Chapter 10.1, Problem 19E

(a)

Section 1:

To determine

To graph: A 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 ‘Number of tornadoes’ to move it to Y variable and ‘Year’ to move it to X variable column.

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

Section 2:

To determine

To explain: The relationship between two variables seems linear.

Answer to Problem 19E

Solution: The variables are in linear relationship.

Explanation of Solution

From the graph obtained in the above section 1, it can be seen that all the points lie near the trend line. So, it can be concluded that the variables are linearly related.

Section 3:

To determine

To explain: Whether there are any outliers or unusual pattern.

Answer to Problem 19E

Solution: There are no outliers.

Explanation of Solution

Outlier is a value that seems to be out on their own from the rest of the data. From the graph obtained in the above section 1, it can be seen that there are no outliers or any unusual patterns in the scatterplot.

(b)

To determine

To find: The least square regression line.

Answer to Problem 19E

Solution: The regression line is:

$\text{Number of tornadoes}=-24517+12.8\text{Year}$

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 ‘Number of tornadoes’ to move it to Y variable and ‘Year’ to move it to X variable column.

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

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

Hence, the obtained regression line is

$\text{Number of tornadoes}=-24517+12.8\text{Year}$

(c)

To determine

To explain: The intercept of regression equation.

Answer to Problem 19E

Solution: The fit is correct because intercept is only used when $x=0$.

Explanation of Solution

The fit is correct because the intercept only describes the model when $x=0$ that in this case is far outside the range of the provided data of annual number of tornadoes in the United States between 1953 and 2014.

(d)

Section 1:

To determine

To find: The residuals.

Answer to Problem 19E

Solution: The residuals are as follows:

Years	Number of tornadoes	Residuals
1953	421	-127.661
1954	550	-11.495
1955	593	18.670
1956	504	-83.164
1957	856	256.001
1958	564	-48.833
1959	604	-21.668
1960	616	-22.502
1961	697	45.663
1962	657	-7.171
1963	464	-213.006
1964	704	14.160
1965	906	203.325
1966	585	-130.509
1967	926	197.656
1968	660	-81.178
1969	608	-146.013
1970	653	-113.847
1971	888	108.318
1972	741	-51.516
1973	1102	296.649
1974	947	128.815
1975	920	88.980
1976	835	-8.854
1977	852	-4.689
1978	788	-81.523
1979	852	-30.358
1980	866	-29.192
1981	783	-125.027
1982	1046	125.139
1983	931	-2.696
1984	907	-39.530
1985	684	-275.365
1986	764	-208.199
1987	656	-329.034
1988	702	-295.868
1989	856	-154.703
1990	1133	109.463
1991	1132	95.628
1992	1298	248.794
1993	1176	113.959
1994	1082	7.125
1995	1235	147.290
1996	1173	72.456
1997	1148	34.621
1998	1449	322.787
1999	1340	200.952
2000	1075	-76.882
2001	1215	50.283
2002	934	-243.551
2003	1374	183.614
2004	1817	613.780
2005	1265	48.945
2006	1103	-125.889
2007	1096	-145.724
2008	1692	437.442
2009	1156	-111.393
2010	1282	1.773
2011	1691	397.938
2012	938	-367.896
2013	907	-411.731
2014	888	-443.565

Explanation of Solution

Calculation: Obtain the residuals using Minitab as follows:

Step 1: Enter the data in Minitab.

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

Step 3: Double click on ‘Number of tornadoes’ to move it response column and ‘Year’ to move it to predictor column.

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

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

Hence, the obtained residuals are shown below:

Years	Number of tornadoes	Residuals
1953	421	-127.661
1954	550	-11.495
1955	593	18.670
1956	504	-83.164
1957	856	256.001
1958	564	-48.833
1959	604	-21.668
1960	616	-22.502
1961	697	45.663
1962	657	-7.171
1963	464	-213.006
1964	704	14.160
1965	906	203.325
1966	585	-130.509
1967	926	197.656
1968	660	-81.178
1969	608	-146.013
1970	653	-113.847
1971	888	108.318
1972	741	-51.516
1973	1102	296.649
1974	947	128.815
1975	920	88.980
1976	835	-8.854
1977	852	-4.689
1978	788	-81.523
1979	852	-30.358
1980	866	-29.192
1981	783	-125.027
1982	1046	125.139
1983	931	-2.696
1984	907	-39.530
1985	684	-275.365
1986	764	-208.199
1987	656	-329.034
1988	702	-295.868
1989	856	-154.703
1990	1133	109.463
1991	1132	95.628
1992	1298	248.794
1993	1176	113.959
1994	1082	7.125
1995	1235	147.290
1996	1173	72.456
1997	1148	34.621
1998	1449	322.787
1999	1340	200.952
2000	1075	-76.882
2001	1215	50.283
2002	934	-243.551
2003	1374	183.614
2004	1817	613.780
2005	1265	48.945
2006	1103	-125.889
2007	1096	-145.724
2008	1692	437.442
2009	1156	-111.393
2010	1282	1.773
2011	1691	397.938
2012	938	-367.896
2013	907	-411.731
2014	888	-443.565

Section 2:

To determine

To graph: The scatterplot of residual versus year.

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 ‘Residuals’ to move it to Y variable and ‘Year’ to move it to X variable column.

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

From the scatter plot, it is clear that the residual plot looks mostly random

(e)

To determine

To explain: That residuals are normal or not.

Answer to Problem 19E

Solution: The residuals are normally distributed.

Explanation of Solution

Graph: 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.

Hence, the obtained graph is shown below:

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

(f)

To determine

To explain: The accuracy of inference based on residual checks.

Answer to Problem 19E

Solution: The inference can be made based on residuals.

Explanation of Solution

From the graph obtained in part (e), it is clear that the residuals of the regression follow normal distribution, so it satisfies the assumption of regression equation. Therefore, inference can be made based on residual checks.