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QuestionCalculate the correlation coefficient for the sets x={28,10,21,17,24,−3,16}x = \{28, 10, 21, 17, 24, -3, 16\} and y={29,21,−8,22,−7,−10,−5}y = \{29, 21, -8, 22, -7, -10, -5\}.

Studdy Solution

STEP 1

Assumptions1. We have two sets of data, xx and yy, each with7 observations. . We are asked to compute the correlation coefficient, which measures the strength and direction of the linear relationship between two variables.

STEP 2

The formula for the correlation coefficient, denoted by rr, is given byr=n(∑xy)−(∑x)(∑y)[n∑x2−(∑x)2][n∑y2−(∑y)2]r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}}where- nn is the number of observations, - ∑xy\sum xy is the sum of the product of corresponding xx and yy values, - ∑x\sum x and ∑y\sum y are the sums of xx and yy values respectively, - ∑x2\sum x^2 and ∑y2\sum y^2 are the sums of the squares of xx and yy values respectively.

STEP 3

First, let's calculate the required sums. We need to calculate ∑x\sum x, ∑y\sum y, ∑x2\sum x^2, ∑y2\sum y^2, and ∑xy\sum xy.

STEP 4

Calculate ∑x\sum x by adding all xx values∑x=28+10+21+17+24−3+16\sum x =28 +10 +21 +17 +24 -3 +16

STEP 5

Calculate ∑y\sum y by adding all yy values∑y=29+21−8+22−7−10−5\sum y =29 +21 -8 +22 -7 -10 -5

STEP 6

Calculate ∑x2\sum x^2 by squaring each xx value and then adding them∑x2=282+102+212+172+242+(−3)2+162\sum x^2 =28^2 +10^2 +21^2 +17^2 +24^2 + (-3)^2 +16^2

STEP 7

Calculate ∑y2\sum y^2 by squaring each yy value and then adding them∑y2=292+212+(−)2+222+(−7)2+(−10)2+(−5)2\sum y^2 =29^2 +21^2 + (-)^2 +22^2 + (-7)^2 + (-10)^2 + (-5)^2

STEP 8

Calculate ∑xy\sum xy by multiplying each pair of corresponding xx and yy values and then adding them∑xy=28⋅29+10⋅21+21⋅(−8)+17⋅22+24⋅(−7)+(−3)⋅(−10)+16⋅(−5)\sum xy =28 \cdot29 +10 \cdot21 +21 \cdot (-8) +17 \cdot22 +24 \cdot (-7) + (-3) \cdot (-10) +16 \cdot (-5)

STEP 9

Now that we have calculated all the required sums, we can substitute these values into the formula for the correlation coefficient.

STEP 10

Substitute the values into the formula and calculate the correlation coefficient.
r=n(∑xy)−(∑x)(∑y)[n∑x2−(∑x)2][n∑y2−(∑y)2]r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}}The correlation coefficient, rr, is the solution to this problem.

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