ISE 362 HOMEWORK FIVE Due Date: Tuesday 11/14
1. Suppose that in a certain chemical process the reaction time y (hour) is related to the temperature (oF) in the chamber in which the reaction takes place according to the simple linear regression model with equation y = 5.00 – 0.01x and σ = 0.50. Suppose five observations are made independently on reaction time, each one for a temperature of 250OF. What is the probability that all five times are between 2.0 and 3.0 hours?
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2. Engineers frequently use models in problem formulation and solution. Sometimes these models are based on our physical, chemical, or engineering science knowledge of the phenomenon, and in such cases we call these models deterministic models. However, there are “many” situations in which two or more variables of interest are related, and the deterministic model relating these variables is unknown. In these cases it is necessary to build a non-deterministic model relating the variables based on observed data. Consider the situation where y is the salt concentration (mg/l) found in surface streams in an upstate watershed and x is the percentage of the watershed area consisting of paved roads. There is no obvious physical mechanism that relates the salt concentration to the roadway area. Suppose that the relationship between salt concentration and roadway area can be described by the simple linear regression model with the true regression line y = 3 + 15x and with σ2 = 4.0. What is the probability that salt concentration exceeds 30 when roadway area is 2.0?
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3. A scatter plot of x = rainfall volume (m3) y = runoff volume (m3) for a particular location are shown below. The accompanying values were read from the plot. Calculate point estimates of the slope and intercept of the true regression line. Calculate a point estimate of the true average runoff volume when rainfall volume is 50. Calculate a point estimate of the standard deviation σ. Also, find the proportion of the observed variation in runoff volume can be attributed to the simple linear regression relationship between runoff and rainfall.
x / 5 / 12 / 14 / 17 / 23 / 30 / 40 / 47y / 4 / 10 / 13 / 15 / 15 / 25 / 27 / 46
x / 55 / 67 / 72 / 81 / 96 / 112 / 127
y / 38 / 46 / 53 / 70 / 82 / 99 / 100
Point Estimates of β1 and β0:
Point Estimate of μy when x = 50:
Point Estimate of σ:
Proportion:
4. The accompanying data was read from the graph below that appeared in a recent study. The independent variable is SO2 deposition rate (mg/m2/day) and the dependent variable is steel weight loss (g/m2). Calculate the equation of the estimated regression line. Also; what percentage of observed variation in steel weight loss can be attributed to the model relationship in combination with variation in deposition rate?
x / 14 / 18 / 40 / 43 / 45 / 112y / 280 / 350 / 470 / 500 / 560 / 1200
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5. The following summary statistics were obtained from a study that used regression analysis to investigate the relationship between pavement deflection and surface temperature of the pavement at various locations on a state highway. Here x = temperature (oF) and y = deflection adjustment factor (y ≥ 0):
n = 15 Σ xi = 1,425 Σ yi = 10.68 Σ xi2 = 139,037.25 Σ xiyi = 987.645
Compute β1, β2 the equation of the estimated regression line. Also, suppose temperature were measured in oC rather than in oF. What would be the new estimated regression line?
(Hint: oF = (9/5) oC + 32; now substitute for the “old x” in terms of the new x.”)
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6. Infestation of crops by insects has long been of great concern to farmers and agricultural scientists. A study reports data on x = age of a cotton plant (days) and y = percentage of damaged squares. Consider the accompanying n = 12 observations:
x / 9 / 12 / 12 / 15 / 18 / 18y / 11 / 12 / 23 / 30 / 29 / 52
x / 21 / 21 / 27 / 30 / 30 / 33
y / 41 / 65 / 60 / 72 / 84 / 93
The summary statistics are: Σ xi = 246, Σ xi2 = 5,742, Σ yi = 572, Σ yi2 = 35,634, Σ xiyi =14,022.
Why is the relationship between x and y not deterministic? Also, determine the equation of the least squares line.
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7. The Turbine Oil Oxidation Test (TOST) and the Rotating Bomb Oxidation Test (RBOT) are two different procedures for evaluating the oxidation stability of steam turbine oils. The accompanying observations on x = TOST time (hr) y = RBOT time (min) for 12 oil specimens have been reported:
TOST / 4200 / 3600 / 3750 / 3675 / 4050 / 2770RBOT / 370 / 340 / 375 / 310 / 350 / 200
TOST / 4870 / 4500 / 3450 / 2700 / 3750 / 3300
RBOT / 400 / 375 / 285 / 225 / 345 / 285
Summary Values: n = 12, Σ xi =44,615, Σ xi2 = 170,355,425, Σ yi = 3,860, Σ yi2 = 1,284,450 Σ xiyi =14,755,500.
Calculate value of the sample correlation coefficient r. Would the value of r be affected if you let x = RBOT time and y = TOST time? (Why)?
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8. Regression analysis can be used to build a model to predict the purity of oxygen produced in a chemical distillation process for a given percentage of hydrocarbons that are present in the main condenser of the distillation unit. This regression model can be used for process optimization, such as finding the percentage of hydrocarbons that maximizes purity, or for process control purposes. The table below represents the data for 20 observations from this type of chemical distillation process. Fit a simple linear regression model to the oxygen purity data. Provide an estimate for the variance of the error term є.
Observation # Hydrocarbon Level(%) Purity(%)
1 0.99 90.01
2 1.02 89.05
3 1.15 91.43
4 1.29 93.74
5 1.46 96.73
6 1.36 94.45
7 0.87 87.59
8 1.23 91.77
9 1.55 99.42
10 1.40 93.65
11 1.19 93.54
12 1.15 92.52
13 0.98 90.56
14 1.01 89.54
15 1.11 89.85
16 1.20 90.39
17 1.26 93.25
18 1.32 93.41
19 1.43 94.98
20 0.95 87.33
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