Chapter 4.2, Problem 13P

Incoine: Medicai Care Let x be per capita income in thousands of dollars. Let y be the number of medical doctors per 10,000 residents. Six small cities in Oregon gave the following information about x and y(based on information from Lifein America's Small Cities by G. S. Thomas, Prometheus Books).

x	8.6	9.3	10.1	8.0	8.3	8.7
y	9.6	18.5	20.9	10.2	11.4	13.1

Complete parts (a) through (e), given $\sum x=53,\sum y=83.7,\sum x^2=471.04,\sum y^2=1276.83,\sum xy=755.89,\text{ and }r\approx \mathrm{0.934.}$ (f) Suppose a small city in Oregon has a per capita income of 10 thousand dollars. What is the predicted number of MDs per 10,000 residents?

To determine

To graph: The scatter diagram.

Explanation of Solution

Given: The data that consists of the variables ‘per capita income in thousands of dollars’ and ‘the number of medical doctors per 10,000 residents’, which are represented by x and y, respectively, are provided.

Graph:

Follow the steps given below in MS Excel to obtain the scatter diagram of the data.

Step 1: Enter the data into an MS Excel sheet. The screenshot is given below.

Step 2: Select the data and click on ‘Insert’. Go to ‘charts’ and select ‘Scatter’ as the chart type.

Step 3: Select the first plot and click the ‘add chart element’ option provided in the left-hand corner of the menu bar. Insert the ‘Axis titles’ and the ‘Chart title’. The scatter plot for the provided data is shown below.

Interpretation: The scatterplot shows that the correlation between the per capita income (x) and the number of medical doctors (y) is positive. So, as x increases (or decreases), the value of y increases (or decreases).

To determine

To test: Whether the provided values of ${\displaystyle \sum x}$, ${\displaystyle \sum y}$, ${\displaystyle \sum x^2}$, ${\displaystyle \sum y^2}$, ${\displaystyle \sum xy}$ and $r$ are correct or not.

Answer to Problem 13P

Solution: The provided values, that is, ${\displaystyle \sum x}=53$, ${\displaystyle \sum y=83.7}$, ${\displaystyle \sum x^2=471.04}$, ${\displaystyle \sum y^2}=1276.83$, ${\displaystyle \sum xy}=755.89$ and $r\approx 0.934$ are correct.

Explanation of Solution

Given: The provided values are ${\displaystyle \sum x}=53$, ${\displaystyle \sum y=83.7}$, ${\displaystyle \sum x^2=471.04}$, ${\displaystyle \sum y^2}=1276.83$ ${\displaystyle \sum xy}=755.89$ and $r\approx 0.934$

Calculation:

To compute ${\displaystyle \sum x}$, ${\displaystyle \sum y}$, ${\displaystyle \sum x^2}$ $,{\displaystyle \sum y^2}$, and ${\displaystyle \sum xy}$, it is easy to organize the data into a table of five columns and then add the values in each column. The table is given below.

$x$	$y$	$x^2$	$y^2$	$xy$
8.6	9.6	73.96	92.16	82.56
9.3	18.5	86.49	342.25	172.05
10.1	20.9	102.01	436.81	211.09
8	10.2	64	104.04	81.6
8.3	11.4	68.89	129.96	94.62
8.7	13.1	75.69	171.61	113.97
${\displaystyle \sum x=53}$	${\displaystyle \sum y=83.7}$	${\displaystyle \sum x^2=471.04}$	${\displaystyle \sum y^2=1276.83}$	${\displaystyle \sum xy=755.89}$

Now, the value of $r$ can be calculated by using the formula below.

$r=\frac{n{\displaystyle \sum xy\text{-}\left({\displaystyle \sum x}\right)\left({\displaystyle \sum y}\right)}}{\sqrt{n{\displaystyle \sum x^2}-{\left({\displaystyle \sum x}\right)}^2}\sqrt{n{\displaystyle \sum y^2}-{\left({\displaystyle \sum y}\right)}^2}}$

Substitute the values in the above formula. Thus:

$\begin{array}{c}r=\frac{6\left(755.89\right)-\left(53\right)\left(83.7\right)}{\sqrt{\left(6\right)\left(471.04\right)-{\left(53\right)}^2}\sqrt{\left(6\right)\left(1276.83\right)-{\left(83.7\right)}^2}}\\ =\frac{4535.34-4436.1}{\sqrt{91}\sqrt{1444}}\\ =\frac{-358}{362.5}\\ \approx 0.934\end{array}$

Thus, the value of $r=0.934$.

Conclusion: The provided values, that is, ${\displaystyle \sum x}=53$, ${\displaystyle \sum y=83.7}$, ${\displaystyle \sum x^2=471.04}$, ${\displaystyle \sum y^2}=1276.83$, ${\displaystyle \sum xy}=755.89$ and $r\approx 0.934$ have been verified.

To determine

To find: The values of $\overline{x},\overline{y}$, a, b and the equation of the least-squares line.

Answer to Problem 13P

Solution: The calculated values are $\overline{x}=8.83,\overline{y}=13.95,a=-36.898$ and $b\approx 5.756$. The equation of the least-squares line is:

$\widehat{y}=-36.898+5.756x$.

Explanation of Solution

Given: The provided values are ${\displaystyle \sum x}=53$, ${\displaystyle \sum y=83.7}$, ${\displaystyle \sum x^2=471.04}$, ${\displaystyle \sum y^2}=1276.83$ and ${\displaystyle \sum xy}=755.89$. The number of pairs of (x,y) in the data set is $n=6$.

Calculation:

The value of $\overline{x}$ can be calculated as follows:

$\begin{array}{c}\overline{x}=\frac{{\displaystyle \sum x}}{n}\\ =\frac{53}{6}\\ =\$8.83\text{ thousand}\end{array}$

The value of $\overline{y}$ can be calculated as follows:

$\begin{array}{c}\overline{y}=\frac{{\displaystyle \sum y}}{n}\\ =\frac{83.7}{6}\\ =13.95\text{ physicians per 10,000}\end{array}$

The value of $b$ can be calculated as follows:

$\begin{array}{c}b=\frac{n{\displaystyle \sum xy}-\left({\displaystyle \sum x}\right)\left({\displaystyle \sum y}\right)}{n{\displaystyle \sum x^2-{\left({\displaystyle \sum x}\right)}^2}}\\ =\frac{6\left(755.89\right)-\left(53\right)\left(83.7\right)}{6\left(471.04\right)-{\left(53\right)}^2}\\ =\frac{99.24}{17.24}\\ \approx 5.756\end{array}$

The value of $a$ can be calculated as follows:

$\begin{array}{c}a=\overline{y}-b\overline{x}\\ =13.95-\left(5.756\times 8.83\right)\\ =13.95-50.8254\\ \approx -36.898\end{array}$

Therefore, the values are $\overline{x}=8.83,\overline{y}=13.95,a=-36.898$ and $b\approx 5.756$.

The general formula of a least-squares line is:

$\widehat{y}=a+bx$

Here, a is the y-intercept and b is the slope.

Substitute the values of a and b in the general equation to get the equation of the least-squares line of the data as follows:

$\begin{array}{c}\widehat{y}=a+bx\\ =-36.898+\left(5.756\right)x\\ =5.756x-36.898\end{array}$

Therefore, the least-squares line equation is $\widehat{y}=-36.898+5.756x$.

To determine

To graph: The least-squares line on the scatter diagram that passes through the point $\left(\overline{x},\overline{y}\right)$.

Explanation of Solution

Given: The data that consists of the variables ‘per capita income’ and ‘the number of medical doctors per 10,000 residents’, which are represented by x and y, respectively, are provided.

Graph:

Follow the steps given below in MS Excel to obtain the scatter diagram of the data.

Step 1: Enter the data into an MS Excel sheet. The screenshot is given below.

Step 2: Select the data and click on ‘Insert’. Go to ‘charts’ and select ‘Scatter’ as the chart type.

Step 3: Select the first plot and click the ‘add chart element’ option provided in the left-hand corner of the menu bar. Insert the ‘Axis titles’ and the ‘Chart title’. The scatter plot for the provided data is shown below.

Step 4: Right click on any data point and select ‘Add Trendline’. In the dialogue box, select ‘linear’ and check ‘Display Equation on Chart’. The scatter diagram with the least-squares line is given below.

Interpretation: The least-squares line passes through the point $\left(\overline{x},\overline{y}\right)=\left(8.83,13.95\right)$ and the scatterplot displays the equation of the least-squares line as $\widehat{y}=-36.898+5.756x$.

To determine

The value of $r^2$ and the percentage of variation in y that can be explained as well as the one that cannot.

Answer to Problem 13P

Solution: The value of $r^2$ is 0.872. The percentage of variation in the number of medical doctors per 10,000 residents can be explained by the corresponding variation in the per capita income in thousands of dollars using the least-squares line, which is 87.2%, while the remaining 12.8% cannot be explained.

Explanation of Solution

Given: The value of the correlation coefficient (r) is $0.934$.

Calculation: The coefficient of determination $\left(r^2\right)$ can be calculated as:

$\begin{array}{c}{r}^2={\left(r\right)}^2\\ ={\left(0.934\right)}^2\\ \approx 0.872\end{array}$

Therefore, the value of $r^2$ is 0.872.

$r^2$ indicates the proportion of the total variation in y that can be explained by using the least-squares line. So, the percentage of variation in y that can be explained is 87.2%.

Further, the proportion of variation in y that cannot be explained can be calculated as:

$\begin{array}{c}1-r^2=1-0.872\\ =0.128\\ =12.8\%\end{array}$

Hence, the percentage of variation in y that cannot be explained is 12.8%.

Interpretation: About 87.2% of the variation in y can be explained by the corresponding variation in x and the least-squares line while the remaining 12.8% of variation cannot be explained.

To determine

To find: The predicted number of MDs (medical doctors) per 10,000 residents.

Answer to Problem 13P

Solution: The predicted value is 20.7 physicians per 10,000 residents.

Explanation of Solution

Given: The least-squares line from part (c) is $\widehat{y}=-36.898+5.756x$.

Calculation:

The predicted value $\left(\widehat{y}\right)$ for $x=10$ can be calculated as follows:

$\begin{array}{c}\widehat{y}=-36.898+5.756x\\ =-36.898+5.756\left(10\right)\\ =-36.898+57.56\\ \approx 20.7\end{array}$

Thus, the value of $\widehat{y}\approx 20.7$

Interpretation: The predicted number of medical doctors per 10,000 residents for a city with a per capita income of 10 thousand dollars is 20.7.