The table below shows the temperature (in degrees) for eight consecutive days as well as the respective number of ice cream cones an ice cream shop sold on each of these days. Temperature 68 77 83 85 89 94 96 99 Number of Cones 403 447 457 465 489 503 543 576 What is the correlation coefficient of the set of data? Round your answer to the nearest thousandth. (1 point) Responses 0.956 0.956 Image with alt text: negative 0 point 972 0.019 0.019 0.508
To find the correlation coefficient, we need to calculate the covariance and standard deviation for both the temperature and the number of cones.
First, let's calculate the means for both sets of data:
Mean of temperature (x̄):
68 + 77 + 83 + 85 + 89 + 94 + 96 + 99 = 681
x̄ = 681 / 8 = 85.125
Mean of number of cones (ȳ):
403 + 447 + 457 + 465 + 489 + 503 + 543 + 576 = 3883
ȳ = 3883 / 8 = 485.375
Next, let's calculate the covariances and standard deviations:
Covariance (cov):
cov = Σ(x - x̄)(y - ȳ) / n
where n is the number of data points
cov = (68 - 85.125)(403 - 485.375) + (77 - 85.125)(447 - 485.375) + ... + (99 - 85.125)(576 - 485.375) / 8
cov = -1694.875
Standard deviation of temperature (σx):
σx = √(Σ(x - x̄)² / n)
σx = √((68 - 85.125)² + (77 - 85.125)² + ... + (99 - 85.125)² / 8)
σx ≈ 11.769
Standard deviation of number of cones (σy):
σy = √(Σ(y - ȳ)² / n)
σy = √((403 - 485.375)² + (447 - 485.375)² + ... + (576 - 485.375)² / 8)
σy ≈ 49.497
Finally, let's calculate the correlation coefficient (r):
r = cov / (σx * σy)
r ≈ -1694.875 / (11.769 * 49.497)
r ≈ -1694.875 / 583.271
r ≈ -2.909
Rounding to the nearest thousandth, the correlation coefficient is -2.909.
Therefore, none of the given options (0.956, -0.972, 0.019, 0.508) are correct.