Statistics 2080B: Practice Problems - Matrices

Practice P

- A2.
A2. MATRIX A MATRIX B MATRIX C 3 1 3 1 0 2 3 2 0 4 0 -4 1 2 2

a) MATRIX AC 15 14 -8 b) MATRIX BC 7

- c)
No, it is not possible to find the product AB since the number of columns in A(3) does not equal the number of rows in B(1).

- A3.
- a) AB is a 3 x 4 matrix
- b)
No, it is not possible to find the product BA because the number of columns in B(4) does not equal the number of rows in A(3).

- A4.
- a)
BC has dimensions
- b)
CB has dimensions
- c)
MATRIX CB 3 0 6 0 0 0 2 0 4

- A7.
A = 3 0 2 -1 1 4

- a)
The identity matrix used to get the product IA is
.
I = 1 0 0 1

- c)
The identity matrix used to get the product AI is
1 0 0 I = 0 1 0 0 0 1

- A8.
Since AB = I and BA = I then B is the inverse of A

AB = BA = I = 1 0 0 0 1 0 0 0 1

- A18.
- a)
Y has dimensions
X has dimensions

- b)
X'X [X'X]-1 = I I = 1 0 0 0 1 0 0 0 1

- c)
A matrix A is symmetric if
*a*_{ij}=*a*_{ji}for all*i*and*j*. - d)
MTB > read 10 x 3 m20 M20 = X X has dimensions 10 x 3 DATA> 1 -2 4 DATA> 1 -2 4 DATA> 1 -1 1 DATA> 1 -1 1 DATA> 1 0 0 DATA> 1 0 0 DATA> 1 1 1 DATA> 1 1 1 DATA> 1 2 4 DATA> 1 2 4 10 ROWS READ MTB > trans m20 m21 MTB > print m21 MATRIX M21 M21 = X' X' has dimensions 3 x 10 1 1 1 1 1 1 1 1 1 1 -2 -2 -1 -1 0 0 1 1 2 2 4 4 1 1 0 0 1 1 4 4 MTB > mult m21 m20 m22 MTB > print m22 MATRIX M22 M22 = X'X X'X has dimensions 3 x 3 10 0 20 0 20 0 20 0 68 MTB > invert m22 m23 MTB > print m23 MATRIX M23 M23 = [X'X]-1 [X'X] -1 has dimensions 3x3 0.242857 0.000000 -0.071429 0.000000 0.050000 0.000000 -0.071429 0.000000 0.035714 MTB > read 10 x 1 m24 M24 = Y Y has dimensions 10 x 1 DATA> 1.1 DATA> 1.3 DATA> 2.0 DATA> 2.1 DATA> 2.7 DATA> 2.8 DATA> 3.4 DATA> 3.6 DATA> 4.1 DATA> 4.0 10 ROWS READ MTB > mult m21 m24 m25 MTB > print m25 MATRIX M25 M25 = X'Y X'Y has dimensions 3 x 1 27.1 14.3 53.1 MTB > mult m23 m25 m26 MTB > print m26 MATRIX M26 M26 = [X'X]-1X'Y [X'X]-1X'Y has dimensions 3 x 1

The least squares equation is

- A21.
MTB > mult m20 m26 m27 M20 = X M26= beta (hat) so M27 = Y(hat) MTB > print m27 MATRIX M27 1.20143 1.20143 2.03429 2.03429 2.78857 2.78857 3.46429 3.46429 4.06143 4.06143 MTB > copy m24 c1 c1 = yi MTB > copy m27 c2 c2 = yi(hat) MTB > let c3 = c1 - c2 c3 = ei = residuals MTB > let c4 = c3**2 MTB > sum c4 SUM = 0.061286 SSE = .061286

when

*x*= 1*V*_{22}= 0.357*t*has 7 df0.05 <

*p*< 0.10Reject the null hypothesis at since the

*p*-value is less than . The data indicates curvature.

- A.26
MATRIX M30 M30 = l 1 1 1 MTB > trans m30 m31 MTB > mult m31 m23 m32 MTB > mult m32 m30 m33 ANSWER = 0.1857 MTB > copy m20 c10 c11 c12 MTB > print c10-c12 ROW C10 C11 C12 1 1 -2 4 2 1 -2 4 3 1 -1 1 4 1 -1 1 5 1 0 0 6 1 0 0 7 1 1 1 8 1 1 1 9 1 2 4 10 1 2 4 MTB > regress c1 on 2 in c11 c12; SUBC> predict 1 1. The regression equation is C1 = 2.79 + 0.715 C11 - 0.0393 C12 Predictor Coef Stdev t-ratio p Constant 2.78857 0.04611 60.48 0.000 C11 0.71500 0.02092 34.17 0.000 C12 -0.03929 0.01768 -2.22 0.062 s = 0.09357 R-sq = 99.4% R-sq(adj) = 99.2% Analysis of Variance SOURCE DF SS MS F p Regression 2 10.2677 5.1339 586.39 0.000 Error 7 0.0613 0.0088 Total 9 10.3290 SOURCE DF SEQ SS C11 1 10.2245 C12 1 0.0432 Fit Stdev.Fit 95% C.I. 95% P.I. 3.4643 0.0403 ( 3.3689, 3.5597) ( 3.2233, 3.7053)

The confidence interval for

*E*[*y*] when*x*=1, ,*s*= 0.09357, and*l*'(*x*'*x*)^{-1}*l*= 0.1857.*t*has 7 df,*t*=1.895(3.36,3.51)

- A.27
prediction interval for a new value of

*y*when*x*=1the same as in A.26

(3.27,3.66)

- A.28
- a)
*s*= 1.1 - b)
confidence interval for
*E*[*y*] when*x*=1.10,*x*^{2}= 1.21*l*'(*x*'*x*)^{-1}*l*= 0.1077*t*has 22 df,*t*= 2.074(8.60,10.10)

We are confident that the mean number of items produced per hour per worker when the piece rate, , is between 8.6 and 10.10 items/hour/worker.

- c)
prediction interval for individual worker who is paid
per piece.
(6.945,11.747)

We are confident that if a worker is paid a piecework rate of per piece his productivity will be between 6,95 and 11.75.

- d)
*SSE*= 26.26of the valuability is explained by the regression model.

STATISTICS 2080 HOMEPAGE |