Chapter 11, Problem 26E

(a)

To determine

To find: The distribution of present and past year total billing and total number of architects, staff, and engineers by using numerical and graphical summaries.

Answer to Problem 26E

Solution: The distribution of present and past year total billing and total number of architects and engineers is positively skewed. The distribution of staff has some missing frequencies but it can be said that it is approximately skewed to right.

Explanation of Solution

Calculation: To show the distribution of present and past year total billing and total number of architects, staff, and engineers, create the histogram. To obtain the histogram by using Minitab, follow the steps below:

Step 1: Open the Minitab worksheet which contains the data.

Step 2: Go to Graph > Histogram Boxplot > Simple.

Step 3: Click OK.

Step 4: Select “ArchBill13, ArchBill12 N_Arch, N_En, and N_Staff” in the column for Graph variables.

Step 5: Go to multiple graphs “In separate panels of the same graph.”

Step 6: Click “OK.” Again click “OK.”

A histogram with the numerical summaries is obtained. The obtained histogram is shown below:

Interpretation: It can be seen from the obtained histogram that the distribution of present and past year total billing and total number of architects and engineers is positively skewed. It means the data are skewed to right. The distribution of staff has some missing frequencies but it can be said that it is approximately skewed to right.

(b)

To determine

To find: The relationship for every variable by using numerical and graphical summaries.

Answer to Problem 26E

Solution: There is a positive correlation between the variables.

Explanation of Solution

Calculation: To show the relationship between the present and past year total billing and total number of architects, staff, and engineers, obtain the scatterplot between the variables. To obtain the scatterplot by using Minitab, follow the steps below:

Step 1: Open the Minitab worksheet which contains the data.

Step 2: Go to Graph> Matrix plot> Simple > OK.

Step 3: Select “ArchBill13, ArchBill12 N_Arch, N_En, and N_Staff” in the column for Graph Variables.

Step 4: Click “OK.”

The scatterplot is obtained for all the variables. Obtained scatterplot is shown below:

To show the relationship between the present and past year total billing and total number of architects, staff and engineers, obtain the correlation between the variables. To obtain the correlation by using Minitab, follow the steps below:

Step 1: Open the Minitab worksheet which contains the data.

Step 2: Go to Stat> Stat> Basic statistics > Correlation.

Step 3: Select “ArchBill13, ArchBill12 N_Arch, N_En, and N_Staff” in the column for Variables.

Step 4: Click “OK.”

The correlation between present year and total number of architects is obtained as 0.962. The correlation between present year and total number of staff is obtained as 0.373. The correlation between present year and total number of engineers is obtained as 0.230. The correlation between past year and total number of engineers is obtained as 0.204. The correlation between past year and total number of staff is obtained as 0.349. The correlation between past year and total number of architects is obtained as 0.959.

Interpretation: The correlation between present year and total number of architects is obtained as 0.962 and the correlation between past year and total number of architects is obtained as 0.959. It indicates that there is a positive correlation between the variables. The value of correlation is near to 1, hence it can be concluded that the variables present year and total number of architects and the variables past year and total number of architects are strong and positively correlated. The correlation between present year and total number of staff is obtained as 0.373, the correlation between present year and total number of engineers is obtained as 0.230, the correlation between past year and total number of engineers is obtained as 0.204, and the correlation between past year and total number of staff is obtained as 0.349. It indicates that there is a positive correlation between the variables. The value of correlation is small, hence it can be concluded that the variables are not strong but positively correlated. The scatterplot plot also represents the similar results.

(c)

To determine

To test: A multiple regression with the fitted equation.

Answer to Problem 26E

Solution: A regression model of present year and total number of architects, staff, and engineers is $\text{ArchBill13}=-\text{0}\text{.284 + 0}\text{.554 N\_Arch + 0}\text{.0020 N\_Eng + 0}\text{.0035 N\_Staff}$. The regression standard error is 1.51993. A regression model of past year and total number of architects, staff, and engineers is $\text{ArchBill12}=-0.750+0.574\text{ N\_Arch}-\text{0}\text{.0033N\_Eng + 0}\text{.0034 N\_Staff}$. The regression standard error is 1.670.

Explanation of Solution

Calculation: To perform the multiple regression by using year and census count as explanatory variables use Minitab and follow the steps given below:

Step 1: Open the Minitab worksheet which contains the data.

Step 2: Go to Stat> Regression > Regression > Fit regression model.

Step 3: Select ArchBill13 in the column for Response and select N_Arch, N_En, and N_Staff in the column for Predictors.

Step 4: Click “OK.”

Step 5: Again go to Stat> Regression > Regression > Fit regression model.

Step 6: Select ArchBill12 in the column for Response and select N_Arch, N_En, and N_Staff in the column for Predictors.

Step 7: Click “OK.”

Conclusion: A regression model of present year and total number of architects, staff, and engineers is $\text{ArchBill13}=-\text{0}\text{.284 + 0}\text{.554 N\_Arch + 0}\text{.0020 N\_Eng + 0}\text{.0035 N\_Staff}$. The regression standard error is 1.51993. The regression standard error is 1.51993. A regression model of past year and total number of architects, staff, and engineers is $\text{ArchBill12}=-0.750+0.574\text{ N\_Arch}-\text{0}\text{.0033N\_Eng + 0}\text{.0034 N\_Staff}$. The regression standard error is 1.670.

(d)

To determine

To find: The residuals.

Answer to Problem 26E

Solution: The table representing the residuals is shown below:

Residuals for past year	Residuals for present year
1.93465	1.03345
$-3.01793$	$\begin{array}{l}-2.43330\hfill \end{array}$
3.86630	4.04527
1.73284	0.64784
0.30066	0.41724
$-1.65429$	$-0.85708$
$-0.71383$	2.40798
0.26357	0.36849
$-2.04027$	$-1.48790$
$-0.47964$	$-0.93279$
0.33508	0.39052
1.49413	1.20884
$-2.91169$	$-1.83608$
1.58345	0.88167
$-0.91173$	$-1.43515$
1.47193	0.66624
$-0.40800$	0.55973
0.21479	$-0.07001$
$-0.48023$	$-0.94503$
$-1.33049$	$-1.57423$
$-0.13342$	$-0.51936$
0.27727	0.65183
0.00292	$-0.10204$
1.06694	0.34126
$-0.463015$	$-1.42741$

The data points are randomly scattered and does not represent any pattern.

Explanation of Solution

Calculation: To obtain the residuals by using Minitab, follow the steps below:

Step 1: Open the Minitab worksheet that contains the data.

Step 2: Go to Stat> Regression > General regression.

Step 3: Select ArchBill12 in the column for Response and select N_Arch, N_En, and N_Staff in the column for Predictors.

Step 4: Click on storage and select “Residuals.” Click on Graphs and select “Residuals versus fits.”

Step 5: Again go to Stat> Regression > General regression.

Step 6: Select ArchBill13 in the column for Response and select N_Arch, N_En, and N_Staff in the column for Predictors.

Step 7: Click on storage and select “Residuals.” Click on Graphs and select “Residuals versus fits.”

Step 8: Click “OK.”

The residuals and the residual plot are obtained. The table representing the residuals is shown below:

Residuals for past year	Residuals for present year
1.93465	1.03345
$-3.01793$	$\begin{array}{l}-2.43330\hfill \end{array}$
3.86630	4.04527
1.73284	0.64784
0.30066	0.41724
$-1.65429$	$-0.85708$
$-0.71383$	2.40798
0.26357	0.36849
$-2.04027$	$-1.48790$
$-0.47964$	$-0.93279$
0.33508	0.39052
1.49413	1.20884
$-2.91169$	$-1.83608$
1.58345	0.88167
$-0.91173$	$-1.43515$
1.47193	0.66624
$-0.40800$	0.55973
0.21479	$-0.07001$
$-0.48023$	$-0.94503$
$-1.33049$	$-1.57423$
$-0.13342$	$-0.51936$
0.27727	0.65183
0.00292	$-0.10204$
1.06694	0.34126
$-0.463015$	$-1.42741$

The obtained residual plot for the past year is shown below:

The obtained residual plot for the present year is shown below:

Interpretation: From the obtained residual plots, it can be seen that the data points are randomly scattered and does not represent any pattern.

(e)

To determine

To find: The predicted total billing for the previous year.

Answer to Problem 26E

Solution: The predicted total billing for the previous year is 1.028.

Explanation of Solution

Calculation: The predicted total billing for the provided data can be obtained by using Minitab. Steps are as follows:

Step 1: Open the Minitab worksheet which contains the data.

Step 2: Go to Stat> Regression > General regression.

Step 3: Select ArchBill12 in the column for Response and select N_Arch, N_En, and N_Staff in the column for Predictors.

Step 4: Click on Options and write 3, 1 and 17 in the column for Prediction intervals for new observations.

Step 5: Click “OK.”

The predicted value is obtained as 1.028.

(f)

To determine

To explain: The use of statistical inference under this setting.

Answer to Problem 26E

Solution: The use of statistical inference under this setting did not justify the data as data does not follow normal distribution. The data are skewed to right.

Explanation of Solution

To obtain the statistical inference under this setting, use Minitab and follow the steps given below:

Step 1: Open the Minitab worksheet that contains the data.

Step 2: Go to Graph > Probability Plot > Single > Click OK.

Step 3: Select ArchBill12, ArchBill13, N_Arch, N_En, and N_Staff in the column for Graph variables.

The obtained Normal quantile plot is shown below:

From the obtained normal quantile plot, it can be seen that the residuals deviate from the line. So, it can be concluded that the data does not follow normal distribution. From the part (a), it can be seen that all the variables are skewed to right.

In the provided problem, it can be seen that all the variables are positively correlated. The variables are correlated and randomly distributed. The data are skewed to right. From the normal quantile plot, it can be seen that the residuals deviate from the line. So, it can be concluded that the data does not follow normal distributed.