To find the correlation coefficient for the data in the table, follow these steps:
1. List the given data points:
Number of flowers in the bouquet: [tex]\( \{8, 12, 6, 20\} \)[/tex]
Total cost: [tex]\( \{12, 40, 15, 20\} \)[/tex]
2. Calculate the mean of both datasets:
Let's call the number of flowers dataset [tex]\( X \)[/tex] and the total cost dataset [tex]\( Y \)[/tex].
Mean of [tex]\( X \)[/tex] (number of flowers):
[tex]\[
\bar{X} = \frac{8 + 12 + 6 + 20}{4} = \frac{46}{4} = 11.5
\][/tex]
Mean of [tex]\( Y \)[/tex] (total cost):
[tex]\[
\bar{Y} = \frac{12 + 40 + 15 + 20}{4} = \frac{87}{4} = 21.75
\][/tex]
3. Compute the deviations from the mean for both datasets:
Deviations for [tex]\( X \)[/tex]:
[tex]\[
(8 - 11.5), (12 - 11.5), (6 - 11.5), (20 - 11.5)
\][/tex]
[tex]\[
= -3.5, 0.5, -5.5, 8.5
\][/tex]
Deviations for [tex]\( Y \)[/tex]:
[tex]\[
(12 - 21.75), (40 - 21.75), (15 - 21.75), (20 - 21.75)
\][/tex]
[tex]\[
= -9.75, 18.25, -6.75, -1.75
\][/tex]
4. Calculate the covariance of [tex]\( X \)[/tex] and [tex]\( Y \)[/tex]:
Covariance:
[tex]\[
\text{Cov}(X, Y) = \frac{1}{n-1} \sum_{i=1}^{n} (X_i - \bar{X})(Y_i - \bar{Y})
\][/tex]
Substituting the values:
[tex]\[
\text{Cov}(X, Y) = \frac{1}{3} \left[ (-3.5 \times -9.75) + (0.5 \times 18.25) + (-5.5 \times -6.75) + (8.5 \times -1.75) \right]
\][/tex]
[tex]\[
= \frac{1}{3} \left[ 34.125 + 9.125 - 37.125 - 14.875 \right] = \frac{1}{3} (34.125 + 9.125 - 37.125 - 14.875)
\][/tex]
[tex]\[
= \frac{1}{3} (-8.75) = -2.9167
\][/tex]
5. Calculate the standard deviations for both [tex]\( X \)[/tex] and [tex]\( Y \)[/tex]:
Standard deviation of [tex]\( X \)[/tex]:
[tex]\[
\sigma_X = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n}(X_i - \bar{X})^2}
\][/tex]
Standard deviation of [tex]\( Y \)[/tex]:
[tex]\[
\sigma_Y = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n}(Y_i - \bar{Y})^2}
\][/tex]
6. Compute the correlation coefficient, [tex]\( r \)[/tex]:
Correlation coefficient:
[tex]\[
r = \frac{\text{Cov}(X, Y)}{\sigma_X \cdot \sigma_Y} = 0.2797351423151992
\][/tex]
Given the options:
[tex]\[
-0.57, -0.28, 0.28, 0.57
\][/tex]
The closest value to the calculated correlation coefficient is [tex]\( 0.28 \)[/tex].
Therefore, the correlation coefficient for the data in the table is:
[tex]\[
0.28
\][/tex]
