Assignment 6, 100 pts: Due on 04/27/2021
1. (10 pts) Referring to risk adjusted control chart, in order to detect a change in the system,
which variables should one monitor, x, y, or z, and why?
2. (25 pts) Consider the example used in the class for linear regression (weight of the card-
board). Using only the first 5 data points, fit a linear regression between weight (y) and
humidity (x) and determine the values of Bo, B₁. Also, write down the matrices Y and X
(see slide #10). Also, determine the values of ŷ and comapre with the corresponding values
of y.
3. (25 pts) Two observable quality characteristic x₁ and ₂ of a production process is recorded
for m = 50 samples with a sample size of n = 25. Assume that mean values of the quality
characteristics and the covariance matrix were determined as:
x
55
30
1
2
3
4
5
6
7
8
and S
(a) Determine the UCL for a Hotelling T2 control chart. Consider a = 0.001.
(b) The first 15 samples of the data is shown in Table 1. The data are sample means of
each quality characteristic. Determine the T² for these 15 samples. Based on the UCL
from part (a), do you see any out of control samples?
Sample Number X1 X2
58
32
60 33
50 27
54
31
63
38
53 30
42
20
55 31
200
130
Table 1: Data for Exercise 4(b)
1
Sample Number
9
10
130
120
11
12
13
14
15
|
X1 X2
46 25
|5049575875 55
29
27
30
33
45
27
4. (15 pts) Consider a T² control chart for monitoring p = 6 quality characteristics. Suppose
that the sample size is n = 3 and there are m = 30 preliminary samples available to estimate
the sample covariance matrix. Find the control limits assuming that a = 0.005. 5. (25 pts) In a large study, 942 women were surveyed and their daily dietary intake of Calcium,
Iron, Protein, Vitamin A, and Vitamin C were recorded. Since we have 5 quality charac-
teristic, p = 5. The sample average and sample covariance matrix for the dietary intake
are:
X =
624.0 mg
11.1 mg
65.8 g
839.6 μg
78.9 mg
and S =
157829.4 940.1 6075.8
102411.1
2383.2
6701.6
137.7
940.
5.8 114.1
6075.8 114.1 934.9 7330.1 477.2
102411.1 2383.2 7330.1 2668452.4 22063.3
6701.6 137.7 477.2
22063.3 5416.3
The recommended intake of these nutrients are:
Ho
=
1000 mg
15 mg
60 g
800 με
75 mg
Conduct hypothesis test (with a = 0.01) to verify if the average dietary intake meets the
recommendations. Although it is not needed, you may check the data related to this problem
on canvas Assignment 6 folder (Women's survey data).
2
