Find the correlation coefficient, r, of the data described below.
The manager of a ski resort in the Alps always worries there won’t be enough snow to keep the resort open into the spring. She decided to see if there was a relationship between the temperature in January and the amount of snow in the spring.
For several years, she recorded the average temperature in January (in Celsius), x. On March 1, she also measured the depth of the snow at the bottom of a particular ski slope (in centimeters), y.
Average temperature (in Celsius) Snow depth (in centimeters)
−6.68 53.08
−1.01 57.31
1.06 45.92
3.53 32.80
4.31 32.91
Round your answer to the nearest thousandth.
r =
To calculate the correlation coefficient ( r ) for the given data, you can follow these steps:
1. Collect the data pairs:
– ( x = ) Average temperature: ([-6.68, -1.01, 1.06, 3.53, 4.31])
– ( y = ) Snow depth: ([53.08, 57.31, 45.92, 32.80, 32.91])
2. Calculate the means of ( x ) and ( y ):
– Mean of ( x ): (bar{x} = frac{(-6.68 – 1.01 + 1.06 + 3.53 + 4.31)}{5} = frac{1.21}{5} approx 0.242)
– Mean of ( y ): (bar{y} = frac{(53.08 + 57.31 + 45.92 + 32.80 + 32.91)}{5} = frac{221.02}{5} approx 44.204)
3. Calculate the covariance of ( x ) and ( y ):
[
text{Cov}(x,y) = frac{sum (x_i – bar{x})(y_i – bar{y})}{n}