Question

the table shows the estimate of tax revenues, in billions of dollars, for a state each year. estimate the correlation coefficient of the data below.
year | 2008 | 2009 | 2010 | 2011 | 2012
tax revenue | 2.24 | 3.96 | 3.85 | 4.28 | 2.32
the correlation coefficient of the data is most likely closest to which of the following?
-0.946
0.078
0.912
-0.823

Explanation:

Step1: Assign Variables

Let \( x \) be the year (2008, 2009, 2010, 2011, 2012) and \( y \) be the tax revenue. We can represent the years as \( x = [0, 1, 2, 3, 4] \) (using 2008 as \( x = 0 \)) and \( y = [2.24, 3.96, 3.85, 4.28, 2.32] \).

Step2: Calculate Means

First, find the mean of \( x \), \( \bar{x} \):
\[ \bar{x} = \frac{0 + 1 + 2 + 3 + 4}{5} = \frac{10}{5} = 2 \]
Then, find the mean of \( y \), \( \bar{y} \):
\[ \bar{y} = \frac{2.24 + 3.96 + 3.85 + 4.28 + 2.32}{5} = \frac{16.65}{5} = 3.33 \]

Step3: Calculate Deviations and Products

For each data point, calculate \( (x_i - \bar{x}) \), \( (y_i - \bar{y}) \), and their product \( (x_i - \bar{x})(y_i - \bar{y}) \), and \( (x_i - \bar{x})^2 \), \( (y_i - \bar{y})^2 \):

- For \( x = 0, y = 2.24 \):
- \( (0 - 2) = -2 \)
- \( (2.24 - 3.33) = -1.09 \)
- Product: \( (-2)(-1.09) = 2.18 \)
- \( (x_i - \bar{x})^2 = (-2)^2 = 4 \)
- \( (y_i - \bar{y})^2 = (-1.09)^2 = 1.1881 \)

- For \( x = 1, y = 3.96 \):
- \( (1 - 2) = -1 \)
- \( (3.96 - 3.33) = 0.63 \)
- Product: \( (-1)(0.63) = -0.63 \)
- \( (x_i - \bar{x})^2 = (-1)^2 = 1 \)
- \( (y_i - \bar{y})^2 = (0.63)^2 = 0.3969 \)

- For \( x = 2, y = 3.85 \):
- \( (2 - 2) = 0 \)
- \( (3.85 - 3.33) = 0.52 \)
- Product: \( (0)(0.52) = 0 \)
- \( (x_i - \bar{x})^2 = 0^2 = 0 \)
- \( (y_i - \bar{y})^2 = (0.52)^2 = 0.2704 \)

- For \( x = 3, y = 4.28 \):
- \( (3 - 2) = 1 \)
- \( (4.28 - 3.33) = 0.95 \)
- Product: \( (1)(0.95) = 0.95 \)
- \( (x_i - \bar{x})^2 = 1^2 = 1 \)
- \( (y_i - \bar{y})^2 = (0.95)^2 = 0.9025 \)

- For \( x = 4, y = 2.32 \):
- \( (4 - 2) = 2 \)
- \( (2.32 - 3.33) = -1.01 \)
- Product: \( (2)(-1.01) = -2.02 \)
- \( (x_i - \bar{x})^2 = 2^2 = 4 \)
- \( (y_i - \bar{y})^2 = (-1.01)^2 = 1.0201 \)

Step4: Sum the Products and Squares

Sum of \( (x_i - \bar{x})(y_i - \bar{y}) \):
\[ 2.18 - 0.63 + 0 - 0.95 - 2.02 = -0.42 \]
Sum of \( (x_i - \bar{x})^2 \):
\[ 4 + 1 + 0 + 1 + 4 = 10 \]
Sum of \( (y_i - \bar{y})^2 \):
\[ 1.1881 + 0.3969 + 0.2704 + 0.9025 + 1.0201 = 3.778 \]

Step5: Calculate Correlation Coefficient

The formula for the correlation coefficient \( r \) is:
\[ r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}} \]
Substitute the values:
\[ r = \frac{-0.42}{\sqrt{10 \times 3.778}} = \frac{-0.42}{\sqrt{37.78}} \approx \frac{-0.42}{6.146} \approx -0.068 \]
Wait, this is close to 0.078. Wait, maybe my year assignment was off. Wait, maybe I should use the actual years as \( x \) values (2008, 2009, 2010, 2011, 2012) instead of 0 - 4. Let's recalculate with \( x = [2008, 2009, 2010, 2011, 2012] \).

Step1 (Revised): Assign Variables Correctly

Let \( x = [2008, 2009, 2010, 2011, 2012] \), \( y = [2.24, 3.96, 3.85, 4.28, 2.32] \)

Step2 (Revised): Calculate Means

\( \bar{x} = \frac{2008 + 2009 + 2010 + 2011 + 2012}{5} = \frac{10050}{5} = 2010 \)
\( \bar{y} = 3.33 \) (same as before)

Step3 (Revised): Deviations and Products

For \( x = 2008, y = 2.24 \):
- \( (2008 - 2010) = -2 \)
- \( (2.24 - 3.33) = -1.09 \)
- Product: \( (-2)(-1.09) = 2.18 \)
- \( (x_i - \bar{x})^2 = 4 \)
- \( (y_i - \bar{y})^2 = 1.1881 \)

For \( x = 2009, y = 3.96 \):
- \( (2009 - 2010) = -1 \)
- \( (3.96 - 3.33) = 0.63 \)
- Product: \( (-1)(0.63) = -0.63 \)
- \( (x_i - \bar{x})^2 = 1 \)
- \( (y_i - \bar{y})^2 = 0.3969 \)

For \( x = 2010, y = 3.85 \):
- \( (2010 - 2010) = 0 \)
- \( (3.85 - 3.33) = 0.52 \)
- Product: \( 0 \)
- \( (x_i - \bar{x})^2 = 0 \)
- \( (y_i - \bar{y})^2 = 0.2704 \)

For \( x = 2011, y = 4.28 \):
- \( (2011 - 2010) = 1 \)
- \( (4.28 - 3.33) = 0.95 \)
- Product: \( 0.95 \)
- \( (x_i - \bar{x})^2 = 1 \)
- \( (y_i - \bar{y})^2 = 0.9025 \)

For \( x = 2012, y = 2.32 \):
- \( (2012 - 2010) = 2 \)
- \( (2.32 - 3.33) = -1.01 \)
- Product: \( -2.02 \)
- \( (x_i - \bar{x})^2 = 4 \)
- \( (y_i - \bar{y})^2 = 1.0201 \)

Step4 (Revised): Sum the Products and Squares

Sum of \( (x_i - \bar{x})(y_i - \bar{y}) \): \( 2.18 - 0.63 + 0 + 0.95 - 2.02 = 0.48 \)? Wait, no, earlier calculation was wrong. Wait, 2.18 - 0.63 is 1.55, +0.95 is 2.5, -2.02 is 0.48. Wait, I must have miscalculated before. Wait, let's recalculate the sum:

2.18 (from 2008) + (-0.63) (2009) + 0 (2010) + 0.95 (2011) + (-2.02) (2012) = 2.18 - 0.63 = 1.55; 1.55 + 0 = 1.55; 1.55 + 0.95 = 2.5; 2.5 - 2.02 = 0.48.

Sum of \( (x_i - \bar{x})^2 \): 4 + 1 + 0 + 1 + 4 = 10 (same as before, since the differences are the same, just shifted by 2010).

Sum of \( (y_i - \bar{y})^2 \): 3.778 (same as before).

Step5 (Revised): Calculate Correlation Coefficient

\[ r = \frac{0.48}{\sqrt{10 \times 3.778}} = \frac{0.48}{\sqrt{37.78}} \approx \frac{0.48}{6.146} \approx 0.078 \]

Answer:

0.078