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QuestionFind the correlation coefficient for the study hours (0, 0, 1, 2, 3, 5, 5, 8) and exam scores (61, 92, 67, 64, 75, 73, 82, 90). Options: -0.5, 0.5, -0.9, 0.9.

Studdy Solution

STEP 1

Assumptions1. We are given a set of paired data points hours spent studying and corresponding exam scores. . We are asked to find the correlation coefficient, which is a measure of the strength and direction of a linear relationship between two variables.
3. The correlation coefficient ranges from -1 to1. A value of1 indicates a perfect positive correlation, -1 a perfect negative correlation, and0 no correlation.

STEP 2

The formula to calculate the correlation coefficient (r) isr=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- n is the number of pairs of scores, - ∑xy\sum xy is the sum of the products of paired scores, - ∑x\sum x and ∑y\sum y are the sums of x and y scores respectively, - ∑x2\sum x^2 and ∑y2\sum y^2 are the sums of squared x and y scores respectively.

STEP 3

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

STEP 4

Calculate ∑x\sum x, which is the sum of the hours spent studying∑x=0+0+1+2+3+++8\sum x =0 +0 +1 +2 +3 + + +8

STEP 5

Calculate ∑y\sum y, which is the sum of the exam scores∑y=61+92+67+64+75+73+82+90\sum y =61 +92 +67 +64 +75 +73 +82 +90

STEP 6

Calculate ∑xy\sum xy, which is the sum of the product of paired scores∑xy=(0×61)+(0×92)+(1×67)+(2×64)+(3×75)+(5×73)+(5×82)+(8×90)\sum xy = (0 \times61) + (0 \times92) + (1 \times67) + (2 \times64) + (3 \times75) + (5 \times73) + (5 \times82) + (8 \times90)

STEP 7

Calculate ∑x2\sum x^2, which is the sum of the squares of hours spent studying∑x2=02+02+12+22+32+52+52+2\sum x^2 =0^2 +0^2 +1^2 +2^2 +3^2 +5^2 +5^2 +^2

STEP 8

Calculate ∑y2\sum y^2, which is the sum of the squares of exam scores∑y2=612+922+672+642+752+732+822+902\sum y^2 =61^2 +92^2 +67^2 +64^2 +75^2 +73^2 +82^2 +90^2

STEP 9

Now, we have all the necessary sums. Let's plug these values into the formula for the correlation coefficient.

STEP 10

Calculate the correlation coefficientr=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]}}

STEP 11

The calculated correlation coefficient will be a value between - and. Compare this value with the given options (-0.5,0.5, -0.9,0.9) to determine which value best approximates the correlation coefficient of the data.

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