Question

(3) 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.
 x 8.2 9.2 10.2 8 8.3 8.7 y 9.9 18.2 21 10.2 11.4 13.1
Complete parts (a) through (e), given Σx = 52.6, Σy = 83.8, Σx2 = 464.5, Σy2 = 1275.86, Σxy = 753.01, and r ≈ 0.974.

(a) Find x, and y. Then find the equation of the least-squares line  = a + bx. (Round your answers for x and y to two decimal places. Round your answers for a and b to three decimal places.)
 x = y = = +  x

(b) Find the value of the coefficient of determination r2. What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line? What percentage is unexplained? (Round your answer for r2 to three decimal places. Round your answers for the percentages to one decimal place.)
 r2 = explained % unexplained %

(c) Suppose a small city in Oregon has a per capita income of 9.3 thousand dollars. What is the predicted number of M.D.s per 10,000 residents? (Round your answer to two decimal places.)
M.D.s per 10,000 residents
Step 1

(a) Least square regression line:

The least squares regression line can be obtained using EXCEL. Enter the values of x and y in two columns, say in columns A and B in an EXCEL sheet, with the first cell containing the data labels, x and y. Then, follow the procedure given below:

• Go to Data > Data Analysis > Regression.
• Enter Input Y Range as \$B\$2:\$B\$9 and Input X Range as \$A\$2:\$A\$9.
• Click OK.

The output is as follows:

Step 2

Now, b is the slope of the regression line, which is approximately 5.44 and, a is the “Intercept”, which is approximately -33.756 (rounded to three decimal places).

Therefore, the equation of the least-squares line is = -33.756 +5.44 x.

Step 3

(b) The value of coefficient of determination:

From the EXCEL output in Part (a), the value of coefficient of determination is R-square is 0.948. That is, 94.8% of the va...

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