A paper examined the relationship between the time that eagles spend aerially searching for food (indicated by the percentage of eagles soaring) and relative food availability. The accompanying data were taken from a scatterplot that appeared in this paper. Use x to denote salmon availability and y to denote the percentage of eagles in the air.

x 0 0 0.2 0.5 0.5 1.0 1.2 1.9 2.6 3.3 4.7 6.5

y 28.0 68.8 26.8 38.3 48.2 30.9 26.7 8.0 4.4 7.2 6.8 6.6

(a) Calculate the correlation coefficient for these data. (Give the answer to three decimal places.)

r =

(b) Because the scatterplot of the original data appeared curved, transforming both the x and y values by taking square roots was suggested. Calculate the correlation coefficient for the variables x and y . (Give the answer to three decimal places.)

r =

a) -0.717

b) -0.835

Step-by-step explanation:

The correlation coefficient r can be computed as

a)

xy          x²      y²      xy

0.028.00.00784.000.00

0.068.80.004733.440.00

0.226.80.04718.245.36

0.538.3  0.251466.8919.15

0.548.20.252323.2424.10

1.030.91.00        954.8130.90

1.226.7         1.44  712.8932.04

1.98.0    3.6164.0015.20

2.64.4    6.76        19.3611.44

3.37.2    10.8951.8423.76

4.76.8       22.09  46.2431.96

6.56.6       42.25  43.5642.90

sumx= 22.4

sumy= 300.7

sumx²= 88.58

sumy²= 11918.5

sumxy= 236.81

r= -0.717

So, the required correlation coefficient is -0.717.

b)

Firstly we take the square root of x and y values and then following calculations are made.

x             y           x²      y²     x*y

0.000005.291500.028.00.00000

0.000008.294580.068.80.00000

0.447215.176870.226.82.31517

0.707116.188700.538.3        4.37607

0.707116.942620.548.24.90918

1.000005.558781.030.95.55878

1.095455.167201.226.7  5.66039

1.378402.828431.98.0  3.89872

1.612452.097622.64.4    3.38231

1.816592.683283.37.2    4.87442

2.167952.607684.76.8    5.65332

2.549512.569056.56.6    6.54981

sumx= 13.4818

sumy= 55.4063

sumx²=  22.4

sumy²= 300.7

sumxy= 47.1782

So, the required correlation coefficient is -0.8353.