Question

A survey of 500 major U.S. manufacturing plants was completed in order to gain information about water pollution near each plant facility. Data have been collected on the amount of WATER POLLUTANTS (PL) found within ½ mile of each plant. (No plants were within 50 miles of each other). The pollutants were measured as the average parts per million based on each gallon of water sampled for each plant. For each plant, the AMOUNT OF WATER (W) used and the AVERAGE RAINFALL (R) was recorded. Plants were classified to be within one of four TYPES (chemical, paper, consumer durable goods, others) and the plant AGE (under 15 years; 15 years or older) was noted. A scale for LOCAL ENVIRONMENTAL REGULATIONS ENFORCEMENT was made (Strong enforcement; Moderate enforcement; Minimal to No Enforcement). Using these data, (20 points) a. Specify a linear model that would allow you test the effects of the impact of each variable above. b. Interpret each parameter in your model. c. Show how you would test the hypothesis that chemical and paper plant pollution is equal. d. How would you test the impact of local regulation enforcement on plant’s pollution

Answer 1

Solution

Part (a)

Let y = Pollutants (in average parts per million) based on each gallon of water sampled for each plant.

x₁ =The AMOUNT OF WATER (W) used

x₂ =The AVERAGE RAINFALL (R)

x₃ =The classification of plant – 1 for chemical, 2 for paper, 3 for consumer durable goods, and 4 for others

x₄ =The age classification of the plant – 1 for under 15 years and 2 for 15 years or older

x₅ =The scale for LOCAL ENVIRONMENTAL REGULATIONS ENFORCEMENT – 1 for Strong enforcement; 2 for Moderate enforcement; and 3 for Minimal to No Enforcement.

Then, the linear regression model is: y = β₀ + β₁x₁ + β₂x₂ + β₃x₃ + β₄x₄ + β₅x₅ + ε, where ε is the error term, which is assumed to be Normally distributed with mean 0 and variance σ². Answer 1

Part (b)

By Principles of Least Squares, the least square estimates, β_icap ‘s for β₀ to β₅ can be obtained.

Interpretation

β_icap for i = 1 to 5, represents the change in the response variable y per unit change in the independent variable x_i, keeping other things constant. In the given scenario, this would show how pollutants are impacted by each of the five causative factors included in the regression. Answer

Part (c)

From the original data, find the mean of y_i’s corresponding to chemical and paper plants and using the significance test, t, test if ybar (chemical plant) = ybar (paper plant). Answer

Part (d)

Using the significance test, t, test if β₅ = 0. Answer

DONE