To find the correlation coefficient, we first need to calculate the means of x and y, as well as the covariance between x and y. Then we can use the formula for r:
Mean of x (x̄) = (43 + 80 + 81 + 84 + 93) / 5 = 76.2
Mean of y (ȳ) = (51 + 55 + 57 + 31 + 31) / 5 = 45
Covariance = Σ[(x - x̄)(y - ȳ)] / (n-1)
Covariance = [(43 - 76.2)(51 - 45) + (80 - 76.2)(55 - 45) + (81 - 76.2)(57 - 45) + (84 - 76.2)(31 - 45) + (93 - 76.2)(31 - 45)] / 4
Covariance = [(-33)(6) + (3.8)(10) + (4.8)(12) + (7.8)(-14) + (16.8)(-14)] / 4
Covariance = [-198 + 38 + 57.6 - 109.2 + (-235.2)] / 4
Covariance = -446.8 / 4
Covariance ≈ -111.7
Now we can calculate the standard deviations of x and y:
σx = √(Σ(x - x̄)² / (n - 1))
σy = √(Σ(y - ȳ)² / (n - 1))
σx = √(((-33)² + (3.8)² + (4.8)² + (7.8)² + (16.8)²) / 4)
σx = √((1089 + 14.44 + 23.04 + 60.84 + 282.24) / 4)
σx = √(1479.56 / 4)
σx ≈ √369.89
σx ≈ 19.23
σy = √((6² + 10² + 12² + (-14)² + (-14)²) / 4)
σy = √(36 + 100 + 144 + 196 + 196) / 4
σy = √(672) / 4
σy ≈ √168
σy ≈ 12.96
Now we can calculate the correlation coefficient, r:
r = Covariance / (σx * σy)
r = -111.7 / (19.23 * 12.96)
r = -111.7 / 249.1
r ≈ -0.448
Therefore, the correlation coefficient, r, is approximately -0.448.
or
Find the correlation coefficient, r, of the data described below.
Playing video games makes Brittany so happy that she thinks it even helps her complete more schoolwork. Brittany's roommate Caden is skeptical, so over the next few days, Caden asks Brittany about her progress on her daily reading.
For each day, Caden notes the number of minutes Brittany spends playing video games, x. He also takes the number of pages Brittany reads and divides it by the total number of pages assigned, y.
Minutes playing video games Percentage of reading assignment
43 51
80 55
81 57
84 31
93 31
Round your answer to the nearest thousandth.
r=
1 answer
1 answer