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.
To find the correlation coefficient, you first need to calculate the mean temperature and the mean number of cones, as well as the standard deviation for each set of data.
Mean temperature:
(68 + 77 + 83 + 85 + 89 + 94 + 96 + 99) / 8 = 458 / 8 = 57.25
Mean number of cones:
(403 + 447 + 457 + 465 + 489 + 503 + 543 + 576) / 8 = 3883 / 8 = 485.375
Standard deviation of temperature:
Step 1: Calculate the difference between each temperature and the mean temperature:
68 - 57.25 = 10.75
77 - 57.25 = 19.75
83 - 57.25 = 25.75
85 - 57.25 = 27.75
89 - 57.25 = 31.75
94 - 57.25 = 36.75
96 - 57.25 = 38.75
99 - 57.25 = 41.75
Step 2: Square each difference:
10.75^2 ≈ 115.5625
19.75^2 ≈ 390.0625
25.75^2 ≈ 664.5625
27.75^2 ≈ 770.0625
31.75^2 ≈ 1008.0625
36.75^2 ≈ 1350.5625
38.75^2 ≈ 1502.5625
41.75^2 ≈ 1743.0625
Step 3: Calculate the sum of these squared differences:
115.5625 + 390.0625 + 664.5625 + 770.0625 + 1008.0625 + 1350.5625 + 1502.5625 + 1743.0625 = 6544.5
Step 4: Divide the sum by (n-1) where n is the number of observations (8):
6544.5 / (8-1) = 6544.5 / 7 = 935.0714
Step 5: Take the square root of the result:
√935.0714 ≈ 30.5831
Standard deviation of number of cones:
Step 1: Calculate the difference between each number of cones and the mean number of cones:
403 - 485.375 = -82.375
447 - 485.375 = -38.375
457 - 485.375 = -28.375
465 - 485.375 = -20.375
489 - 485.375 = 3.625
503 - 485.375 = 17.625
543 - 485.375 = 57.625
576 - 485.375 = 90.625
Step 2: Square each difference:
(-82.375)^2 ≈ 6790.3906
(-38.375)^2 ≈ 1477.2656
(-28.375)^2 ≈ 805.2656
(-20.375)^2 ≈ 414.7656
3.625^2 ≈ 13.1406
17.625^2 ≈ 310.6406
57.625^2 ≈ 3322.2656
90.625^2 ≈ 8219.0156
Step 3: Calculate the sum of these squared differences:
6790.3906 + 1477.2656 + 805.2656 + 414.7656 + 13.1406 + 310.6406 + 3322.2656 + 8219.0156 = 21252.1006
Step 4: Divide the sum by (n-1) where n is the number of observations (8):
21252.1006 / (8-1) = 21252.1006 / 7 = 3036.0144
Step 5: Take the square root of the result:
√3036.0144 ≈ 55.1302
Product of the differences:
(10.75 * -82.375) + (19.75 * -38.375) + (25.75 * -28.375) + (27.75 * -20.375) + (31.75 * 3.625) + (36.75 * 17.625) + (38.75 * 57.625) + (41.75 * 90.625)
= -885.3125 - 759.0625 - 729.5625 - 566.0625 + 115.0625 + 647.0625 + 2231.1875 + 3782.40625
= 2626.90625
Correlation coefficient:
2626.90625 / (8-1) / (30.5831 * 55.1302)
= 2626.90625 / 7 / (30.5831 * 55.1302)
= 52.153125 / (30.5831 * 55.1302)
≈ 52.153125 / 1685.7424662
≈ 0.030935
Rounded to the nearest thousandth, the correlation coefficient is approximately 0.031.