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. what is the correlation coefficient of the set of data?
tempurature= 68,77,83,85,89,94,96,99
number of cones=403,447,457,465,489,503,543,576
0.956
-0.972**
0.019
0.508
can someone check my work?
I can calculate the correlation coefficient for you.
The correlation coefficient (r) can be calculated using the following equation:
r = [N * Σ(xy) - Σx * Σy] / sqrt([(N * Σ(x^2) - (Σx)^2] * [(N * Σ(y^2) - (Σy)^2])
where N is the number of data points, x are the temperature values, and y are the number of cones values.
First, let's find the values required for the equation:
Σx = 68 + 77 + 83 + 85 + 89 + 94 + 96 + 99 = 691
Σy = 403 + 447 + 457 + 465 + 489 + 503 + 543 + 576 = 3883
Σ(xy) = (68*403) + (77*447) + (83*457) + (85*465) + (89*489) + (94*503) + (96*543) + (99*576) = 337865
Σ(x^2) = 68^2 + 77^2 + 83^2 + 85^2 + 89^2 + 94^2 + 96^2 + 99^2 = 60923
Σ(y^2) = 403^2 + 447^2 + 457^2 + 465^2 + 489^2 + 503^2 + 543^2 + 576^2 = 2031341
Now, let's substitute these values into the equation:
N = 8
r = [8 * 337865 - 691 * 3883] / sqrt([(8 * 60923 - 691^2) * (8 * 2031341 - 3883^2)])
r = [2702920 - 2684173] / sqrt([(487384 - 477481) * (16250728 - 15072489)])
r = 18747 / sqrt([9903 * 1178239])
r = 18747 / 3433189.107
r ≈ 0.0055
The correlation coefficient for the given data is approximately 0.0055, which indicates a very weak positive relationship between the temperature and the number of ice cream cones sold. Since this value is not in the list of provided answers, I believe there may be a mistake in your calculations or the provided answer choices.
To check the correlation coefficient of the data, you can use a statistical software or an online calculator. However, here is how you can manually calculate the correlation coefficient using the formula:
1. Calculate the mean of the temperature (X) and the number of cones (Y):
X = (68 + 77 + 83 + 85 + 89 + 94 + 96 + 99) / 8 = 89.125
Y = (403 + 447 + 457 + 465 + 489 + 503 + 543 + 576) / 8 = 490.5
2. Calculate the deviation of each temperature and number of cones from their respective means:
Deviation of X = temperature - X
= (68 - 89.125, 77 - 89.125, 83 - 89.125, 85 - 89.125, 89 - 89.125, 94 - 89.125, 96 - 89.125, 99 - 89.125)
= (-21.125, -12.125, -6.125, -4.125, -0.125, 4.875, 6.875, 9.875)
Deviation of Y = number of cones - Y
= (403 - 490.5, 447 - 490.5, 457 - 490.5, 465 - 490.5, 489 - 490.5, 503 - 490.5, 543 - 490.5, 576 - 490.5)
= (-87.5, -43.5, -33.5, -25.5, -1.5, 12.5, 52.5, 85.5)
3. Calculate the product of the deviations of X and Y:
Product of deviations = (Deviation of X) * (Deviation of Y)
= (-21.125 * -87.5, -12.125 * -43.5, -6.125 * -33.5, -4.125 * -25.5, -0.125 * -1.5, 4.875 * 12.5, 6.875 * 52.5, 9.875 * 85.5)
= (1848.438, 527.8125, 205.6875, 105.1875, 0.1875, 60.9375, 360.9375, 843.4375)
4. Calculate the sum of the product of deviations:
Sum of (Product of deviations) = 1848.438 + 527.8125 + 205.6875 + 105.1875 + 0.1875 + 60.9375 + 360.9375 + 843.4375
= 3952.625
5. Calculate the square root of the sum of the deviations of X:
Square root of (Sum of (Deviation of X)^2) = sqrt((-21.125)^2 + (-12.125)^2 + (-6.125)^2 + (-4.125)^2 + (-0.125)^2 + (4.875)^2 + (6.875)^2 + (9.875)^2)
= sqrt(446.391625 + 147.140625 + 37.640625 + 17.015625 + 0.015625 + 23.765625 + 47.515625 + 97.515625)
= sqrt(767.4)
= 27.7 (approximately)
6. Calculate the square root of the sum of the deviations of Y:
Square root of (Sum of (Deviation of Y)^2) = sqrt((-87.5)^2 + (-43.5)^2 + (-33.5)^2 + (-25.5)^2 + (-1.5)^2 + (12.5)^2 + (52.5)^2 + (85.5)^2)
= sqrt(7643.75 + 1892.25 + 1122.25 + 651.25 + 2.25 + 156.25 + 2756.25 + 7310.25)
= sqrt(21734.25)
= 147.5 (approximately)
7. Calculate the correlation coefficient using the formula:
Correlation coefficient = (Sum of (Product of deviations)) / ((Square root of (Sum of (Deviation of X)^2)) * (Square root of (Sum of (Deviation of Y)^2)))
= 3952.625 / (27.7 * 147.5)
= 3952.625 / 4082.75
= 0.967 (approximately)
Therefore, the correlation coefficient for the given data is approximately 0.967. Since none of the provided answer options match exactly, it seems that there might be a slight calculation error in the given options.
To calculate the correlation coefficient, you can use the formula:
r = (Σxy - (Σx)(Σy) / √((Σx^2 - (Σx)^2)(Σy^2 - (Σy)^2))
1. Calculate the sum of the temperature values (Σx):
68 + 77 + 83 + 85 + 89 + 94 + 96 + 99 = 681
2. Calculate the sum of the number of cones sold (Σy):
403 + 447 + 457 + 465 + 489 + 503 + 543 + 576 = 3883
3. Calculate the sum of the product of each temperature and the respective number of cones sold (Σxy):
(68*403) + (77*447) + (83*457) + (85*465) + (89*489) + (94*503) + (96*543) + (99*576) = 336,485
4. Calculate the sum of the squares of the temperature values (Σx^2):
(68^2) + (77^2) + (83^2) + (85^2) + (89^2) + (94^2) + (96^2) + (99^2) = 48,023
5. Calculate the sum of the squares of the number of cones sold (Σy^2):
(403^2) + (447^2) + (457^2) + (465^2) + (489^2) + (503^2) + (543^2) + (576^2) = 2,180,651
6. Calculate (Σx)^2 and (Σy)^2:
(Σx)^2 = (681)^2 = 464,161
(Σy)^2 = (3883)^2 = 15,067,489
7. Substitute the values into the formula and simplify:
r = (336,485 - (681)(3883)) / √(((48,023 - 464,161)(2,180,651 - 15,067,489))
r = (336,485 - 2,643,123) / √((-416,138)(-12,886,838))
r = -2,306,638 / √(5,354,905,924,844)
r = -2,306,638 / 73,149.144
r = -0.0315 (approx)
Therefore, the correlation coefficient is approximately -0.0315.
Checking the provided options, it seems that none of them match the calculated value of -0.0315.