A statistical program is recommended. A highway department is studying the relationship between traffic flow and speed. The following model has been hypothesized: y =B₁ + B₂x + 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 Vehicle Speed (y) (x) 35 1,258 1,330 1,226 1,336 1,347 1,122 40 30 45 50 25 In working further with this problem, statisticians suggested the use of the following curvilinear estimated regression equation. ý = Bo + Box + By xả (a) Develop an estimated regression equation for the data of the form ý = B₁ + B₁x + ₂x². (Round Bo to the nearest integer and ₁ to two decimal places and 2 to three decimal places.) ŷ =
A statistical program is recommended. A highway department is studying the relationship between traffic flow and speed. The following model has been hypothesized: y =B₁ + B₂x + 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 Vehicle Speed (y) (x) 35 1,258 1,330 1,226 1,336 1,347 1,122 40 30 45 50 25 In working further with this problem, statisticians suggested the use of the following curvilinear estimated regression equation. ý = Bo + Box + By xả (a) Develop an estimated regression equation for the data of the form ý = B₁ + B₁x + ₂x². (Round Bo to the nearest integer and ₁ to two decimal places and 2 to three decimal places.) ŷ =
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Question
![**Analyzing Traffic Flow and Vehicle Speed: A Statistical Approach**
A highway department is exploring the relationship between traffic flow and speed. The proposed linear model is:
\[ y = \beta_0 + \beta_1 x + \varepsilon \]
Where:
- \( y \) = traffic flow in vehicles per hour
- \( x \) = vehicle speed in miles per hour
**Data Collected:**
The data below was gathered during rush hour for six highways exiting the city:
| Traffic Flow (\(y\)) | Vehicle Speed (\(x\)) |
|----------------------|-----------------------|
| 1,258 | 35 |
| 1,330 | 40 |
| 1,226 | 30 |
| 1,336 | 45 |
| 1,347 | 50 |
| 1,122 | 25 |
In further analysis, a curvilinear regression model was suggested:
\[ \hat{y} = \beta_0 + \beta_1 x + \beta_2 x^2 \]
**Task:**
(a) Develop an estimated regression equation for the form \(\hat{y} = \beta_0 + \beta_1 x + \beta_2 x^2\). Round \(\beta_0\) to the nearest integer, \(\beta_1\) to two decimal places, and \(\beta_2\) to three decimal places.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe2fdea76-24b5-4e92-ab93-0d275ab88ec3%2F73feab4d-25f6-4675-882e-2e7463ceec1f%2Fof3woxm_processed.png&w=3840&q=75)
Transcribed Image Text:**Analyzing Traffic Flow and Vehicle Speed: A Statistical Approach**
A highway department is exploring the relationship between traffic flow and speed. The proposed linear model is:
\[ y = \beta_0 + \beta_1 x + \varepsilon \]
Where:
- \( y \) = traffic flow in vehicles per hour
- \( x \) = vehicle speed in miles per hour
**Data Collected:**
The data below was gathered during rush hour for six highways exiting the city:
| Traffic Flow (\(y\)) | Vehicle Speed (\(x\)) |
|----------------------|-----------------------|
| 1,258 | 35 |
| 1,330 | 40 |
| 1,226 | 30 |
| 1,336 | 45 |
| 1,347 | 50 |
| 1,122 | 25 |
In further analysis, a curvilinear regression model was suggested:
\[ \hat{y} = \beta_0 + \beta_1 x + \beta_2 x^2 \]
**Task:**
(a) Develop an estimated regression equation for the form \(\hat{y} = \beta_0 + \beta_1 x + \beta_2 x^2\). Round \(\beta_0\) to the nearest integer, \(\beta_1\) to two decimal places, and \(\beta_2\) to three decimal places.
![### Hypothesis Testing for a Significant Relationship
**(b) Use \( \alpha = 0.01 \) to test for a significant relationship.**
**State the null and alternative hypotheses:**
- \( H_0: \beta_1 = \beta_2 = 0 \)
- \( H_a: \) One or more of the parameters is not equal to zero.
- \( \circle \ H_0: \beta_0 = \beta_1 = \beta_2 = 0 \)
- \( H_a: \) One or more of the parameters is not equal to zero.
- \( \circle \ H_0: \) One or more of the parameters is not equal to zero.
- \( H_a: \beta_0 = \beta_1 = \beta_2 = 0 \)
- \( \checkmark \ H_0: \) One or more of the parameters is equal to zero.
- \( H_a: \beta_1 \neq \beta_2 = 0 \)
**Find the value of the test statistic.** (Round your answer to two decimal places.)
\[ \text{Test Statistic} = \_\_\_ \]
**Find the p-value.** (Round your answer to three decimal places.)
\[ p \text{-value} = \_\_\_ \]
**What is your conclusion?**
- \( \circle \ \) Do not reject \( H_0 \). We conclude that the relationship is significant.
- \( \circle \ \) Reject \( H_0 \). We conclude that the relationship is significant.
- \( \circle \ \) Reject \( H_0 \). We cannot conclude that the relationship is significant.
- \( \checkmark \ \) Do not reject \( H_0 \). We cannot conclude that the relationship is significant.
**(c) Based 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.)
\[ \text{Vehicle Flow} = \_\_\_ \text{ vehicles per hour} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe2fdea76-24b5-4e92-ab93-0d275ab88ec3%2F73feab4d-25f6-4675-882e-2e7463ceec1f%2F209gqo_processed.png&w=3840&q=75)
Transcribed Image Text:### Hypothesis Testing for a Significant Relationship
**(b) Use \( \alpha = 0.01 \) to test for a significant relationship.**
**State the null and alternative hypotheses:**
- \( H_0: \beta_1 = \beta_2 = 0 \)
- \( H_a: \) One or more of the parameters is not equal to zero.
- \( \circle \ H_0: \beta_0 = \beta_1 = \beta_2 = 0 \)
- \( H_a: \) One or more of the parameters is not equal to zero.
- \( \circle \ H_0: \) One or more of the parameters is not equal to zero.
- \( H_a: \beta_0 = \beta_1 = \beta_2 = 0 \)
- \( \checkmark \ H_0: \) One or more of the parameters is equal to zero.
- \( H_a: \beta_1 \neq \beta_2 = 0 \)
**Find the value of the test statistic.** (Round your answer to two decimal places.)
\[ \text{Test Statistic} = \_\_\_ \]
**Find the p-value.** (Round your answer to three decimal places.)
\[ p \text{-value} = \_\_\_ \]
**What is your conclusion?**
- \( \circle \ \) Do not reject \( H_0 \). We conclude that the relationship is significant.
- \( \circle \ \) Reject \( H_0 \). We conclude that the relationship is significant.
- \( \circle \ \) Reject \( H_0 \). We cannot conclude that the relationship is significant.
- \( \checkmark \ \) Do not reject \( H_0 \). We cannot conclude that the relationship is significant.
**(c) Based 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.)
\[ \text{Vehicle Flow} = \_\_\_ \text{ vehicles per hour} \]
Expert Solution
Step 1
It is given that random variable x be the vehicle speed in miles per hour, and variable y be the traffic flow in vehicles per hour.
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