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.

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 $4.8515\text{and}\text{0}\text{.35247}$, the standard error for the coefficient is $1.065$ and $90\%$ confidence levels for the coefficient $a_0\text{and}{a}_1$ is,

$\left[3.6951,6.0080\right]\text{and}\left[0.2501,0.4548\right]$

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,

Now, the coefficient of normal equation is calculated as,

Now, the normal equation of y is calculated as,

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

Therefore, the value of coefficientis $4.8515\text{and}\text{0}\text{.35247}$.

Now, for calculating the standard error follow these commands,

Now calculate $r^2$ value,

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

Now for calculating $90\%$ confidence levels for the coefficient, first calculate the inverse of ${\left[Z\right]}^T\left[Z\right]$ as,

Then the standard error of each coefficient,

Thevalue of $\text{TINV}\left(0.1,10-2\right)=\text{TINV}\left(0.1,8\right)=1.8595$ is determined by the statistic t value calculator. Therefore, the $90\%$ confidence levels for the coefficient is,

Hence, the $90\%$ confidence levels for the coefficient $a_0\text{and}{a}_1$ is $\left[3.6951,6.0080\right]\text{and}\left[0.2501,0.4548\right]$ respectively.

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 $-11.4887,7.1438,-1.0412\text{and}\text{0}\text{.0467}$, the standard error for the coefficient is $0.57$ and $90\%$ confidence levels for the coefficient $a_0,a_1,a_2\text{and}{a}_3$ is,

$\left[-20.0253,-2.9521\right],\left[3.0378,11.2498\right],\left[-1.6302,-0.4522\right]\text{and}\left[0.0208,0.0726\right]$.

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,

Now, the coefficient of normal equation is calculated as,

Now, the normal equation of y is calculated as,

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

Therefore, the value of coefficient is $-11.4887,7.1438,-1.0412\text{and}\text{0}\text{.0467}$.

Now, for calculating the standard error follow these commands,

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

Now for calculating $90\%$ confidence levels for the coefficient, first calculate the inverse of ${\left[Z\right]}^T\left[Z\right]$ as,

Then the standard error of each coefficient,

The value of $\text{TINV}\left(0.1,8-4\right)=\text{TINV}\left(0.1,4\right)=2.13185$ is determined by the statistic t value calculator. Therefore, the $90\%$ confidence levels for the coefficient is,

And,

Hence, the $90\%$ confidence levels for the coefficient $a_0,a_1,a_2\text{and}{a}_3$ is,

$\left[-20.0253,-2.9521\right],\left[3.0378,11.2498\right],\left[-1.6302,-0.4522\right]\text{and}\left[0.0208,0.0726\right]$ respectively.