Draw a scatter diagram of the data and compute the correlation coefficient. Determine if there is a linear relation between x and y.

Given data:
- \(s_x = \_\_\_\_\_\)
- \(s_y = \_\_\_\_\_\)

Values:
- \(X: 2, 4, 6, 6, 7\)
- \(Y: 4, 8, 10, 13, 20\)

Question

College

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Answer :

prabaase87 prabaase87 · Answer 1 · 2025-05-05T03:01:31-04:00

There is a strong positive linear relationship between x and y, with a correlation coefficient of approximately 0.874.

To draw a scatter diagram of the given data, we plot the x-values on the horizontal axis and the y-values on the vertical axis:

X: 2, 4, 6, 6, 7

Y: 4, 8, 10, 13, 20

Now, let's plot these points on a graph:

From the scatter diagram, we can observe that there is a general positive trend between x and y. As x increases, y tends to increase as well.

To compute the correlation coefficient, we need to calculate the sums and products of x and y:

n = 5

Σx = 2 + 4 + 6 + 6 + 7 = 25

Σy = 4 + 8 + 10 + 13 + 20 = 55

Σxy = (2 * 4) + (4 * 8) + (6 * 10) + (6 * 13) + (7 * 20) = 222

Σx^2 = (2^2) + (4^2) + (6^2) + (6^2) + (7^2) = 135

Σy^2 = (4^2) + (8^2) + (10^2) + (13^2) + (20^2) = 909

Now, we can use the formula to calculate the correlation coefficient:

r = (nΣxy - ΣxΣy) / sqrt((nΣx^2 - (Σx)^2)(nΣy^2 - (Σy)^2))

r = (5 * 222 - (25 * 55)) / sqrt((5 * 135 - (25)^2)(5 * 909 - (55)^2))

After evaluating the above expression, we find that the correlation coefficient, r, is approximately 0.874.

Since the correlation coefficient is positive and close to 1, we can conclude that there is a strong positive linear relationship between x and y.

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