Question

find the correlation coefficient, r, of the data described below.
grape growers in france think they will have a massive crop this year and want to know how this will influence demand for grape juice.
they compared how many grapes were produced (in millions of metric tons), x, and the amount of french grape juice that was sold in previous years (in billions of liters), y.
round your answer to the nearest thousandth.
r =

Explanation:

Step1: Calculate necessary sums

First, we list the data points:
\(x\): \(3.31, 3.38, 3.40, 3.88, 4.92, 5.52\)
\(y\): \(5.29, 5.48, 5.41, 5.56, 5.65, 5.93\)

Calculate \(n = 6\) (number of data points).

Calculate \(\sum x\):
\(3.31 + 3.38 + 3.40 + 3.88 + 4.92 + 5.52 = 24.41\)

Calculate \(\sum y\):
\(5.29 + 5.48 + 5.41 + 5.56 + 5.65 + 5.93 = 33.32\)

Calculate \(\sum xy\):
\((3.31\times5.29)+(3.38\times5.48)+(3.40\times5.41)+(3.88\times5.56)+(4.92\times5.65)+(5.52\times5.93)\)
\(= 17.5099 + 18.5224 + 18.394 + 21.5728 + 27.798 + 32.7336 = 136.5307\)

Calculate \(\sum x^2\):
\(3.31^2 + 3.38^2 + 3.40^2 + 3.88^2 + 4.92^2 + 5.52^2\)
\(= 10.9561 + 11.4244 + 11.56 + 15.0544 + 24.2064 + 30.4704 = 103.6717\)

Calculate \(\sum y^2\):
\(5.29^2 + 5.48^2 + 5.41^2 + 5.56^2 + 5.65^2 + 5.93^2\)
\(= 27.9841 + 30.0304 + 29.2681 + 30.9136 + 31.9225 + 35.1649 = 185.2836\)

Step2: Use the correlation coefficient formula

The formula for the correlation coefficient \(r\) is:

$$ r = \frac{n\sum xy - \sum x \sum y}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} $$

First, calculate the numerator:
\(n\sum xy - \sum x \sum y = 6\times136.5307 - 24.41\times33.32\)
\(= 819.1842 - 813.3412 = 5.843\)

Calculate the denominator's first part: \(n\sum x^2 - (\sum x)^2 = 6\times103.6717 - (24.41)^2\)
\(= 622.0302 - 595.8481 = 26.1821\)

Calculate the denominator's second part: \(n\sum y^2 - (\sum y)^2 = 6\times185.2836 - (33.32)^2\)
\(= 1111.7016 - 1109.2224 = 2.4792\)

Multiply the two parts of the denominator: \(26.1821\times2.4792 \approx 64.91\)

Take the square root: \(\sqrt{64.91} \approx 8.0567\)

Now, calculate \(r\):
\(r = \frac{5.843}{8.0567} \approx 0.725\)

Answer:

\(0.725\)