Which of the following is the best estimate of the correlation coefficient for the data shown below? (1, 18), (2, 12), (3, 20), (4, 16), (5, 22), (6, 30), (7, 28), (8, 40), (9, 34), (10, 29), (11, 33), (12, 39), (13, 44), (14, 39), (15, 28)
A 0.1
B. 0.3
C. 0.5
D. 0.8
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To find the best estimate of the correlation coefficient for the given data, we need to calculate the correlation coefficient.
The correlation coefficient, often denoted by "r," measures the strength and direction of the linear relationship between two variables. It ranges from -1 to +1, where -1 indicates a perfect negative linear relationship, +1 indicates a perfect positive linear relationship, and 0 indicates no linear relationship.
To calculate the correlation coefficient, we can use the following formula:
r = (Σ((X - X̄) * (Y - Ȳ))) / (sqrt(Σ((X - X̄)^2)) * sqrt(Σ((Y - Ȳ)^2)))
Let's calculate the correlation coefficient step by step:
1. Calculate the means (X̄ and Ȳ) for each set of data.
X̄ = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15) / 15 = 8
Ȳ = (18 + 12 + 20 + 16 + 22 + 30 + 28 + 40 + 34 + 29 + 33 + 39 + 44 + 39 + 28) / 15 = 28.8
2. Calculate the deviations for each set of data. The deviation is the difference between the data value and the mean.
Deviation for X: (X - X̄)
Deviation for Y: (Y - Ȳ)
Here are the deviations:
(-7, -10.8), (-6, -16.8), (-5, -8.8), (-4, -12.8), (-3, -6.8), (-2, 1.2), (-1, -0.8), (0, 11.2), (1, 5.2), (2, 0.2), (3, 4.2), (4, 10.2), (5, 15.2), (6, 10.2), (7, -0.8)
3. Calculate the product of the deviations for each pair of X and Y.
Product = (Deviation for X) * (Deviation for Y)
Here are the products:
59.6, 100.8, 44, 51.2, 20.4, -2.4, 0.8, 0, 5.2, 0.4, 12.6, 41, 76.8, 62.4, 3.2
4. Sum up the products and the squared deviations.
Sum of products = Σ((X - X̄) * (Y - Ȳ)) = 468.6
Sum of squared deviations for X = Σ((X - X̄)^2) = 154.6667
Sum of squared deviations for Y = Σ((Y - Ȳ)^2) = 3021.3333
5. Calculate the square root of the sum of squared deviations for X and Y.
sqrt(154.6667) ≈ 12.44
sqrt(3021.3333) ≈ 54.98
6. Substitute the values into the correlation coefficient formula:
r = (Σ((X - X̄) * (Y - Ȳ))) / (sqrt(Σ((X - X̄)^2)) * sqrt(Σ((Y - Ȳ)^2)))
r ≈ (468.6) / (12.44 * 54.98) ≈ 0.605
Therefore, the best estimate of the correlation coefficient for the given data is approximately 0.605.
Since none of the answer choices match, it seems there might be a mistake in the given answer options.