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 0.57 0.28

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

| \( x \) | \( y \) | \( x - \bar{x} \) | \( y - \bar{y} \) | \( (x - \bar{x})(y - \bar{y}) \) | \( (x - \bar{x})^2 \) | \( (y - \bar{y})^2 \) |
|--------|--------|------------------|------------------|---------------------------------|----------------------|----------------------|
| 8 | 12 | -3.5 | -9.75 | 34.125 | 12.25 | 95.0625 |
| 12 | 40 | 0.5 | 18.25 | 9.125 | 0.25 | 333.0625 |
| 6 | 15 | -5.5 | -6.75 | 37.125 | 30.25 | 45.5625 |
| 20 | 20 | 8.5 | -1.75 | -14.875 | 72.25 | 3.0625 |

Step4: Sum the columns

\( \sum (x - \bar{x})(y - \bar{y}) = 34.125 + 9.125 + 37.125 - 14.875 = 65.5 \)
\( \sum (x - \bar{x})^2 = 12.25 + 0.25 + 30.25 + 72.25 = 115 \)
\( \sum (y - \bar{y})^2 = 95.0625 + 333.0625 + 45.5625 + 3.0625 = 476.75 \)

Step5: Calculate correlation coefficient

\[ r = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sqrt{\sum (x - \bar{x})^2 \sum (y - \bar{y})^2}} = \frac{65.5}{\sqrt{115 \times 476.75}} \]
\( \sqrt{115 \times 476.75} \approx \sqrt{54826.25} \approx 234.15 \)
\( r \approx \frac{65.5}{234.15} \approx 0.28 \) (positive, as positive deviations sometimes multiply to positive) Wait, earlier sign? Wait, recalculate \( (x - \bar{x})(y - \bar{y}) \):
8: (-3.5)(-9.75)=34.125 (positive)
12: (0.5)(18.25)=9.125 (positive)
6: (-5.5)(-6.75)=37.125 (positive)
20: (8.5)(-1.75)=-14.875 (negative)
Total: 34.125+9.125=43.25; 43.25+37.125=80.375; 80.375-14.875=65.5 (positive). Wait, but when I calculate the correlation, maybe I made a mistake. Wait, let's use the formula for correlation coefficient:

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

Step6: Alternative formula (easier)

Calculate \( \sum x = 46 \), \( \sum y = 87 \), \( \sum xy = (8×12)+(12×40)+(6×15)+(20×20) = 96 + 480 + 90 + 400 = 1066 \)
\( \sum x^2 = 8^2 + 12^2 + 6^2 + 20^2 = 64 + 144 + 36 + 400 = 644 \)
\( \sum y^2 = 12^2 + 40^2 + 15^2 + 20^2 = 144 + 1600 + 225 + 400 = 2369 \)

\[ r = \frac{4×1066 - 46×87}{\sqrt{[4×644 - 46^2][4×2369 - 87^2]}} \]

Numerator: \( 4264 - 4002 = 262 \)

Denominator:
First part: \( 2576 - 2116 = 460 \)
Second part: \( 9476 - 7569 = 1907 \)
Denominator: \( \sqrt{460×1907} \approx \sqrt{877220} \approx 936.6 \)

Wait, this is different. I must have messed up the first method. Wait, no, the first method's mean calculation was wrong? Wait, 8+12+6+20=46? 8+12=20, 6+20=26, 20+26=46. Correct. 12+40=52, 15+20=35, 52+35=87. Correct.

Wait, the alternative formula:

\( n = 4 \)
\( \sum xy = 8×12 + 12×40 + 6×15 + 20×20 = 96 + 480 + 90 + 400 = 1066 \)
\( \sum x = 46 \), \( \sum y = 87 \)
\( \sum x^2 = 64 + 144 + 36 + 400 = 644 \)
\( \sum y^2 = 144 + 1600 + 225 + 400 = 2369 \)

Numerator: \( 4×1066 - 46×87 = 4264 - 4002 = 262 \)

Denominator:
\( \sqrt{(4×644 - 46^2)(4×2369 - 87^2)} = \sqrt{(2576 - 2116)(9476 - 7569)} = \sqrt{460×1907} \approx \sqrt{877220} \approx 936.6 \)

Then \( r = 262 / 936.6 ≈ 0.28 \). Ah, so the correct correlation coefficient is approximately 0.28. Wait, but the options have 0.28 (positive) and -0.28. Wait, why? Because in the first method, I think I messed up the sign of (y - \bar{y}) for 20: 20 - 21.75 = -1.75 (correct), and x - \bar{x} for 20 is 20 - 11.5 = 8.5 (correct). So (8.5)(-1.75) = -14.875 (correct). The other products: 8: (-3.5)(-9.75)=34.125, 12: (0.5)(18.25)=9.125, 6: (-5.5)(-6.75)=37.125. Sum: 34.125+9.125=43.25, +37.125=80.375, -14.875=65.5. Then \( \sum (x - \bar{x})^2 = (-3.5)^2 + 0.5^2 + (-5.5)^2 + 8.5^2 = 12.25 + 0.25 + 30.25 + 72.25 = 115 \) (correct). \( \sum (y - \bar{y})^2 = (-9.75)^2 + 18.25^2 + (-6.75)^2 + (-1.75)^2 = 95.0625 + 333.0625 + 45.5625 + 3.0625 = 476.75 \) (correct). Then \( r = 65.5 / \sqrt{115×476.75} = 65.5 / \sqrt{54826.25} = 65.5 / 234.15 ≈ 0.28 \). So the correlation coefficient is approximately 0.28 (positive, since numerator is positive). Wait, but the options have 0.28 and -0.28. Wait, maybe I made a mistake in the sign. Wait, let's check the scatter: as x increases, y doesn't have a strong trend. 8→12, 12→40 (increase), 6→15 (decrease from 12→15? No, 6 is less than 8, 15 is more than 12? Wait, 6 flowers cost $15, which is more than 8 flowers' $12. So the relationship is weak and positive? Wait, the calculation gives 0.28, so the answer is 0.28.

Answer:

0.28