The following data represents the scores of two variables, x and y: x: 10, 15, 20, 25, 30 y: 20, 25, 30, 35, 40 The correlation coefficient between x and y is:a.0.7b.0.8c.0.9d.1
Question
The following data represents the scores of two variables, x and y: x: 10, 15, 20, 25, 30 y: 20, 25, 30, 35, 40 The correlation coefficient between x and y is:a.0.7b.0.8c.0.9d.1
Solution
To find the correlation coefficient between variables x and y, we can use the formula for Pearson's correlation coefficient.
Step 1: Calculate the mean of x and y. Mean of x = (10 + 15 + 20 + 25 + 30) / 5 = 20 Mean of y = (20 + 25 + 30 + 35 + 40) / 5 = 30
Step 2: Calculate the deviations from the mean for both x and y. Deviation of x = (10 - 20), (15 - 20), (20 - 20), (25 - 20), (30 - 20) = -10, -5, 0, 5, 10 Deviation of y = (20 - 30), (25 - 30), (30 - 30), (35 - 30), (40 - 30) = -10, -5, 0, 5, 10
Step 3: Calculate the product of the deviations for each pair of values. Product of deviations = (-10 * -10), (-5 * -5), (0 * 0), (5 * 5), (10 * 10) = 100, 25, 0, 25, 100
Step 4: Calculate the sum of the product of deviations. Sum of product of deviations = 100 + 25 + 0 + 25 + 100 = 250
Step 5: Calculate the standard deviation of x and y. Standard deviation of x = sqrt(((-10)^2 + (-5)^2 + 0^2 + 5^2 + 10^2) / 5) = sqrt((100 + 25 + 0 + 25 + 100) / 5) = sqrt(250 / 5) = sqrt(50) = 7.07 Standard deviation of y = sqrt(((-10)^2 + (-5)^2 + 0^2 + 5^2 + 10^2) / 5) = sqrt((100 + 25 + 0 + 25 + 100) / 5) = sqrt(250 / 5) = sqrt(50) = 7.07
Step 6: Calculate the correlation coefficient. Correlation coefficient = (Sum of product of deviations) / (Standard deviation of x * Standard deviation of y) Correlation coefficient = 250 / (7.07 * 7.07) = 250 / 49.99 = 5
Therefore, the correlation coefficient between x and y is 5.
None of the given options (a. 0.7, b. 0.8, c. 0.9, d. 1) match the calculated correlation coefficient of 5.
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