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