Question

Calculate the correlation coefficient of the following data: x y 1 16 4 20 8 15 6 22 2 9 A.) .14 B.) 0.37 C.) -0.37 D.) -.14

Answer 1

The correlation coefficient of the given data is approximately 0.37.

Step-by-step explanation:

To calculate the correlation coefficient, several intermediate steps are involved.

Given data:

`\[x: 1, 4, 8, 6, 2\]`

`\[y: 16, 20, 15, 22, 9\]`

Step 1: Calculate the means of x and y.

`\[ \text{Mean of x (} \bar{x} \text{)} = (1 + 4 + 8 + 6 + 2)/(5) = (21)/(5) = 4.2 \]`

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`[ \text{Mean of y (} \bar{y} \text{)} = (16 + 20 + 15 + 22 + 9)/(5) = (82)/(5) = 16.4 \]`

Step 2: Calculate the differences from the means for each x and y value.

`\[ \text{Differences for x:} \]`

`\[ 1 - 4.2 = -3.2, \, 4 - 4.2 = -0.2, \, 8 - 4.2 = 3.8, \, 6 - 4.2 = 1.8, \, 2 - 4.2 = -2.2 \]`

`\[ \text{Differences for y:} \]`

`\[ 16 - 16.4 = -0.4, \, 20 - 16.4 = 3.6, \, 15 - 16.4 = -1.4, \, 22 - 16.4 = 5.6, \, 9 - 16.4 = -7.4 \]`

Step 3: Multiply the differences together for each pair of values.

`\[ \text{Product of differences:} \]`

`\[ (-3.2)(-0.4) = 1.28, \, (-0.2)(3.6) = -0.72, \, (3.8)(-1.4) = -5.32, \, (1.8)(5.6) = 10.08, \, (-2.2)(-7.4) = 16.28 \]`

Step 4: Calculate the sum of the products and the sum of the squares of the differences for x and y separately.

`\[ \text{Sum of products of differences} = 1.28 + (-0.72) + (-5.32) + 10.08 + 16.28 = 21.6 \]`

`\[ \text{Sum of squares of differences for x:} \]`

`\[ (-3.2)^2 + (-0.2)^2 + (3.8)^2 + (1.8)^2 + (-2.2)^2 = 10.24 + 0.04 + 14.44 + 3.24 + 4.84 = 32.8 \]`

`\[ \text{Sum of squares of differences for y:} \]`

`\[ (-0.4)^2 + (3.6)^2 + (-1.4)^2 + (5.6)^2 + (-7.4)^2 = 0.16 + 12.96 + 1.96 + 31.36 + 54.76 = 101.2 \]`

Step 5: Plug these values into the correlation coefficient formula:

`\[ \text{Correlation coefficient} = \frac{\text{Sum of products of differences}}{\sqrt{\text{Sum of squares of differences for x} * \text{Sum of squares of differences for y}}} \]`

`\[ = (21.6)/(√(32.8 * 101.2)) \approx (21.6)/(√(3322.56)) \approx (21.6)/(57.62) \approx 0.3748 \]`

Rounded to three decimal places, the correlation coefficient is approximately 0.375 or 0.37.