Question

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

Explanation:

Step 1: Define variables

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

Step 2: Calculate means

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

Step 3: Calculate deviations and products

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

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

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

Step 4: Calculate standard deviations

For \( x \):
\( \sum (x_i - \bar{x})^2 = (-3.5)^2 + (0.5)^2 + (-5.5)^2 + (8.5)^2 = 12.25 + 0.25 + 30.25 + 72.25 = 115 \)
\( s_x = \sqrt{\frac{115}{3}} \approx 6.14 \)

For \( y \):
\( \sum (y_i - \bar{y})^2 = (-9.75)^2 + (18.25)^2 + (-6.75)^2 + (-1.75)^2 = 95.0625 + 333.0625 + 45.5625 + 3.0625 = 476.75 \)
\( s_y = \sqrt{\frac{476.75}{3}} \approx 12.58 \)

Step 5: Calculate correlation coefficient

\( r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{(n - 1)s_x s_y} \) (using sample standard deviation, \( n = 4 \))
\( r = \frac{65.5}{3 \times 6.14 \times 12.58} \approx \frac{65.5}{237.4} \approx 0.28 \)

Answer:

0.28