Question

What is the correlation coefficient between the variables? Round to three decimal places. Enter your answer in the box. r≈ Age Score 14 360 25 293 19 315 35 212 20 315 12 331 5 404

Answer 1

The correlation coefficient measures the strength and direction of the linear relationship between two variables. In this case, the correlation coefficient is approximately 0.413.

Step-by-step explanation:

The correlation coefficient measures the strength and direction of the linear relationship between two variables. To find the correlation coefficient, we can use the formula:

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

Using the provided data:

Average of the X values (Age): (14 + 25 + 19 + 35 + 20 + 12 + 5) / 7 = 19.71

Average of the Y values (Score): (360 + 293 + 315 + 212 + 315 + 331 + 404) / 7 = 317.57

Σxy: (14 * 360) + (25 * 293) + (19 * 315) + (35 * 212) + (20 * 315) + (12 * 331) + (5 * 404) = 769211

(Σx)(Σy): (14 + 25 + 19 + 35 + 20 + 12 + 5)(360 + 293 + 315 + 212 + 315 + 331 + 404) = 2021818

(nΣx^2) - (Σx)^2: (7 * [(14^2) + (25^2) + (19^2) + (35^2) + (20^2) + (12^2) + (5^2)]) - ([(14 + 25 + 19 + 35 + 20 + 12 + 5)^2]) = 3514

(nΣy^2) - (Σy)^2: (7 * [(360^2) + (293^2) + (315^2) + (212^2) + (315^2) + (331^2) + (404^2)]) - ([(360 + 293 + 315 + 212 + 315 + 331 + 404)^2]) = 2384668

Using these values, we can calculate the correlation coefficient:

r = [(7 * 769211) - (2021818)] / sqrt([(7 * 3514) - (2021818)][(7 * 2384668) - (2021818)]) = 0.413

Therefore, the correlation coefficient between the variables is approximately 0.413.

Answer 2

There are two data sets x and y.

X includes = 14 25 19 35 20 12 5

Y includes = 360 293 315 212 315 331 404

to solve for the correlation coefficient, we need to get the following values step by step

Step 1: Find the mean of each set.

The mean of X = 18.571

The mean of Y = 318.571

Step 2: Subtract the mean of X from every value X value (denote this with letter a). Do the same for y (denote this with letter b).

The mean of X subtracted from every X value (a):

14 - 18.571 = -4.571

25 - 18.571 = 6.429

19 - 18.571 = 0.429

35 - 18.571 = 16.429

20 - 18.571 = 1.429

12 - 18.571 = -6.571

5 - 18.571 = -13.571

The mean of Y subtracted from every value of Y (b):

360 - 318.571 = 41.429

293 - 318.571 = -25.571

315 - 318.571 = -3.571

212 = 318.571 = -106.571

315 - 318.571 = -3.571

331 - 318.571 = 12.429

404 - 318.571 = 85.429

Step 3: Calculate: a * b, a^2 and b^2 of every value.

For a*b

-189.388

-164.388

-1.531

-1750.816

-5.102

-81.673

-1159.388

Sum: -3352.286

For a²

20.898

41.327

0.184

269.898

2.041

43.184

184.184

Sum: 561.714

For b²

1716.327

653.898

12.755

11357.469

12.755

154.469

7298.041

Sum: 21205.714

Step 4: Solve using this formula

r = ∑a * b / √((a²)(b²))

r = -3352.286 / √((561.714)(21205.714))

= -0.9713

The correlation coefficient is -0.971