Question

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.

Answer 1

Answer:

The percentage of variation esplained by the model is given by the determination coefficient, on this case:

$R^2 = 0.934^2 =0.872$

And we have 87.2% of the variation explained by the linear model given.

$\hat y = 5.756(8.5) -36.895=12.031$

And we have 12.031 doctors per 10000 residents.

Explanation:

Assuming the following dataset:

x                 y

8.6           9.6

9.3           18.5

10.1          20.9

8.0           10.2

8.3           11.4

8.7            13.1

Assuming this question: "The data has a correlation coefficient of r = 0.934. Calculate the regression line for this  data. What percentage ofvariation is explained by the regression line? Predict the number of doctors per 10,000 residents in a town with a per capita income of $8500."

We want a linear model like this:

$y = mx +b$

Where m represent the slope and b the intercept for the linear model. And we cna find the slope and b with the following formulas:

$m = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 -(\sum x)^2}$

$b = \frac{\sum y}{n} -m \frac{\sum x}{n}$

And from the dataset we have the following values:

$n= 6, \sum x =53, \sum y = 83.7 , \sum xy = 755.89, \sum x^2 = 471.04$

And replacing into the equation for m we got:

$m =\frac{6(755.89) - (53)(83.7)}{6(471.04) -(53)^2}=5.756$

And the intercept:

$b = \frac{83.7}{6}-36.895 5.756 \frac{53}{6}=-36.895$

And then the linear model is given by:

$\hat y = 5.756 x -36.895$

We can find the estimation replacing x = 8.5 into the linear model and we got:

$\hat y = 5.756(8.5) -36.895=12.031$

And we have 12.031 doctors per 10000 residents.

The percentage of variation esplained by the model is given by the determination coefficient, on this case:

$R^2 = 0.934^2 =0.872$

And we have 87.2% of the variation explained by the linear model given.