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)
1,257	35
1,330	40
1,224	30
1,333	45
1,351	50
1,123	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 =

(b) Use α = 0.01 to test for a significant relationship. State the null and alternative hypotheses.

H₀: b₁ = b₂ = 0
H_a: One or more of the parameters is not equal to zero.

H₀: One or more of the parameters is not equal to zero.
H_a: b₀ = b₁ = b₂ = 0   

  H₀: One or more of the parameters is not equal to zero.
H_a: b₁ = b₂ = 0

H₀: b₀ = b₁ = b₂ = 0
H_a: One or more of the parameters is not equal to zero.

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

Find the p-value. (Round your answer to three decimal places.)

p-value =

What is your conclusion?

Reject H₀. We conclude that the relationship is significant.

Do not reject H₀. We conclude that the relationship is significant.   

Do not reject H₀. We cannot conclude that the relationship is significant.

Reject H₀. We cannot conclude that the relationship is significant.

(c) 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

Answer 1

(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.)

Ans: The regression equation is

ŷ =430.1 + 37.56 x - 0.382 x^2

(b) Use α = 0.01 to test for a significant relationship. State the null and alternative hypotheses.

Ans:

H0: b1 = b2 = 0
Ha: One or more of the parameters is not equal to zero.

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

Ans: F=76.00

Find the p-value. (Round your answer to three decimal places.)

p-value =0.003

What is your conclusion?

Ans:

Reject H0. We conclude that the relationship is significant.

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

Ans: The model predicts the traffic flow in vehicles per hour at a speed of 38 miles per hour

ŷ =430.1 + 37.56*38 - 0.382 *38^2= 1305.772 vehicles per hour