Recompute the regression fits from Probs. (a) 17.3 and (b) 17.17, using the matrix approach. Estimate the standard errors and develop 90% confidence intervals for the coefficients.

Question

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Answer 1

Textbook Question

Chapter 17, Problem 22P

Recompute the regression fits from Probs. (a) 17.3 and (b) 17.17, using the matrix approach. Estimate the standard errors and develop 90% confidence intervals for the coefficients.

(a)

Expert Solution

To determine

To calculate: The coefficient of the regression fit equation of the given databy the use of matrix approach and then calculate standard error and $90%$ confidence levels for the coefficient.

x	0	2	4	6	9	11	12	15	17	19
y	5	6	7	6	9	8	7	10	12	12

Answer to Problem 22P

Solution:

The value of coefficient by the use of matrix approach is $4.8515 and 0.35247$ , the standard error for the coefficient is $1.065$ and $90%$ confidence levels for the coefficient $a0 and a1$ is,

$[3.6951, 6.0080] and [0.2501, 0.4548]$

Explanation of Solution

Given Information:

The data is,

x	0	2	4	6	9	11	12	15	17	19
y	5	6	7	6	9	8	7	10	12	12

Calculation:

The coefficient of the regression fit equation can be found out by the Matrix approach by following the below steps,

Step 1. First make Z matrix that contain column of ones in the first column and in the second column x value is shown.

Step 2. Now the coefficient of normal equation is made.

Step 3. Now compute the normal equation of the right hand side that is y data.

Step 4. The coefficient of the model can be found out by taking inverse of the result obtained in step 2 and step 3.

The following MATLAB will perform the desired steps,

% x data is given.

X=[024691112151719]';

% y data is given.

Y=[5676987101212]';

On the command window, write this command as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 1

Now, the coefficient of normal equation is calculated as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 2

Now, the normal equation of y is calculated as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 3

Finally, the coefficient of the model can be found out,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 4

Therefore, the value of coefficientis $4.8515 and 0.35247$ .

Now, for calculating the standard error follow these commands,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 5

Now calculate $r2$ value,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 6

The standard error of the given data is $1.065$ .

Now for calculating $90%$ confidence levels for the coefficient, first calculate the inverse of $[Z]T[Z]$ as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 7

Then the standard error of each coefficient,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 8

Thevalue of $TINV(0.1,10−2)=TINV(0.1,8)=1.8595$ is determined by the statistic t value calculator. Therefore, the $90%$ confidence levels for the coefficient is,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 9

Hence, the $90%$ confidence levels for the coefficient $a0 and a1$ is $[3.6951, 6.0080] and [0.2501, 0.4548]$ respectively.

(b)

Expert Solution

To determine

To calculate: The coefficient of the regression fit cubic equation of the given databy the use of matrix approach and then calculate standard error and $90%$ confidence levels for the coefficient.

x	3	4	5	7	8	9	11	12
y	1.6	3.6	4.4	3.4	2.2	2.8	3.8	4.6

Answer to Problem 22P

Solution:

The value of coefficient by the use of matrix approach is $−11.4887, 7.1438, −1.0412 and 0.0467$ , the standard error for the coefficient is $0.57$ and $90%$ confidence levels for the coefficient $a0,a1,a2 and a3$ is,

$[−20.0253,−2.9521], [3.0378,11.2498],[−1.6302,−0.4522] and[0.0208, 0.0726]$ .

Explanation of Solution

Given Information:

The data is,

x	3	4	5	7	8	9	11	12
y	1.6	3.6	4.4	3.4	2.2	2.8	3.8	4.6

Calculation:

The coefficient of the regression fit equation can be found out by the Matrix approach by following the below steps,

Step 1. First make Z matrix that contain column of ones in the first column and in the second column x value is shown.

Step 2. Now the coefficient of normal equation is made.

Step 3. Now compute the normal equation of the right hand side that is y data.

Step 4. The coefficient of the model can be found out by taking inverse of the result obtained in step 2 and step 3.

The following MATLAB will perform the desired steps,

% x data is given.

X=[3457891112]';

% y data is given.

Y=[1.63.64.43.42.22.83.84.6]';

On the command window, write this commandfor cubic equation as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 10

Now, the coefficient of normal equation is calculated as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 11

Now, the normal equation of y is calculated as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 12

Finally, the coefficient of the model can be found out,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 13

Therefore, the value of coefficient is $−11.4887, 7.1438, −1.0412 and 0.0467$ .

Now, for calculating the standard error follow these commands,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 14

The standard error of the given data is $0.57$ .

Now for calculating $90%$ confidence levels for the coefficient, first calculate the inverse of $[Z]T[Z]$ as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 15

Then the standard error of each coefficient,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 16

The value of $TINV(0.1,8−4)=TINV(0.1,4)=2.13185$ is determined by the statistic t value calculator. Therefore, the $90%$ confidence levels for the coefficient is,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 17

And,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 18

Hence, the $90%$ confidence levels for the coefficient $a0,a1,a2 and a3$ is,

$[−20.0253,−2.9521], [3.0378,11.2498],[−1.6302,−0.4522] and[0.0208, 0.0726]$ respectively.

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Answer 2

Textbook Question

Chapter 17, Problem 22P

Recompute the regression fits from Probs. (a) 17.3 and (b) 17.17, using the matrix approach. Estimate the standard errors and develop 90% confidence intervals for the coefficients.

(a)

Expert Solution

To determine

To calculate: The coefficient of the regression fit equation of the given databy the use of matrix approach and then calculate standard error and $90%$ confidence levels for the coefficient.

x	0	2	4	6	9	11	12	15	17	19
y	5	6	7	6	9	8	7	10	12	12

Answer to Problem 22P

Solution:

The value of coefficient by the use of matrix approach is $4.8515 and 0.35247$ , the standard error for the coefficient is $1.065$ and $90%$ confidence levels for the coefficient $a0 and a1$ is,

$[3.6951, 6.0080] and [0.2501, 0.4548]$

Explanation of Solution

Given Information:

The data is,

x	0	2	4	6	9	11	12	15	17	19
y	5	6	7	6	9	8	7	10	12	12

Calculation:

The coefficient of the regression fit equation can be found out by the Matrix approach by following the below steps,

Step 1. First make Z matrix that contain column of ones in the first column and in the second column x value is shown.

Step 2. Now the coefficient of normal equation is made.

Step 3. Now compute the normal equation of the right hand side that is y data.

Step 4. The coefficient of the model can be found out by taking inverse of the result obtained in step 2 and step 3.

The following MATLAB will perform the desired steps,

% x data is given.

X=[024691112151719]';

% y data is given.

Y=[5676987101212]';

On the command window, write this command as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 1

Now, the coefficient of normal equation is calculated as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 2

Now, the normal equation of y is calculated as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 3

Finally, the coefficient of the model can be found out,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 4

Therefore, the value of coefficientis $4.8515 and 0.35247$ .

Now, for calculating the standard error follow these commands,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 5

Now calculate $r2$ value,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 6

The standard error of the given data is $1.065$ .

Now for calculating $90%$ confidence levels for the coefficient, first calculate the inverse of $[Z]T[Z]$ as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 7

Then the standard error of each coefficient,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 8

Thevalue of $TINV(0.1,10−2)=TINV(0.1,8)=1.8595$ is determined by the statistic t value calculator. Therefore, the $90%$ confidence levels for the coefficient is,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 9

Hence, the $90%$ confidence levels for the coefficient $a0 and a1$ is $[3.6951, 6.0080] and [0.2501, 0.4548]$ respectively.

(b)

Expert Solution

To determine

To calculate: The coefficient of the regression fit cubic equation of the given databy the use of matrix approach and then calculate standard error and $90%$ confidence levels for the coefficient.

x	3	4	5	7	8	9	11	12
y	1.6	3.6	4.4	3.4	2.2	2.8	3.8	4.6

Answer to Problem 22P

Solution:

The value of coefficient by the use of matrix approach is $−11.4887, 7.1438, −1.0412 and 0.0467$ , the standard error for the coefficient is $0.57$ and $90%$ confidence levels for the coefficient $a0,a1,a2 and a3$ is,

$[−20.0253,−2.9521], [3.0378,11.2498],[−1.6302,−0.4522] and[0.0208, 0.0726]$ .

Explanation of Solution

Given Information:

The data is,

x	3	4	5	7	8	9	11	12
y	1.6	3.6	4.4	3.4	2.2	2.8	3.8	4.6

Calculation:

The coefficient of the regression fit equation can be found out by the Matrix approach by following the below steps,

Step 1. First make Z matrix that contain column of ones in the first column and in the second column x value is shown.

Step 2. Now the coefficient of normal equation is made.

Step 3. Now compute the normal equation of the right hand side that is y data.

Step 4. The coefficient of the model can be found out by taking inverse of the result obtained in step 2 and step 3.

The following MATLAB will perform the desired steps,

% x data is given.

X=[3457891112]';

% y data is given.

Y=[1.63.64.43.42.22.83.84.6]';

On the command window, write this commandfor cubic equation as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 10

Now, the coefficient of normal equation is calculated as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 11

Now, the normal equation of y is calculated as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 12

Finally, the coefficient of the model can be found out,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 13

Therefore, the value of coefficient is $−11.4887, 7.1438, −1.0412 and 0.0467$ .

Now, for calculating the standard error follow these commands,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 14

The standard error of the given data is $0.57$ .

Now for calculating $90%$ confidence levels for the coefficient, first calculate the inverse of $[Z]T[Z]$ as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 15

Then the standard error of each coefficient,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 16

The value of $TINV(0.1,8−4)=TINV(0.1,4)=2.13185$ is determined by the statistic t value calculator. Therefore, the $90%$ confidence levels for the coefficient is,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 17

And,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 18

Hence, the $90%$ confidence levels for the coefficient $a0,a1,a2 and a3$ is,

$[−20.0253,−2.9521], [3.0378,11.2498],[−1.6302,−0.4522] and[0.0208, 0.0726]$ respectively.

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Answer 3

Textbook Question

Chapter 17, Problem 22P

Recompute the regression fits from Probs. (a) 17.3 and (b) 17.17, using the matrix approach. Estimate the standard errors and develop 90% confidence intervals for the coefficients.

(a)

Expert Solution

To determine

To calculate: The coefficient of the regression fit equation of the given databy the use of matrix approach and then calculate standard error and $90%$ confidence levels for the coefficient.

x	0	2	4	6	9	11	12	15	17	19
y	5	6	7	6	9	8	7	10	12	12

Answer to Problem 22P

Solution:

The value of coefficient by the use of matrix approach is $4.8515 and 0.35247$ , the standard error for the coefficient is $1.065$ and $90%$ confidence levels for the coefficient $a0 and a1$ is,

$[3.6951, 6.0080] and [0.2501, 0.4548]$

Explanation of Solution

Given Information:

The data is,

x	0	2	4	6	9	11	12	15	17	19
y	5	6	7	6	9	8	7	10	12	12

Calculation:

The coefficient of the regression fit equation can be found out by the Matrix approach by following the below steps,

Step 1. First make Z matrix that contain column of ones in the first column and in the second column x value is shown.

Step 2. Now the coefficient of normal equation is made.

Step 3. Now compute the normal equation of the right hand side that is y data.

Step 4. The coefficient of the model can be found out by taking inverse of the result obtained in step 2 and step 3.

The following MATLAB will perform the desired steps,

% x data is given.

X=[024691112151719]';

% y data is given.

Y=[5676987101212]';

On the command window, write this command as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 1

Now, the coefficient of normal equation is calculated as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 2

Now, the normal equation of y is calculated as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 3

Finally, the coefficient of the model can be found out,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 4

Therefore, the value of coefficientis $4.8515 and 0.35247$ .

Now, for calculating the standard error follow these commands,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 5

Now calculate $r2$ value,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 6

The standard error of the given data is $1.065$ .

Now for calculating $90%$ confidence levels for the coefficient, first calculate the inverse of $[Z]T[Z]$ as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 7

Then the standard error of each coefficient,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 8

Thevalue of $TINV(0.1,10−2)=TINV(0.1,8)=1.8595$ is determined by the statistic t value calculator. Therefore, the $90%$ confidence levels for the coefficient is,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 9

Hence, the $90%$ confidence levels for the coefficient $a0 and a1$ is $[3.6951, 6.0080] and [0.2501, 0.4548]$ respectively.

(b)

Expert Solution

To determine

To calculate: The coefficient of the regression fit cubic equation of the given databy the use of matrix approach and then calculate standard error and $90%$ confidence levels for the coefficient.

x	3	4	5	7	8	9	11	12
y	1.6	3.6	4.4	3.4	2.2	2.8	3.8	4.6

Answer to Problem 22P

Solution:

The value of coefficient by the use of matrix approach is $−11.4887, 7.1438, −1.0412 and 0.0467$ , the standard error for the coefficient is $0.57$ and $90%$ confidence levels for the coefficient $a0,a1,a2 and a3$ is,

$[−20.0253,−2.9521], [3.0378,11.2498],[−1.6302,−0.4522] and[0.0208, 0.0726]$ .

Explanation of Solution

Given Information:

The data is,

x	3	4	5	7	8	9	11	12
y	1.6	3.6	4.4	3.4	2.2	2.8	3.8	4.6

Calculation:

The coefficient of the regression fit equation can be found out by the Matrix approach by following the below steps,

Step 1. First make Z matrix that contain column of ones in the first column and in the second column x value is shown.

Step 2. Now the coefficient of normal equation is made.

Step 3. Now compute the normal equation of the right hand side that is y data.

Step 4. The coefficient of the model can be found out by taking inverse of the result obtained in step 2 and step 3.

The following MATLAB will perform the desired steps,

% x data is given.

X=[3457891112]';

% y data is given.

Y=[1.63.64.43.42.22.83.84.6]';

On the command window, write this commandfor cubic equation as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 10

Now, the coefficient of normal equation is calculated as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 11

Now, the normal equation of y is calculated as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 12

Finally, the coefficient of the model can be found out,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 13

Therefore, the value of coefficient is $−11.4887, 7.1438, −1.0412 and 0.0467$ .

Now, for calculating the standard error follow these commands,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 14

The standard error of the given data is $0.57$ .

Now for calculating $90%$ confidence levels for the coefficient, first calculate the inverse of $[Z]T[Z]$ as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 15

Then the standard error of each coefficient,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 16

The value of $TINV(0.1,8−4)=TINV(0.1,4)=2.13185$ is determined by the statistic t value calculator. Therefore, the $90%$ confidence levels for the coefficient is,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 17

And,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 18

Hence, the $90%$ confidence levels for the coefficient $a0,a1,a2 and a3$ is,

$[−20.0253,−2.9521], [3.0378,11.2498],[−1.6302,−0.4522] and[0.0208, 0.0726]$ respectively.

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Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 4

Textbook Question

Chapter 17, Problem 22P

Recompute the regression fits from Probs. (a) 17.3 and (b) 17.17, using the matrix approach. Estimate the standard errors and develop 90% confidence intervals for the coefficients.

(a)

Expert Solution

To determine

To calculate: The coefficient of the regression fit equation of the given databy the use of matrix approach and then calculate standard error and $90%$ confidence levels for the coefficient.

x	0	2	4	6	9	11	12	15	17	19
y	5	6	7	6	9	8	7	10	12	12

Answer to Problem 22P

Solution:

The value of coefficient by the use of matrix approach is $4.8515 and 0.35247$ , the standard error for the coefficient is $1.065$ and $90%$ confidence levels for the coefficient $a0 and a1$ is,

$[3.6951, 6.0080] and [0.2501, 0.4548]$

Explanation of Solution

Given Information:

The data is,

x	0	2	4	6	9	11	12	15	17	19
y	5	6	7	6	9	8	7	10	12	12

Calculation:

The coefficient of the regression fit equation can be found out by the Matrix approach by following the below steps,

Step 1. First make Z matrix that contain column of ones in the first column and in the second column x value is shown.

Step 2. Now the coefficient of normal equation is made.

Step 3. Now compute the normal equation of the right hand side that is y data.

Step 4. The coefficient of the model can be found out by taking inverse of the result obtained in step 2 and step 3.

The following MATLAB will perform the desired steps,

% x data is given.

X=[024691112151719]';

% y data is given.

Y=[5676987101212]';

On the command window, write this command as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 1

Now, the coefficient of normal equation is calculated as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 2

Now, the normal equation of y is calculated as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 3

Finally, the coefficient of the model can be found out,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 4

Therefore, the value of coefficientis $4.8515 and 0.35247$ .

Now, for calculating the standard error follow these commands,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 5

Now calculate $r2$ value,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 6

The standard error of the given data is $1.065$ .

Now for calculating $90%$ confidence levels for the coefficient, first calculate the inverse of $[Z]T[Z]$ as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 7

Then the standard error of each coefficient,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 8

Thevalue of $TINV(0.1,10−2)=TINV(0.1,8)=1.8595$ is determined by the statistic t value calculator. Therefore, the $90%$ confidence levels for the coefficient is,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 9

Hence, the $90%$ confidence levels for the coefficient $a0 and a1$ is $[3.6951, 6.0080] and [0.2501, 0.4548]$ respectively.

(b)

Expert Solution

To determine

To calculate: The coefficient of the regression fit cubic equation of the given databy the use of matrix approach and then calculate standard error and $90%$ confidence levels for the coefficient.

x	3	4	5	7	8	9	11	12
y	1.6	3.6	4.4	3.4	2.2	2.8	3.8	4.6

Answer to Problem 22P

Solution:

The value of coefficient by the use of matrix approach is $−11.4887, 7.1438, −1.0412 and 0.0467$ , the standard error for the coefficient is $0.57$ and $90%$ confidence levels for the coefficient $a0,a1,a2 and a3$ is,

$[−20.0253,−2.9521], [3.0378,11.2498],[−1.6302,−0.4522] and[0.0208, 0.0726]$ .

Explanation of Solution

Given Information:

The data is,

x	3	4	5	7	8	9	11	12
y	1.6	3.6	4.4	3.4	2.2	2.8	3.8	4.6

Calculation:

The coefficient of the regression fit equation can be found out by the Matrix approach by following the below steps,

Step 1. First make Z matrix that contain column of ones in the first column and in the second column x value is shown.

Step 2. Now the coefficient of normal equation is made.

Step 3. Now compute the normal equation of the right hand side that is y data.

Step 4. The coefficient of the model can be found out by taking inverse of the result obtained in step 2 and step 3.

The following MATLAB will perform the desired steps,

% x data is given.

X=[3457891112]';

% y data is given.

Y=[1.63.64.43.42.22.83.84.6]';

On the command window, write this commandfor cubic equation as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 10

Now, the coefficient of normal equation is calculated as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 11

Now, the normal equation of y is calculated as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 12

Finally, the coefficient of the model can be found out,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 13

Therefore, the value of coefficient is $−11.4887, 7.1438, −1.0412 and 0.0467$ .

Now, for calculating the standard error follow these commands,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 14

The standard error of the given data is $0.57$ .

Now for calculating $90%$ confidence levels for the coefficient, first calculate the inverse of $[Z]T[Z]$ as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 15

Then the standard error of each coefficient,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 16

The value of $TINV(0.1,8−4)=TINV(0.1,4)=2.13185$ is determined by the statistic t value calculator. Therefore, the $90%$ confidence levels for the coefficient is,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 17

And,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 18

Hence, the $90%$ confidence levels for the coefficient $a0,a1,a2 and a3$ is,

$[−20.0253,−2.9521], [3.0378,11.2498],[−1.6302,−0.4522] and[0.0208, 0.0726]$ respectively.

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Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 5

Textbook Question

Answer 6

Textbook Question

Answer 7

(a)

Expert Solution

To determine

To calculate: The coefficient of the regression fit equation of the given databy the use of matrix approach and then calculate standard error and $90%$ confidence levels for the coefficient.

x	0	2	4	6	9	11	12	15	17	19
y	5	6	7	6	9	8	7	10	12	12

Answer to Problem 22P

Solution:

The value of coefficient by the use of matrix approach is $4.8515 and 0.35247$ , the standard error for the coefficient is $1.065$ and $90%$ confidence levels for the coefficient $a0 and a1$ is,

$[3.6951, 6.0080] and [0.2501, 0.4548]$

Explanation of Solution

Given Information:

The data is,

x	0	2	4	6	9	11	12	15	17	19
y	5	6	7	6	9	8	7	10	12	12

Calculation:

The coefficient of the regression fit equation can be found out by the Matrix approach by following the below steps,

Step 1. First make Z matrix that contain column of ones in the first column and in the second column x value is shown.

Step 2. Now the coefficient of normal equation is made.

Step 3. Now compute the normal equation of the right hand side that is y data.

Step 4. The coefficient of the model can be found out by taking inverse of the result obtained in step 2 and step 3.

The following MATLAB will perform the desired steps,

% x data is given.

X=[024691112151719]';

% y data is given.

Y=[5676987101212]';

On the command window, write this command as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 1

Now, the coefficient of normal equation is calculated as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 2

Now, the normal equation of y is calculated as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 3

Finally, the coefficient of the model can be found out,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 4

Therefore, the value of coefficientis $4.8515 and 0.35247$ .

Now, for calculating the standard error follow these commands,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 5

Now calculate $r2$ value,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 6

The standard error of the given data is $1.065$ .

Now for calculating $90%$ confidence levels for the coefficient, first calculate the inverse of $[Z]T[Z]$ as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 7

Then the standard error of each coefficient,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 8

Thevalue of $TINV(0.1,10−2)=TINV(0.1,8)=1.8595$ is determined by the statistic t value calculator. Therefore, the $90%$ confidence levels for the coefficient is,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 9

Hence, the $90%$ confidence levels for the coefficient $a0 and a1$ is $[3.6951, 6.0080] and [0.2501, 0.4548]$ respectively.

(b)

Expert Solution

To determine

To calculate: The coefficient of the regression fit cubic equation of the given databy the use of matrix approach and then calculate standard error and $90%$ confidence levels for the coefficient.

x	3	4	5	7	8	9	11	12
y	1.6	3.6	4.4	3.4	2.2	2.8	3.8	4.6

Answer to Problem 22P

Solution:

The value of coefficient by the use of matrix approach is $−11.4887, 7.1438, −1.0412 and 0.0467$ , the standard error for the coefficient is $0.57$ and $90%$ confidence levels for the coefficient $a0,a1,a2 and a3$ is,

$[−20.0253,−2.9521], [3.0378,11.2498],[−1.6302,−0.4522] and[0.0208, 0.0726]$ .

Explanation of Solution

Given Information:

The data is,

x	3	4	5	7	8	9	11	12
y	1.6	3.6	4.4	3.4	2.2	2.8	3.8	4.6

Calculation:

The coefficient of the regression fit equation can be found out by the Matrix approach by following the below steps,

Step 1. First make Z matrix that contain column of ones in the first column and in the second column x value is shown.

Step 2. Now the coefficient of normal equation is made.

Step 3. Now compute the normal equation of the right hand side that is y data.

Step 4. The coefficient of the model can be found out by taking inverse of the result obtained in step 2 and step 3.

The following MATLAB will perform the desired steps,

% x data is given.

X=[3457891112]';

% y data is given.

Y=[1.63.64.43.42.22.83.84.6]';

On the command window, write this commandfor cubic equation as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 10

Now, the coefficient of normal equation is calculated as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 11

Now, the normal equation of y is calculated as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 12

Finally, the coefficient of the model can be found out,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 13

Therefore, the value of coefficient is $−11.4887, 7.1438, −1.0412 and 0.0467$ .

Now, for calculating the standard error follow these commands,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 14

The standard error of the given data is $0.57$ .

Now for calculating $90%$ confidence levels for the coefficient, first calculate the inverse of $[Z]T[Z]$ as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 15

Then the standard error of each coefficient,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 16

The value of $TINV(0.1,8−4)=TINV(0.1,4)=2.13185$ is determined by the statistic t value calculator. Therefore, the $90%$ confidence levels for the coefficient is,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 17

And,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 18

Hence, the $90%$ confidence levels for the coefficient $a0,a1,a2 and a3$ is,

$[−20.0253,−2.9521], [3.0378,11.2498],[−1.6302,−0.4522] and[0.0208, 0.0726]$ respectively.

Answer 8

(a)

Expert Solution

To determine

To calculate: The coefficient of the regression fit equation of the given databy the use of matrix approach and then calculate standard error and $90%$ confidence levels for the coefficient.

x	0	2	4	6	9	11	12	15	17	19
y	5	6	7	6	9	8	7	10	12	12

Answer to Problem 22P

Solution:

The value of coefficient by the use of matrix approach is $4.8515 and 0.35247$ , the standard error for the coefficient is $1.065$ and $90%$ confidence levels for the coefficient $a0 and a1$ is,

$[3.6951, 6.0080] and [0.2501, 0.4548]$

Explanation of Solution

Given Information:

The data is,

x	0	2	4	6	9	11	12	15	17	19
y	5	6	7	6	9	8	7	10	12	12

Calculation:

The coefficient of the regression fit equation can be found out by the Matrix approach by following the below steps,

Step 1. First make Z matrix that contain column of ones in the first column and in the second column x value is shown.

Step 2. Now the coefficient of normal equation is made.

Step 3. Now compute the normal equation of the right hand side that is y data.

Step 4. The coefficient of the model can be found out by taking inverse of the result obtained in step 2 and step 3.

The following MATLAB will perform the desired steps,

% x data is given.

X=[024691112151719]';

% y data is given.

Y=[5676987101212]';

On the command window, write this command as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 1

Now, the coefficient of normal equation is calculated as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 2

Now, the normal equation of y is calculated as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 3

Finally, the coefficient of the model can be found out,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 4

Therefore, the value of coefficientis $4.8515 and 0.35247$ .

Now, for calculating the standard error follow these commands,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 5

Now calculate $r2$ value,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 6

The standard error of the given data is $1.065$ .

Now for calculating $90%$ confidence levels for the coefficient, first calculate the inverse of $[Z]T[Z]$ as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 7

Then the standard error of each coefficient,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 8

Thevalue of $TINV(0.1,10−2)=TINV(0.1,8)=1.8595$ is determined by the statistic t value calculator. Therefore, the $90%$ confidence levels for the coefficient is,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 9

Hence, the $90%$ confidence levels for the coefficient $a0 and a1$ is $[3.6951, 6.0080] and [0.2501, 0.4548]$ respectively.

Answer 9

(a)

Expert Solution

Answer 10

(a)

Expert Solution

Answer 11

Expert Solution

Answer 12

To determine

To calculate: The coefficient of the regression fit equation of the given databy the use of matrix approach and then calculate standard error and $90%$ confidence levels for the coefficient.

x	0	2	4	6	9	11	12	15	17	19
y	5	6	7	6	9	8	7	10	12	12

Answer 13

Answer to Problem 22P

Solution:

The value of coefficient by the use of matrix approach is $4.8515 and 0.35247$ , the standard error for the coefficient is $1.065$ and $90%$ confidence levels for the coefficient $a0 and a1$ is,

$[3.6951, 6.0080] and [0.2501, 0.4548]$

Answer 14

Explanation of Solution

Given Information:

The data is,

x	0	2	4	6	9	11	12	15	17	19
y	5	6	7	6	9	8	7	10	12	12

Calculation:

The coefficient of the regression fit equation can be found out by the Matrix approach by following the below steps,

Step 1. First make Z matrix that contain column of ones in the first column and in the second column x value is shown.

Step 2. Now the coefficient of normal equation is made.

Step 3. Now compute the normal equation of the right hand side that is y data.

Step 4. The coefficient of the model can be found out by taking inverse of the result obtained in step 2 and step 3.

The following MATLAB will perform the desired steps,

% x data is given.

X=[024691112151719]';

% y data is given.

Y=[5676987101212]';

On the command window, write this command as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 1

Now, the coefficient of normal equation is calculated as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 2

Now, the normal equation of y is calculated as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 3

Finally, the coefficient of the model can be found out,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 4

Therefore, the value of coefficientis $4.8515 and 0.35247$ .

Now, for calculating the standard error follow these commands,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 5

Now calculate $r2$ value,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 6

The standard error of the given data is $1.065$ .

Now for calculating $90%$ confidence levels for the coefficient, first calculate the inverse of $[Z]T[Z]$ as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 7

Then the standard error of each coefficient,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 8

Thevalue of $TINV(0.1,10−2)=TINV(0.1,8)=1.8595$ is determined by the statistic t value calculator. Therefore, the $90%$ confidence levels for the coefficient is,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 9

Hence, the $90%$ confidence levels for the coefficient $a0 and a1$ is $[3.6951, 6.0080] and [0.2501, 0.4548]$ respectively.

Answer 15

(b)

Expert Solution

To determine

To calculate: The coefficient of the regression fit cubic equation of the given databy the use of matrix approach and then calculate standard error and $90%$ confidence levels for the coefficient.

x	3	4	5	7	8	9	11	12
y	1.6	3.6	4.4	3.4	2.2	2.8	3.8	4.6

Answer to Problem 22P

Solution:

The value of coefficient by the use of matrix approach is $−11.4887, 7.1438, −1.0412 and 0.0467$ , the standard error for the coefficient is $0.57$ and $90%$ confidence levels for the coefficient $a0,a1,a2 and a3$ is,

$[−20.0253,−2.9521], [3.0378,11.2498],[−1.6302,−0.4522] and[0.0208, 0.0726]$ .

Explanation of Solution

Given Information:

The data is,

x	3	4	5	7	8	9	11	12
y	1.6	3.6	4.4	3.4	2.2	2.8	3.8	4.6

Calculation:

The coefficient of the regression fit equation can be found out by the Matrix approach by following the below steps,

Step 1. First make Z matrix that contain column of ones in the first column and in the second column x value is shown.

Step 2. Now the coefficient of normal equation is made.

Step 3. Now compute the normal equation of the right hand side that is y data.

Step 4. The coefficient of the model can be found out by taking inverse of the result obtained in step 2 and step 3.

The following MATLAB will perform the desired steps,

% x data is given.

X=[3457891112]';

% y data is given.

Y=[1.63.64.43.42.22.83.84.6]';

On the command window, write this commandfor cubic equation as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 10

Now, the coefficient of normal equation is calculated as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 11

Now, the normal equation of y is calculated as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 12

Finally, the coefficient of the model can be found out,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 13

Therefore, the value of coefficient is $−11.4887, 7.1438, −1.0412 and 0.0467$ .

Now, for calculating the standard error follow these commands,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 14

The standard error of the given data is $0.57$ .

Now for calculating $90%$ confidence levels for the coefficient, first calculate the inverse of $[Z]T[Z]$ as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 15

Then the standard error of each coefficient,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 16

The value of $TINV(0.1,8−4)=TINV(0.1,4)=2.13185$ is determined by the statistic t value calculator. Therefore, the $90%$ confidence levels for the coefficient is,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 17

And,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 18

Hence, the $90%$ confidence levels for the coefficient $a0,a1,a2 and a3$ is,

$[−20.0253,−2.9521], [3.0378,11.2498],[−1.6302,−0.4522] and[0.0208, 0.0726]$ respectively.

Answer 16

(b)

Expert Solution

Answer 17

(b)

Expert Solution

Answer 18

Expert Solution

Answer 19

To determine

To calculate: The coefficient of the regression fit cubic equation of the given databy the use of matrix approach and then calculate standard error and $90%$ confidence levels for the coefficient.

x	3	4	5	7	8	9	11	12
y	1.6	3.6	4.4	3.4	2.2	2.8	3.8	4.6

Answer 20

Answer to Problem 22P

Solution:

The value of coefficient by the use of matrix approach is $−11.4887, 7.1438, −1.0412 and 0.0467$ , the standard error for the coefficient is $0.57$ and $90%$ confidence levels for the coefficient $a0,a1,a2 and a3$ is,

$[−20.0253,−2.9521], [3.0378,11.2498],[−1.6302,−0.4522] and[0.0208, 0.0726]$ .

Answer 21

Explanation of Solution

Given Information:

The data is,

x	3	4	5	7	8	9	11	12
y	1.6	3.6	4.4	3.4	2.2	2.8	3.8	4.6

Calculation:

The coefficient of the regression fit equation can be found out by the Matrix approach by following the below steps,

Step 1. First make Z matrix that contain column of ones in the first column and in the second column x value is shown.

Step 2. Now the coefficient of normal equation is made.

Step 3. Now compute the normal equation of the right hand side that is y data.

Step 4. The coefficient of the model can be found out by taking inverse of the result obtained in step 2 and step 3.

The following MATLAB will perform the desired steps,

% x data is given.

X=[3457891112]';

% y data is given.

Y=[1.63.64.43.42.22.83.84.6]';

On the command window, write this commandfor cubic equation as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 10

Now, the coefficient of normal equation is calculated as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 11

Now, the normal equation of y is calculated as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 12

Finally, the coefficient of the model can be found out,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 13

Therefore, the value of coefficient is $−11.4887, 7.1438, −1.0412 and 0.0467$ .

Now, for calculating the standard error follow these commands,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 14

The standard error of the given data is $0.57$ .

Now for calculating $90%$ confidence levels for the coefficient, first calculate the inverse of $[Z]T[Z]$ as,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 15

Then the standard error of each coefficient,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 16

The value of $TINV(0.1,8−4)=TINV(0.1,4)=2.13185$ is determined by the statistic t value calculator. Therefore, the $90%$ confidence levels for the coefficient is,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 17

And,

Numerical Methods for Engineers, Chapter 17, Problem 22P , additional homework tip 18

Hence, the $90%$ confidence levels for the coefficient $a0,a1,a2 and a3$ is,

$[−20.0253,−2.9521], [3.0378,11.2498],[−1.6302,−0.4522] and[0.0208, 0.0726]$ respectively.

Answer 22

Ch. 17 - Given these data 8.8 9.5 9.8 9.4 10.0 9.4 10.1 9.2...Ch. 17 - Given these data 29.65 28.55 28.65 30.15 29.35...Ch. 17 - 17.3 Use least-squares regression to fit a...Ch. 17 - 17.4 Use least-squares regression to fit a...Ch. 17 - 17.5 Using the same approach as was employed to...Ch. 17 - Use least-squares regression to fit a straight...Ch. 17 - Fit the following data with (a) A...Ch. 17 - Fit the following data with the power model...Ch. 17 - 17.9 Fit an exponential model...Ch. 17 - 17.10 Rather than using the base-e exponential...

Recompute the regression fits from Probs. (a) 17.3 and (b) 17.17, using the matrix approach. Estimate the standard errors and develop 90% confidence intervals for the coefficients.

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Answer to Problem 22P

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Answer to Problem 22P

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