Question

In: Statistics and Probability

Three variables are used to monitor a chemical process. They are X1 percentage impurities, X2 temperature...

Three variables are used to monitor a chemical process. They are X1 percentage impurities, X2 temperature F, and X3 concentration. The initial sample of 14 observations is listed in the following table. (a) Plot the data on a matrix scatter plot and comment on any observed collinearity. (b) Perform principal components analysis in Minitab using the correlation matrix option. Identify the principal components that explain the majority of the variation. Use the first two principal components. Compute the principal components scores of the first two principal components and construct a matrix scatter plot of the z-

Sample X1 X2 X3
1 16.92 84.50 43.26
2 16.80 58.21 43.44
3 17.38 84.23 42.74
4 16.80 85.74 43.60
5 16.82 85.53 43.18
6 16.71 85.49 43.72
7 17.07 84.29 42.83
8 16.83 85.32 43.41
9 16.71 85.78 44.28
10 16.88 84.87 43.09
11 16.73 85.68 44.00
12 16.87 85.03 43.78
13 17.60 84.10 42.11
14 16.80 85.38 43.48

scores to determine if the selected principal components appear to be randomly distributed? (c) Create standardized z-scores principal components individuals control chart.

Solutions

Expert Solution

Three variables are used to monitor a chemical process. The Variables are X1 percentage impurities, X2 temperature F, and X3 concentration.

(a) Plot the data on a matrix scatter plot:

## The variables X1 and X3 are negatively correlated.

(b) Principal components analysis in Minitab using the correlation matrix option:

Principal Component Analysis: X1, X2, X3

Eigenanalysis of the Correlation Matrix

Eigenvalue 1.8814 1.0034 0.1152
Proportion 0.627 0.334 0.038
Cumulative 0.627 0.962 1.000


Variable PC1 PC2 PC3
X1 0.708 -0.026 0.706
X2 0.034 -0.997 -0.071
X3 -0.706 -0.074 0.704

## First two principal components explain the majority of the variation.


Scree Plot of X1, ..., X3:


Score Plot of X1, ..., X3

Matrix scatter plot of the z-scores:

  ## The selected principal components appear to be randomly distributed.

(c) Standardized z-scores principal components individuals control chart:


Related Solutions

Let X1 and X2 be independent standard normal variables X1 ∼ N(0, 1) and X2 ∼...
Let X1 and X2 be independent standard normal variables X1 ∼ N(0, 1) and X2 ∼ N(0, 1). 1) Let Y1 = X12 + X12 and Y2 = X12− X22 . Find the joint p.d.f. of Y1 and Y2, and the marginal p.d.f. of Y1. Are Y1 and Y2 independent? 2) Let W = √X1X2/(X12 +X22) . Find the p.d.f. of W.
Consider a capital budgeting formulation where the binary variables x1, x2, and x3 are used to...
Consider a capital budgeting formulation where the binary variables x1, x2, and x3 are used to represent the acceptance (xi = 1) or rejection (xi = 0) of each alternative. The requirements that at least one, but not more than two, out of the three alternatives can be accepted can be represented by which of the following constraints? a. x1 + x2 ≤ 1 and x1 + x3 ≤ 1 b. x1 + x2 + x3 ≥ 2 and x1...
Consider a capital budgeting formulation where the binary variables x1, x2, and x3 are used to...
Consider a capital budgeting formulation where the binary variables x1, x2, and x3 are used to represent the acceptance (xi = 1) or rejection (xi = 0) of each alternative. The requirements that at least one, but not more than two, out of the three alternatives can be accepted can be represented by which of the following constraints? a. x1 + x2 ≤ 1 and x1 + x3 ≤ 1 b. x1 + x2 + x3 ≥ 2 and x1...
Consider a capital budgeting formulation where the binary variables x1 and x2 are used to represent...
Consider a capital budgeting formulation where the binary variables x1 and x2 are used to represent the acceptance (xi = 1) or rejection (xi = 0) of each alternative. Which of the following constraints shows that investing in alternative 1 is contingent upon investing in alternative 2? a. x1 + x2 ≤ 1 b. x1 + x2 ≥ 1 c. x1 + x2 ≥ 1 d. x1 ≥ x2
Consider the following three consumption bundles (X1,X2)=(10,10) ; (X1,X2)=(15,10) ; (X1,X2)=(3000,8).
Answer each of the following statements True/False/Uncertain. Give a full explanation of your answer including graphs where appropriate. (When in doubt, always include a fully labeled graph.)A) Consider the following three consumption bundles (X1,X2)=(10,10) ; (X1,X2)=(15,10) ; (X1,X2)=(3000,8). Non-satiation implies that (15,10) is preferred to (10,10) but does not imply that (3000,8) is preferred to (10,10).B) It is not theoretically possible for two indifference curves to cross if the preference relations they are based on satisfy the assumptions of completeness,...
let X1, X2, X3 be random variables that are defined as X1 = θ + ε1...
let X1, X2, X3 be random variables that are defined as X1 = θ + ε1 X2 = 2θ + ε2 X3 = 3θ + ε3 ε1, ε2, ε3 are independent and the mean and variance are the following random variable E(ε1) = E(ε2) = E(ε3) = 0 Var(ε1) = 4 Var(ε2) = 6 Var(ε3) = 8 What is the Best Linear Unbiased Estimator(BLUE) when estimating parameter θ from the three samples X1, X2, X3
Let X1, X2, X3 be continuous random variables with joint pdf f(X1, X2, X3)= 2 if...
Let X1, X2, X3 be continuous random variables with joint pdf f(X1, X2, X3)= 2 if 1<X1<2 -1<X2<0 -X2-1<X3<0                         0 otherwise Find Cov(X2, X3)
. Let X1,X2,and X3 be three random variables with means, variances, and correlation coefficients, denoted by...
. Let X1,X2,and X3 be three random variables with means, variances, and correlation coefficients, denoted by μ1, μ2, μ3; σ² 1,σ² 2,σ² 3; and ρ12, ρ13, ρ23, respectively. For constants b2 and b3, suppose E(X1−μ1|x2, x3) = b2(x2−μ2)+b3 Determine b2 and b3 in terms of the variances and the correlation coefficients
Let X1,X2,X3 be i.i.d. N(0,1) random variables. Suppose Y1 = X1 + X2 + X3, Y2...
Let X1,X2,X3 be i.i.d. N(0,1) random variables. Suppose Y1 = X1 + X2 + X3, Y2 = X1 −X2, Y3 =X1 −X3. Find the joint pdf of Y = (Y1,Y2,Y3)′ using : Multivariate normal distribution properties.
Consider independent random variables X1, X2, and X3 such that X1 is a random variable having...
Consider independent random variables X1, X2, and X3 such that X1 is a random variable having mean 1 and variance 1, X2 is a random variable having mean 2 and variance 4, and X3 is a random variable having mean 3 and variance 9. (a) Give the value of the variance of X1 + (1/2)X2 + (1/3)X3 (b) Give the value of the correlation of Y = X1- X2 and Z = X2 + X3.
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT