Chapter 4.3, Problem 26E

Imports and exports: The following table presents the U.S. imports and exports (in billions of dollars) for each of 29 months.

1. Compute the least-squares regression line for predicting exports (y) from imports (x).
2. Compute the coefficient of determination.
3. The months two lowest exports are months 1 and 2. when the exports were 168.1 and 166.6, respectively. Remove these points and compute the least-squares regression line. Is die result noticeably different?
4. Compute the coefficient of determination for the data set months 1 and 2 removed.
5. Two economists decide to study the relationship between imports and exports. One uses data from months 1 through 29 and the other uses data from months 3 through 29. For which data set will the proportion of variance explained by the least-squares regression line be greater?

To determine

To find: The least-square regression line for the given data set.

Answer to Problem 26E

The least square regression line of the given data set is,

  $\widehat{y}=1063.2+0.3599x$

Explanation of Solution

Given:

The import and exports of United States within $29$ months are given in the following table. All the values are in billions of dollars.

  $\begin{array}{cccccc}\text{Month}& \text{Imports}& \text{Exports}& \text{Month}& \text{Imports}& \text{Exports}\\ 1& 215.9& 168.1& 16& 230.9& 184.3\\ 2& 211.8& 166.6& 17& 230.5& 184.2\\ 3& 217.7& 174.3& 18& 227.6& 185.2\\ 4& 218.1& 175.9& 19& 226.8& 183.4\\ 5& 223.6& 176.2& 20& 226.1& 182.1\\ 6& 224.2& 173.2& 21& 228.4& 186.8\\ 7& 224.9& 179.5& 22& 225.3& 182.7\\ 8& 224.6& 179.9& 23& 231.6& 185.2\\ 9& 225.7& 181.2& 24& 227.0& 188.7\\ 10& 226.6& 180.5& 25& 229.4& 186.7\\ 11& 226.1& 178.3& 26& 231.0& 187.1\\ 12& 230.5& 179.1& 27& 222.3& 185.2\\ 13& 230.9& 179.5& 28& 227.7& 187.6\\ 14& 225.8& 182.1& 29& 232.1& 187.1\\ 15& 234.3& 186.5& & & \end{array}$

Calculation:

The least-square regression is given by the formula,

  $\widehat{y}=b_0+b_{1}x$

Where $b_1=r\frac{s_y}{s_x}$ and $b_0=\overline{y}-b_1\overline{x}$

  $r$ is the correlation coefficient.

  $s_x$ is the standard deviation of $x$ .

  $s_y$ is the standard deviation of $y$ .

The correlation coefficient is given by the formula,

  $r=\frac{1}{n-1}{\displaystyle \sum \left(\frac{x-\overline{x}}{s_x}\right)\left(\frac{y-\overline{y}}{s_y}\right)}$

Considering the exports as $y$ and the imports as $x$ , the following statistics should be calculated for the calculations of slope and the intercept. Here MINITAB is used.

  

The correlation coefficient can be obtained by the following table.

x	y	Zx	Zy	ZxZy
215.9	168.1	-1.99175	-2.32277	4.626396
211.8	166.6	-2.79101	-2.58707	7.220549
217.7	174.3	-1.64086	-1.23035	2.018834
218.1	175.9	-1.56288	-0.94844	1.482294
223.6	176.2	-0.49071	-0.89558	0.439465
224.2	173.2	-0.37374	-1.42417	0.532271
224.9	179.5	-0.23728	-0.31412	0.074536
224.6	179.9	-0.29576	-0.24365	0.072062
225.7	181.2	-0.08133	-0.01459	0.001187
226.6	180.5	0.094118	-0.13793	-0.01298
226.1	178.3	-0.00335	-0.52556	0.001762
230.5	179.1	0.854389	-0.3846	-0.3286
230.9	179.5	0.932365	-0.31412	-0.29288
225.8	182.1	-0.06184	0.143989	-0.0089
234.3	186.5	1.595165	0.919257	1.466367
230.9	184.3	0.932365	0.531623	0.495667
230.5	184.2	0.854389	0.514003	0.439158
227.6	185.2	0.289059	0.6902	0.199509
226.8	183.4	0.133106	0.373045	0.049655
226.1	182.1	-0.00335	0.143989	-0.00048
228.4	186.8	0.445012	0.972116	0.432603
225.3	182.7	-0.15931	0.249707	-0.03978
231.6	185.2	1.068824	0.6902	0.737703
227	188.7	0.172094	1.306891	0.224908
229.4	186.7	0.639953	0.954496	0.610833
231	187.1	0.951859	1.024975	0.975632
222.3	185.2	-0.74413	0.6902	-0.5136
227.7	187.6	0.308553	1.113074	0.343442
232.1	187.1	1.166295	1.024975	1.195423

The sum of $Z_x{Z}_y$ can be obtained by,

  $22.4403$

Hence, the correlation coefficient is,

  $\begin{array}{c}r=\frac{1}{29-1}\times 22.4403\\ =\frac{22.4403}{28}\\ r=0.8015\end{array}$

Then, the coefficient $b_1$ should be,

  $\begin{array}{c}{b}_1=r\frac{s_y}{s_x}\\ =0.8015\times \frac{5.6755}{5.1298}\\ b_1=0.8868\end{array}$

Therefore,

  $\begin{array}{c}{b}_0=\overline{y}-b_1\overline{x}\\ =181.2828-0.8868\times 226.1172\\ =181.2828-200.5217\\ b_0=-19.239\end{array}$

Conclusion:

The least square regression line is found to be,

  $\widehat{y}=-19.239+0.8868x$

To determine

To calculate: The coefficient of determination.

Answer to Problem 26E

The coefficient of determination is found to be,

  $0.6425$

Explanation of Solution

Calculation:

The correlation coefficient $r$ has been computed in the part (a) as,

  $r=0.8015$

The coefficient of determination is calculated by taking the square of the correlation coefficient.

Therefore, the coefficient of determination should be,

  $r^2={\left(0.8015\right)}^2$

Simplifying the square,

  $\begin{array}{c}{r}^2=0.8015\times 0.8015\\ =0.6425\end{array}$

Conclusion:

Therefore, the coefficient of determination is found to be $0.6425$ .

To determine

To find:The least-square regression line without considering the lowest two exports.

Answer to Problem 26E

The least square regression line of the given data set is,

  $\widehat{y}=23.6335+0.6990x$

Explanation of Solution

Calculation:

From the all $29$ months, first and the second months have the minimum exports. Without considering these two data, there are $27$ ordered pairs for imports and exports.

The statistics should be calculated again for the current data set, imports and exports of last $27$ months.

  

The calculations can be completed using a table.

x	y	Zx	Zy	ZxZy
217.7	174.3	-2.36254	-1.85805	4.389725
218.1	175.9	-2.26121	-1.48713	3.362707
223.6	176.2	-0.86789	-1.41758	1.230299
224.2	173.2	-0.71589	-2.11306	1.512718
224.9	179.5	-0.53856	-0.65255	0.351434
224.6	179.9	-0.61456	-0.55982	0.344039
225.7	181.2	-0.33589	-0.25844	0.086808
226.6	180.5	-0.10789	-0.42072	0.045393
226.1	178.3	-0.23456	-0.93074	0.218314
230.5	179.1	0.880098	-0.74528	-0.65592
230.9	179.5	0.98143	-0.65255	-0.64043
225.8	182.1	-0.31056	-0.0498	0.015465
234.3	186.5	1.842756	0.970245	1.787925
230.9	184.3	0.98143	0.460224	0.451678
230.5	184.2	0.880098	0.437042	0.384639
227.6	185.2	0.145437	0.668869	0.097279
226.8	183.4	-0.05723	0.251579	-0.0144
226.1	182.1	-0.23456	-0.0498	0.01168
228.4	186.8	0.348102	1.039794	0.361955
225.3	182.7	-0.43722	0.0893	-0.03904
231.6	185.2	1.158762	0.668869	0.77506
227	188.7	-0.00656	1.480266	-0.00971
229.4	186.7	0.601433	1.016611	0.611424
231	187.1	1.006763	1.109342	1.116845
222.3	185.2	-1.19722	0.668869	-0.80078
227.7	187.6	0.170771	1.225256	0.209238
232.1	187.1	1.285427	1.109342	1.425979

The sum the values in the right most column is found to be,

  $16.6303$

Hence, the correlation coefficient is,

  $\begin{array}{c}r=\frac{1}{27-1}\times 16.6303\\ =\frac{16.6303}{26}\\ r=0.6396\end{array}$

Then, the coefficient $b_1$ should be,

  $\begin{array}{c}{b}_1=r\frac{s_y}{s_x}\\ =0.6396\times \frac{4.3135}{3.943}\\ b_1=0.6990\end{array}$

Therefore,

  $\begin{array}{c}{b}_0=\overline{y}-b_1\overline{x}\\ =182.3148-0.6990\times 227.0259\\ =182.3148-158.6813\\ b_0=23.6335\end{array}$

Conclusion:

The least square regression line is found to be,

  $\widehat{y}=23.6335+0.6990x$

To determine

To calculate: The coefficient of determination for the last $27$ months.

Answer to Problem 26E

The coefficient of determination is found to be,

  $0.4091$

Explanation of Solution

Calculation:

The correlation coefficient $r$ without considering the first two months has been computed in the part (a) as,

  $r=0.6396$

The coefficient of determination is calculated by taking the square of the correlation coefficient.

  $r^2={\left(0.6396\right)}^2$

Simplifying the square,

  $\begin{array}{c}{r}^2=0.6396\times 0.6396\\ =0.4091\end{array}$

Conclusion:

Therefore, the coefficient of determination is found to be $0.4091$ .

To determine

To find: The data set which

Answer to Problem 26E

The data set with all $29$ months has greater proportion of explained variance.

Explanation of Solution

Calculation:

The coefficient of determination for the all $29$ months has been calculated in the part(a) as, $0.6425$ .

Then, the coefficient of determination for the $27$ months without considering the first two months that have minimum exports has been calculated in the part (d) as, $0.4091$ .

The coefficient of determination denotes the proportion of variance that is explained by the least-square regression line.

For the all $29$ months, $64.25\%$ of variance is explained by the least-square regression line while only $40.91\%$ of variance of the $27$ months is explained.

Conclusion:

Since $64.25\%>40.91\%$ , the data set with all $29$ months has greater proportion of explained variance than the data set with $27$ months excluding first two months.