /CONSTANT /MODEL=QUADRATIC /PLOT FIT. 

  

Model Summary and Parameter Estimates
Dependent Variable:   Y
Equation	Model Summary	Parameter Estimates
R Square	F	df1	df2	Sig.	Constant	b1	b2
Quadratic	.979	69.456	2	3	.003	415.714	38.359	-.396
The independent variable is X.

(a) An estimated regression equation for the data of the form ŷ = 416 + 38.36x - 0.396x2

The value of the test statistic ie. F is 69.46

Base on the model, if vehicle speed in miles per hour. is 38 miles per hour, then

traffic flow in vehicles per hour is :1301.86.

Problem 2

A study of emergency service facilities investigated the relationship between the number of facilities and the average distance traveled to provide the emergency service. The following table gives the data collected.

* Curve Estimation. TSET NEWVAR=NONE. CURVEFIT /VARIABLES=Number_facilities WITH Average_distance /CONSTANT /MODEL=QUADRATIC /PLOT FIT. 

Model Summary and Parameter Estimates
Dependent Variable:   Number_facilities
Equation	Model Summary	Parameter Estimates
R Square	F	df1	df2	Sig.	Constant	b1	b2
Quadratic	.987	114.846	2	3	.001	53.941	-64.345	22.494
The independent variable is Average_distance.

An estimated regression equation for the data corresponding to a second-order model with one predictor variable.

lets, assume, ŷ = estimated number of facilitites , and X -predictor i.e. average distance

ŷ = 53.9410 - 64.3450x +22.4940x2