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Context: FACTOR (Questions 1-3) The following equations define the basic Factor Analysis model and the questions below will relate to equation 9-2. X1 - HI = elIFi + 612Fat... + limFm + 81 X2 - 12 = 621F1 + 622F2 + ... + 62mFm + 82 (9-1) Xp - up = epIF1 + ep2F2 + .. . + epmFm+ Ep or, in matrix notation, X- u = L F + E (px1) (9-2) (pxm)(mx1) (px1) Question 1 FACTOR The model assumptions include that e and F are independent! O True O False Question 2 FACTOR Cov(X, F) > L True\fContext: PRINCIPAL (Questions 4 & 5) One method of decomposing a covariance matrix into a product of L and L' is to use PC's The spectral decomposition of (2-16) provides us with one factoring of the covariance ma- trix E. Let E have eigenvalue-eigenvector pairs ();, e;) with A| 2 12 2 .. . 2 Ap 2 0. Then I = Mjeje; + Azeez + ... + Apepep Vei Viz ez (9-10) = [VAjel | VAZ ez | ... VApep] VApep This fits the prescribed covariance structure for the factor analysis model having as many factors as variables (m = p) and specific variances w; = 0 for all i. The load- ing matrix has jth column given by VA, e;. That is, we can write E = LL'+ 0 = LL' (9-11) ( pxp ) (pxp)(pxp) (pxp) Apart from the scale factor VA,, the factor loadings on the jth factor are the coeffi- cients for the jth principal component of the population. Suppose that the population covariance matrix _ for measurements of fish is given by: LENGTH WEIGHT DDT LENGTH 47 1696 85 WEIGHT 1696 141787 -441 DDT 85 -441 9678 The following questions relate to the PC's technique and this Covariance.Question 4 PRINCIPAL Find the eigen values and eigenvectors of 3 and create the the pXp matrix L using the PC method above. Submit the numerical value of |L[1,1]| (the absolute value) to 4 decimal places [| Question 5 PRINCIPAL Make a function that will create L (the loadings matrix) using the covariance matrix - submit the |L[3,3]| to 4 decimal places! [| Context: SAMPLE PRINCIPLE (Questions 6-10) To use the PC's method of factor analysis on a sample we need to select m = 1 criterion determine the number of PC's to retain and submit the numerical value (integer) below!Question 7 SAMPLE PRINCIPLE Use m = 2 (this may not be the best choice) and L made from the correlation matrix R. df