Question

the table shows the number of flowers in four bouquets and the total cost of each bouquet.
cost of bouquets
number of flowers in the bouquet | total cost
8 | $12
12 | $40
6 | $15
20 | $20
what is the correlation coefficient for the data in the table?
-0.57
-0.28
0.28
0.57

Explanation:

Step1: Identify variables

Let \( x \) be the number of flowers (8, 12, 6, 20) and \( y \) be the total cost (\( \$12, \$40, \$15, \$20 \)).

Step2: Calculate means

\( \bar{x} = \frac{8 + 12 + 6 + 20}{4} = \frac{46}{4} = 11.5 \)
\( \bar{y} = \frac{12 + 40 + 15 + 20}{4} = \frac{87}{4} = 21.75 \)

Step3: Calculate deviations and products

For each pair \((x_i, y_i)\):

• \( (8, 12) \): \( (8 - 11.5) = -3.5 \), \( (12 - 21.75) = -9.75 \), product \( (-3.5)(-9.75) = 34.125 \)
• \( (12, 40) \): \( (12 - 11.5) = 0.5 \), \( (40 - 21.75) = 18.25 \), product \( (0.5)(18.25) = 9.125 \)
• \( (6, 15) \): \( (6 - 11.5) = -5.5 \), \( (15 - 21.75) = -6.75 \), product \( (-5.5)(-6.75) = 37.125 \)
• \( (20, 20) \): \( (20 - 11.5) = 8.5 \), \( (20 - 21.75) = -1.75 \), product \( (8.5)(-1.75) = -14.875 \)

Sum of products: \( 34.125 + 9.125 + 37.125 - 14.875 = 65.5 \)

Step4: Calculate standard deviations

For \( x \):
\( \sum (x_i - \bar{x})^2 = (-3.5)^2 + 0.5^2 + (-5.5)^2 + 8.5^2 = 12.25 + 0.25 + 30.25 + 72.25 = 115 \)
\( s_x = \sqrt{\frac{115}{3}} \approx \sqrt{38.333} \approx 6.191 \)

For \( y \):
\( \sum (y_i - \bar{y})^2 = (-9.75)^2 + 18.25^2 + (-6.75)^2 + (-1.75)^2 = 95.0625 + 333.0625 + 45.5625 + 3.0625 = 476.75 \)
\( s_y = \sqrt{\frac{476.75}{3}} \approx \sqrt{158.9167} \approx 12.606 \)

Step5: Calculate correlation coefficient

\( r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}} = \frac{65.5}{\sqrt{115 \times 476.75}} \)
\( \sqrt{115 \times 476.75} \approx \sqrt{54826.25} \approx 234.15 \)
\( r \approx \frac{65.5}{234.15} \approx 0.28 \)

Answer:

0.28