Satellite applications motivated the development of a silver-zinc battery. Table (1) contains failure data collected to characterize the performance of the battery during its life cycle. Use these data.

Table (1) (table.csv)

Special note: Please solve it using R. Also try to elaborate without skipping steps.

Link to download the full questions: https://drive.google.com/file/d/1Qvnp9Q0axA8jKW9Yk4MnK5xEU11q2CRM/view?usp=sharing

X1	X 2	X 3	X 4	X 5	y
0.375	3.13	60.0	40	2.00	101
1.000	3.13	76.8	30	1.99	141
1.000	3.13	60.0	20	2.00	96
1.000	3.13	60.0	20	1.98	125
1.625	3.13	43.2	10	2.01	43
1.625	3.13	60.0	20	2.00	16
1.625	3.13	60.0	20	2.02	188
0.375	5.00	76.8	10	2.01	10
1.000	5.00	43.2	10	1.99	3
1.000	5.00	43.2	30	2.01	386
1.000	5.00	100.0	2	2.00	45
1.625	5.00	76.8	10	1.99	2
0.375	1.25	76.8	10	2.01	76
1.000	1.25	43.2	10	1.99	78
1.000	1.25	76.8	30	2.00	160
1.000	1.25	60.0	0	2.00	3
1.625	1.25	43.2	30	1.99	216
1.625	1.25	60.0	20	2.00	73
0.375	3.13	76.8	30	1.99	314
0.375	3.13	60.0	20	2.00	170

X1 - Charge rate (amps) X2 - Discharge rate (amps) X3 - Depth of discharge (% of rated ampere-hours) X4 - Temparature (0C) X5 - End of Charge Voltage (Volts) Y - Cycles to failure

Q1. Obtain a scatterplot matrix of the 6 variables using the plot() function. Q2. Obtain the mean vector, the variance-covariance matrix (Sm), and the correlation matrix of the 4 variables, using the mean(), cov(), and cor() functions. What is the angle 0 between the Height and Weight vectors? Q3. Obtain the eigenvalues and eigenvectors of the variance-covariance matrix, demonstrate the spectral decomposition of the variance-covariance matrix, and obtain its square-root matrix. Q4. Obtain the following matrices and vectors directly and use them to reconstruct the covariance and correlation matrices. *12 MP X21 X2p Let X = [y1 32 . . ye] where y = . . . X. X1 X2 Obtain: X = d =y- xil, i=1,..., P (1, - -J, x =[d, d . d, ] X3 Find S= 1 X X Find the generalized Sample Variance: S n-1 n Show (X1,, )'d, =0 (to "machine rounding")Q7. Determine the first three linear combinations of X with the first two eigenvector of S and hence determine their mean, variance and all pairs of correlation coefficient. Q9. Test the normality of each variable separately. Also, Test the bivariate normality of the first two variables. Q10. Make Box-Cox transformation of non-normal variables. (If possible transform the data by multivariate Box-Cox transformation). Q11. Test the following hypothesis of mean vector by Hoteling 7 test at 5% level of significance.1.5 4 60 25 2 Hy 110