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?
A. -0.57
B. -0.28
C. 0.28
D. 0.57
Answer (2)
Calculate the means of x and y: x ˉ = 11.5 , y ˉ = 21.75 .
Calculate the standard deviations of x and y: s x ≈ 6.1914 , s y ≈ 12.603 .
Calculate the covariance of x and y: co v ( x , y ) ≈ 21.833 .
Calculate the correlation coefficient: r ≈ 0.28 . The answer is 0.28 .
Explanation
Understanding the Problem We are given a table with the number of flowers in four bouquets and the total cost of each bouquet. Our goal is to find the correlation coefficient for the data in the table. The correlation coefficient measures the strength and direction of a linear relationship between two variables. A positive correlation indicates that as one variable increases, the other tends to increase as well. A negative correlation indicates that as one variable increases, the other tends to decrease. The correlation coefficient ranges from -1 to 1.
Setting up the Calculation Let's denote the number of flowers as x and the total cost as y . The data points are (8, 12), (12, 40), (6, 15), and (20, 20). We need to calculate the correlation coefficient r using the formula: r = s x ∗ s y co v ( x , y ) where co v ( x , y ) is the covariance of x and y , s x is the standard deviation of x , and s y is the standard deviation of y .
Calculating the Means First, let's calculate the means of x and y :
x ˉ = 4 8 + 12 + 6 + 20 = 4 46 = 11.5 y ˉ = 4 12 + 40 + 15 + 20 = 4 87 = 21.75
Calculating Standard Deviation of x Next, we calculate the standard deviations of x and y :
s x = n − 1 ∑ i = 1 n ( x i − x ˉ ) 2 s y = n − 1 ∑ i = 1 n ( y i − y ˉ ) 2
s x = 4 − 1 ( 8 − 11.5 ) 2 + ( 12 − 11.5 ) 2 + ( 6 − 11.5 ) 2 + ( 20 − 11.5 ) 2 s x = 3 ( − 3.5 ) 2 + ( 0.5 ) 2 + ( − 5.5 ) 2 + ( 8.5 ) 2 s x = 3 12.25 + 0.25 + 30.25 + 72.25 = 3 115 ≈ 6.1914
Calculating Standard Deviation of y s y = 4 − 1 ( 12 − 21.75 ) 2 + ( 40 − 21.75 ) 2 + ( 15 − 21.75 ) 2 + ( 20 − 21.75 ) 2 s y = 3 ( − 9.75 ) 2 + ( 18.25 ) 2 + ( − 6.75 ) 2 + ( − 1.75 ) 2 s y = 3 95.0625 + 333.0625 + 45.5625 + 3.0625 = 3 476.75 ≈ 12.603
Calculating the Covariance Now, let's calculate the covariance of x and y :
co v ( x , y ) = n − 1 ∑ i = 1 n ( x i − x ˉ ) ( y i − y ˉ ) co v ( x , y ) = 4 − 1 ( 8 − 11.5 ) ( 12 − 21.75 ) + ( 12 − 11.5 ) ( 40 − 21.75 ) + ( 6 − 11.5 ) ( 15 − 21.75 ) + ( 20 − 11.5 ) ( 20 − 21.75 ) co v ( x , y ) = 3 ( − 3.5 ) ( − 9.75 ) + ( 0.5 ) ( 18.25 ) + ( − 5.5 ) ( − 6.75 ) + ( 8.5 ) ( − 1.75 ) co v ( x , y ) = 3 34.125 + 9.125 + 37.125 − 14.875 = 3 65.5 ≈ 21.833
Calculating the Correlation Coefficient Finally, we calculate the correlation coefficient r :
r = s x ∗ s y co v ( x , y ) = 6.1914 ∗ 12.603 21.833 ≈ 77.99 21.833 ≈ 0.28
Final Answer The calculated correlation coefficient is approximately 0.28. Comparing this to the given options, the closest value is 0.28.
Alternative Calculation Alternatively, using the python calculation tool, the correlation coefficient is approximately 0.2797, which is closest to 0.28.
Examples
Understanding correlation coefficients is very useful in finance. For example, you might want to know how the price of one stock correlates with the price of another stock. If two stocks have a strong positive correlation, they tend to move in the same direction. If they have a strong negative correlation, they tend to move in opposite directions. This information can be used to diversify a portfolio and reduce risk. Another example is in marketing, where you can analyze the correlation between advertising spending and sales revenue to optimize your marketing strategy.
The correlation coefficient for the data is approximately 0.28, so the closest option is C. 0.28. This coefficient indicates a weak positive correlation between the number of flowers and the total cost of bouquets. Therefore, the selected answer option is C.
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