High School

Dear beloved readers, welcome to our website! We hope your visit here brings you valuable insights and meaningful inspiration. Thank you for taking the time to stop by and explore the content we've prepared for you.
------------------------------------------------ Find the value of the linear correlation coefficient \( r \).

x: 22.6, 36.6, 15.6, 35.0, 17.5

y: 7, 6, 6, 2, 6

Group of answer choices:

A. 0

B. -0.478

C. 0.537

D. -0.537

Answer :

To find the linear correlation coefficient, also known as Pearson’s correlation coefficient [tex]r[/tex], we need to use the formula:

[tex]r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}}[/tex]

where:

  • [tex]n[/tex] is the number of data points,
  • [tex]\sum xy[/tex] is the sum of the product of paired scores,
  • [tex]\sum x[/tex] is the sum of [tex]x[/tex]-scores,
  • [tex]\sum y[/tex] is the sum of [tex]y[/tex]-scores,
  • [tex]\sum x^2[/tex] is the sum of the squares of [tex]x[/tex]-scores,
  • [tex]\sum y^2[/tex] is the sum of the squares of [tex]y[/tex]-scores.

First, let's list the data:

[tex]x: 22.6, 36.6, 15.6, 35.0, 17.5[/tex]

[tex]y: 7, 6, 6, 2, 6[/tex]

Next, calculate the necessary sums:

  1. [tex]\sum x = 22.6 + 36.6 + 15.6 + 35.0 + 17.5 = 127.3[/tex]

  2. [tex]\sum y = 7 + 6 + 6 + 2 + 6 = 27[/tex]

  3. [tex]\sum xy = (22.6 \times 7) + (36.6 \times 6) + (15.6 \times 6) + (35.0 \times 2) + (17.5 \times 6) = 158.2 + 219.6 + 93.6 + 70 + 105 = 646.4[/tex]

  4. [tex]\sum x^2 = (22.6^2) + (36.6^2) + (15.6^2) + (35.0^2) + (17.5^2) = 510.76 + 1339.56 + 243.36 + 1225 + 306.25 = 3624.93[/tex]

  5. [tex]\sum y^2 = 7^2 + 6^2 + 6^2 + 2^2 + 6^2 = 49 + 36 + 36 + 4 + 36 = 161[/tex]

Substitute these values into the formula:

[tex]r = \frac{5(646.4) - (127.3)(27)}{\sqrt{[5(3624.93) - 127.3^2][5(161) - 27^2]}}[/tex]

[tex]r = \frac{3232 - 3437.1}{\sqrt{[18124.65 - 16210.29][805-729]}}[/tex]

[tex]r = \frac{-205.1}{\sqrt{1914.36 \times 76}}[/tex]

[tex]r = \frac{-205.1}{\sqrt{145892.16}}[/tex]

[tex]r = \frac{-205.1}{381.906} \approx -0.537[/tex]

The calculated value of the linear correlation coefficient [tex]r[/tex] is approximately [tex]-0.537[/tex], which corresponds to the negative correlation between the [tex]x[/tex] and [tex]y[/tex] values. The negative sign indicates that as [tex]x[/tex] increases, [tex]y[/tex] tends to decrease. Thus, the correct answer is -0.537.