the table shows the number of flowers in four bouquets and the total cost of each bouquet.\ncost of…

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 6.14 ) (using ( n - 1 ) for sample)

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 )

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}} )
( \sum (x_i - \bar{x})^2 = 115 ), ( \sum (y_i - \bar{y})^2 = 476.75 )
Denominator: ( \sqrt{115 \times 476.75} \approx \sqrt{54826.25} \approx 234.15 )
( r = \frac{65.5}{234.15} \approx 0.28 ) (Wait, no—wait, recalculating sum of products: Wait, earlier sum was 65.5? Wait, no, let's recheck:
Wait (8,12): (-3.5)(-9.75)=34.125
(12,40): (0.5)(18.25)=9.125
(6,15): (-5.5)(-6.75)=37.125
(20,20): (8.5)(-1.75)= -14.875
Sum: 34.125 + 9.125 = 43.25; 43.25 + 37.125 = 80.375; 80.375 -14.875 = 65.5. Correct.
Denominator: sqrt(115 * 476.75) = sqrt(115476.75). 115476.75 = 115*(400 + 76.75) = 46000 + 8826.25 = 54826.25. sqrt(54826.25) ≈ 234.15.
65.5 / 234.15 ≈ 0.28? Wait, but wait, maybe I made a mistake. Wait, no—wait, let's use another approach. Alternatively, use calculator steps. Wait, maybe the correct value is approximately 0.28? Wait, no, wait, let's check with actual calculation:

Wait, maybe my mean calculation was wrong? Wait, x values: 8,12,6,20. Sum: 8+12=20, 20+6=26, 26+20=46. Mean x: 46/4=11.5. Correct.
y values: 12,40,15,20. Sum: 12+40=52, 52+15=67, 67+20=87. Mean y: 87/4=21.75. Correct.

Deviations:
x: 8-11.5=-3.5, 12-11.5=0.5, 6-11.5=-5.5, 20-11.5=8.5.
y: 12-21.75=-9.75, 40-21.75=18.25, 15-21.75=-6.75, 20-21.75=-1.75.

Now, sum of (x-x̄)(y-ȳ):
(-3.5)(-9.75) = 34.125
(0.5)(18.25) = 9.125
(-5.5)(-6.75) = 37.125
(8.5)(-1.75) = -14.875
Total: 34.125 + 9.125 = 43.25; 43.25 + 37.125 = 80.375; 80.375 -14.875 = 65.5. Correct.

Sum of (x-x̄)^2: (-3.5)^2 + 0.5^2 + (-5.5)^2 + 8.5^2 = 12.25 + 0.25 + 30.25 + 72.25 = 115. Correct.

Sum of (y-ȳ)^2: (-9.75)^2 + 18.25^2 + (-6.75)^2 + (-1.75)^2 = 95.0625 + 333.0625 + 45.5625 + 3.0625 = 476.75. Correct.

Now, r = 65.5 / sqrt(115 * 476.75) = 65.5 / sqrt(54826.25) = 65.5 / 234.15 ≈ 0.28. Wait, but the options include 0.28. Wait, but maybe I made a mistake? Wait, no—wait, let's check with another method. Let's use the formula for correlation coefficient with n=4.

Alternatively, use the formula:

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

Calculate ( \sum x = 46 ), ( \sum y = 87 ), ( \sum xy = (812) + (1240) + (615) + (2020) = 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 )

Now, numerator: ( 41066 - 4687 = 4264 - 4002 = 262 )
Denominator: ( \sqrt{[4644 - 46^2][42369 - 87^2]} )
Calculate ( 4644 = 2576 ), ( 46^2 = 2116 ), so first part: ( 2576 - 2116 = 460 )
Second part: ( 42369 = 9476 ), ( 87^2 = 7569 ), so ( 9476 - 7569 = 1907 )
Denominator: ( \sqrt{460 * 1907} = \sqrt{877220} \approx 936.6 )
Wait, this is different! Wait, what's wrong here? Oh no! I see the mistake. Earlier, when calculating ( \sum (x_i - \bar{x})(y_i - \bar{y}) ), I used the sample formula (dividing by n-1), but the correlation coefficient formula uses the population or sample? Wait, no—wait, the formula ( r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}} ) is correct for the sample correlation? No, wait, actually, the Pearson correlation coefficient formula 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]}} )

Let's recalculate with this formula:

( n = 4 )
( \sum x = 8 + 12 + 6 + 20 = 46 )
( \sum y = 12 + 40 + 15 + 20 = 87 )
( \sum xy = 812 + 1240 + 615 + 2020 = 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 )

Numerator: ( 41066 - 4687 = 4264 - 4002 = 262 )

Denominator:
First term inside sqrt: ( 4644 - 46^2 = 2576 - 2116 = 460 )
Second term inside sqrt: ( 42369 - 87^2 = 9476 - 7569 = 1907 )
Denominator: ( \sqrt{460 * 1907} = \sqrt{877220} \approx 936.6 )

Now, ( r = 262 / 936.6 \approx 0.28 ). Wait, same result! Because 262 / 936.6 ≈ 0.28. So the correlation coefficient is approximately 0.28. Wait, but the options have 0.28. Wait, but let's check the sign. All the products: when x increases, y sometimes increases, sometimes not. Let's see the data:

Number of flowers: 8 (12), 12 (40), 6 (15), 20 (20). So as x increases from 6 to 8 to 12 to 20, y goes 15, 12, 40, 20. So it's a weak positive correlation. So r should be positive, around 0.28. So the correct option is 0.28. Wait, but earlier when I used the deviation method, I got 65.5 / 234.15 ≈ 0.28, and with the formula, 262 / 936.6 ≈ 0.28. So that's correct.

Answer:

0.28