Regression Task

1) A culvert in a storm drain is the contact surface between the surface runoff and the drain conductor. The sewer device improves the pollutant suppressing properties. Evidence of various devices under controlled conditions is reported in the article "An evaluation of the urban stormwater pollutant demoral efficiency of catch basin inserts" (Water Envir. Res., 2005; 500-510). The following variables are defined: x = filtered quantity (thousands of liters) and y = total percentage of suspended solids removed.

x 23 45 68 91 114 136 159 182 205 228

y 53.3 26.9 54.8 33.8 29.9 8.2 17.2 12.2 3.2 11.1

a) Draw a scatter diagram of the data.

b) Obtain the equation of the least squares line.

c) What proportion of the observed variation in the elimination percentage can beattributed to the model relationship

2. Cardio-respiratory health is widely recognized as an important component of general physical well-being. The direct measurement of the maximum oxygen inhalation (VO2 max) is the best individual measurement, although it is slow and expensive, so it is desired to have a prediction equation from other measurements that are easy to determine. Consider the variables:

y = VO2 max (L / min)

x1 = weight (kg)

x2 = age (years)

x3 = time required to walk one mile (min)

x4 = heart rate (beats per min)

A possible model for male students is given by:

y = 5 + 0.01x1-0-05x2-0.13x3-0.01x4 with phi = 0.4

a) interpret beta1 and beta3

b) What is the expected value of the maximum oxygen inhalation with a weight of 76 kg, 20 years of age, 1 minute walk and a heart rate of 140 beats / min?

3. The electrical energy consumed each month by a chemical plant is thought to be related to the average ambient temperature x1, the number of days in month x2, the average product purity x3, and the tons of product produced x4. Historical data from the previous year is available and is shown in the following table.

Y 240 236 290 274 301 316 300 296 267 276 288 261

X1 25 31 45 60 65 72 80 84 75 60 50 38

X2 24 21 24 25 25 26 25 25 24 25 25 23

X3 91 90 88 87 91 94 87 86 88 91 90 89

X4 100 95 110 88 94 99 97 96 110 105 100 98

a) Fit a multiple linear regression model for the given data.

b) Forecast the value of energy consumption for a month in which X1 = 75 F, X2 = 24 days, X3 = 90%, and X4 = 98 tons.

4. The attached data for y = energy production (w) and x = temperature difference (K) were provided in the article "Comparison of Energy and Exergy Efficiency for Solar Box and Parabolic Cookers" (J. of Energy Engr., 2007 : 53-62).

x 23.2 23.5 23.52 24.3 25.1 26.2 27.4 28.1 29.3 30.6 31.5 32.01 32.63 33.23 33.62 34.18 35.43 35.62 36.16 36.23 36.89 37.9 39.1 41.66

y 3.78 4.12 4.24 5.35 5.87 6.02 6.12 6.41 6.62 6.43 6.13 5.92 5.64 5.45 5.21 4.98 4.65 4.5 4.34 4.03 3.92 3.65 3.02 2.89

a) Fit the cubic regression model:

y = beta0 + beta1 * x + beta2 * x ^ 2 + beta3 * x ^ 3

b) What proportion of the observed variation in energy production can be attributed to the model ratio?

c) Fit a quadratic model and compare the proportion explained by the model with respect to that obtained in b)