Chapter 10, Problem 49E

(a)

To determine

To find: The variables IBI and Forest using numerical method.

Answer to Problem 49E

Solution: The obtained result can be shown in tabular form as follows:

Variable	Mean	Standard deviation
Forest	39.39	32.20
IBI	65.94	18.28

Explanation of Solution

Calculation: Calculate the average and standard deviation of IBI and Forest using Minitab as follows:

Step 1: Enter the data in Minitab.

Step 2: Go to Graphs > Histogram > Simple histogram.

Step 3: Double click on ‘Forest’ and ‘IBI’ to move it to variables column.

Step 4: Click on ‘Statistics’ and check the box for mean and standard deviation.

Step 5: Click ‘OK’ twice to obtain the result.

Results are obtained as:

Variable	Mean	Standard deviation
Forest	39.39	32.20
IBI	65.94	18.28

To determine

To find: The variables IBI and Forest using graphical method.

Answer to Problem 49E

Solution: The graph of Forest is right skewed and the graph of IBI is left skewed.

Explanation of Solution

Graph: Construct the histograms to check the skewness using Minitab as follows:

Step 1: Click on Graphs --> Histogram. Select simple histogram.

Step 2: Double click on ‘Forest’ and ‘IBI’ to move it to variables column.

Step 3: Click ‘OK’ to obtain the result.

Interpretation: The graph of Forest is right skewed and the graph of IBI is left skewed.

(b)

To determine

To graph: A scatterplot.

Explanation of Solution

Graph: Construct a scatter plot as follows:

Step 1: Enter the data in Minitab.

Step 2: Click on Graph --> Scatterplot. Select scatterplot with regression.

Step 3: Double click on ‘BAC’ to move it Y variable and ‘Beer’ to move it to X variable column.

Step 4: Click ‘Ok’ twice to obtain the graph.

The scatter plot is obtained as:

Interpretation: The graph shows weak linear relationship between IBI and Forest with no unusual activity.

(c)

To determine

To explain: The statistical model for simple linear regression.

Answer to Problem 49E

Solution: The model is $\underset{\_}{\text{IBI}\left(\widehat{y}\right)={\beta }_0+{\beta }_1\text{Forest}\left(x\right)+\epsilon }$.

Explanation of Solution

The statistical model for the provided problem can be shown as follows:

$\text{IBI}\left(\widehat{y}\right)={\beta }_0+{\beta }_1\text{Forest}\left(x\right)+\epsilon$

Where,

$\begin{array}{l}\text{IBI is the response varibale}\\ {\beta }_0\text{ is the inetrcept}\\ {\beta }_1\text{ is the slope}\\ \text{Forest is the explanatory variable}\\ \epsilon \text{ is the error term}\end{array}$

(d)

To determine

To explain: The null and alternate hypotheses.

Answer to Problem 49E

Solution: The null and alternative hypotheses are:

$\begin{array}{l}{H}_0:{\beta }_1=0\\ H_1:{\beta }_1\ne 0\end{array}$

Explanation of Solution

Consider the null and alternate hypothesis as follows:

$\begin{array}{c}{H}_0:\left(\begin{array}{l}\text{There is no significant relationship}\\ \text{between IBI and Forest}\end{array}\right)\\ H_1:\left(\begin{array}{l}\text{There is a significant relationship}\\ \text{between IBI and Forest}\end{array}\right)\end{array}$

So, the null and alternative hypothesis can be stated as:

$\begin{array}{l}{H}_0:{\beta }_1=0\\ H_1:{\beta }_1\ne 0\end{array}$

(e)

To determine

To test: The least square regression analysis.

Answer to Problem 49E

Solution: The obtained output represents that the p-value is greater than 0.05. So there is no enough evidence for the linearity in the regression line.

Explanation of Solution

Calculation: Obtain the regression line using Minitab as follows:

Step 1: Enter the data in Minitab.

Step 2: Click on Stat --> Regression --> Regression.

Step 3: Double click on ‘IBI’ to move it response column and ‘Forest’ to move it to predictor column.

Step 4: Click ‘Ok’ to obtain the result.

Conclusion: From the obtained output the value of test statistic is 1.92 and the p-value is 0.061. Since the p-value is greater than 0.05, it can be concluded that there is no enough evidence for the linearity in the regression line

(f)

To determine

To find: The residuals.

Answer to Problem 49E

Solution: The residuals are as follows:

Forest	IBI	Residuals
0	47	-12.9072
0	76	16.0928
9	33	-28.2854
17	78	15.4895
25	62	-1.7355
33	78	13.0394
47	33	-34.1045
59	64	-4.9420
79	83	10.9953
95	67	-7.4548
0	61	1.0928
3	85	24.6334
10	46	-15.4386
17	53	-9.5105
31	55	-9.6543
39	71	5.1206
49	59	-8.4107
63	41	-28.5546
80	82	9.8422
95	56	-18.4548
0	39	-20.9072
3	89	28.6334
10	32	-29.4386
18	43	-19.6636
32	29	-35.8075
41	55	-11.1857
49	81	13.5893
68	82	11.6798
86	82	8.9234
100	85	9.7795
0	59	-0.9072
7	74	13.0208
11	80	18.4083
21	88	24.8770
33	29	-35.9606
43	58	-8.4919
52	71	3.1299
75	60	-11.3922
89	86	12.4640
100	91	15.7795
0	72	12.0928
8	89	27.8677
14	80	17.9489
22	84	20.7238
33	54	-10.9606
43	71	4.5081
52	75	7.1299
79	84	11.9953
90	79	5.3109

Explanation of Solution

Calculation: Obtain the regression line using Minitab as follows:

Step 1: Enter the data in Minitab.

Step 2: Click on Stat --> Regression --> Regression.

Step 3: Double click on ‘IBI’ to move it response column and ‘Forest’ to move it to predictor column.

Step 4: Click on ‘Storage’ and check the box for residuals.

Step 5: Click ‘Ok’ twice to obtain the result.

To determine

To graph: The scatterplot.

Explanation of Solution

Graph: Construct a scatterplot using Minitab as follows:

Step 1: Enter the data in Minitab.

Step 2: Click on Graph --> Scatterplot. Select scatterplot with regression.

Step 3: Double click on ‘Forest’ to move it X variable and ‘Residuals’ to move it to Y variable column.

Step 4: Click ‘Ok’ to obtain the graph.

The scatter plot is obtained as:

Interpretation: The graph shows that there is more variation for small $x$.

To determine

To explain: Whether there is something unusual.

Answer to Problem 49E

Solution: No, there is nothing unusual.

Explanation of Solution

The observations in the obtained residuals plot are scatters randomly over the line of origin. There is no pattern observed in the plot. So it can be concluded that the errors are independent.

(g)

To determine

To find: That residuals are normal or not.

Answer to Problem 49E

Solution: The residuals are approximately normally distributed.

Explanation of Solution

Construct the probability plot for residuals to test for the normality using Minitab as follows:

Step 1: Click on Stat --> Descriptive statistics --> Normality test.

Step 2: Double click on ‘Residuals’ to move it to the variable column.

Step 3: Click ‘OK’ to obtain the graph.

The graph is obtained.

Interpretation: All the points lie near the trend line. Therefore, it can be concluded that residuals are approximately normally distributed.

(h)

To determine

To explain: If the assumptions of statistical inference in satisfied or not.

Answer to Problem 49E

Solution: The assumptions are not reasonable.

Explanation of Solution

Consider the solutions of part (c). Since, the residuals are independent and normally distributed with 0 mean and unknown standard deviation. Therefore, residuals are not normal but left skewed. Thus, it can be concluded that the assumptions for statistical inference are not reasonably satisfied.