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 of the data, we can use the formula:
\[r = \frac{n\sum{xy} - \sum{x}\sum{y}}{\sqrt{(n\sum{x^2} - (\sum{x})^2)(n\sum{y^2} - (\sum{y})^2)}}\]
where \(n\) is the number of data points, \(\sum{xy}\) is the sum of the products of the x and y values, \(\sum{x}\) is the sum of the x values, \(\sum{y}\) is the sum of the y values, \(\sum{x^2}\) is the sum of the squared x values, and \(\sum{y^2}\) is the sum of the squared y values.
First, we can calculate some of the necessary values:
\(\sum{x} = 68 + 77 + 83 + 85 + 89 + 94 + 96 + 99 = 691\)
\(\sum{y} = 403 + 447 + 457 + 465 + 489 + 503 + 543 + 576 = 3783\)
\(\sum{x^2} = 68^2 + 77^2 + 83^2 + 85^2 + 89^2 + 94^2 + 96^2 + 99^2 = 50664\)
\(\sum{y^2} = 403^2 + 447^2 + 457^2 + 465^2 + 489^2 + 503^2 + 543^2 + 576^2 = 1639968\)
\(\sum{xy} = (68)(403) + (77)(447) + (83)(457) + (85)(465) + (89)(489) + (94)(503) + (96)(543) + (99)(576) = 696868\)
Now, we can substitute these values into the formula to find the correlation coefficient:
\[r = \frac{8(696868) - (691)(3783)}{\sqrt{(8(50664) - (691)^2)(8(1639968) - (3783)^2)}}\]
Simplifying further:
\[r = \frac{5574944 - 2619033}{\sqrt{(405312 - 4799281)(13199744 - 14283729)}}\]
\[r = \frac{2955911}{\sqrt{(-4393969)(-1083992)}}\]
\[r = \frac{2955911}{\sqrt{4774312070888}}\]
\[r \approx 0.948\]
Rounding to the nearest thousandth, the correlation coefficient is approximately 0.948.