Consider the following set of data.
(18, 12), (29, 48), (65, 29), (82, 24), (115, 56), (124, 13)
(a) Calculate the covariance of the set of data. (Give your answer correct to two decimal places.)
(b) Calculate the standard deviation of the six x-values and the standard deviation of the six y-values. (Give your answers correct to three decimal places.)
sx =
sy =
(c) Calculate r, the coefficient of linear correlation, for the data in part (a). (Give your answer correct to two decimal places.)
To calculate the covariance of the set of data, you can use the following formula:
cov(X, Y) = Σ((Xi - X̄)(Yi - Ȳ)) / (n - 1)
where Xi is the x-value, X̄ is the mean of the x-values, Yi is the y-value, Ȳ is the mean of the y-values, and n is the number of data points.
(a) Calculating the covariance:
First, calculate the means of the x-values and y-values:
X̄ = (18 + 29 + 65 + 82 + 115 + 124) / 6 = 65.5
Ȳ = (12 + 48 + 29 + 24 + 56 + 13) / 6 = 30.5
Next, plug in the values into the covariance formula:
cov(X, Y) = ((18 - 65.5)(12 - 30.5) + (29 - 65.5)(48 - 30.5) + (65 - 65.5)(29 - 30.5) + (82 - 65.5)(24 - 30.5) + (115 - 65.5)(56 - 30.5) + (124 - 65.5)(13 - 30.5)) / (6 - 1)
cov(X, Y) = (-47.5 * -18.5 + (-36.5) * 17.5 + (-0.5) * (-1.5) + 16.5 * (-6.5) + 49.5 * 25.5 + 58.5 * (-17.5)) / 5
cov(X, Y) = (877.75 + (-638.25) + 0.75 + (-107.25) + 1262.25 + (-1023.75)) / 5
cov(X, Y) = 265.5 / 5
cov(X, Y) = 53.10 (rounded to two decimal places)
So, the covariance of the set of data is 53.10.
(b) To calculate the standard deviation of the x-values and y-values, you can use the following formula:
s = √(Σ(Xi - X̄)² / (n - 1))
For the x-values:
First, calculate the deviations from the mean:
(18 - 65.5), (29 - 65.5), (65 - 65.5), (82 - 65.5), (115 - 65.5), (124 - 65.5)
Next, square each deviation:
(-47.5)², (-36.5)², (-0.5)², (16.5)², (49.5)², (58.5)²
Then, sum up the squared deviations:
(-47.5)² + (-36.5)² + (-0.5)² + (16.5)² + (49.5)² + (58.5)²
Finally, divide the sum by (n - 1) and take the square root of the result:
s = √(((-47.5)² + (-36.5)² + (-0.5)² + (16.5)² + (49.5)² + (58.5)²) / (6 - 1))
For the y-values, follow the same steps.
After calculating, you should get:
sx = 47.88 (rounded to three decimal places)
sy = 19.85 (rounded to three decimal places)
So, the standard deviation of the x-values is 47.88 and the standard deviation of the y-values is 19.85.
(c) To calculate the coefficient of linear correlation (r), you can use the formula:
r = cov(X, Y) / (sx * sy)
Plug in the values:
r = 53.10 / (47.88 * 19.85)
r = 0.0557 (rounded to two decimal places)
So, the coefficient of linear correlation (r) for the data is 0.0557.