Question

The following data gave X = the water content of snow on April 1 and Y = the yield from April to July (in inches) on the Snake River watershed in Wyoming for 1919 to 1935. (The data were taken from an article in Research Notes,Vol. 61, 1950, Pacific Northwest Forest Range Experiment Station, Oregon). x 23.1 32.8 31.8 32.0 30.4 24.0 39.5 24.2 52.5 37.9 30.5 25.1 12.4 35.1 31.5 21.1 27.6 y 10.5 16.7 18.2 17.0 16.3 10.5 23.1 12.4 24.9 22.8 14.1 12.9 8.8 17.4 14.9 10.5 16.1 (a) Estimate the correlation between Y and X. Round your answer to 3 decimal places

Answer 1

Answer:

r=0.933

Step-by-step explanation:

X = the water content of snow on April 1

Y = the yield from April to July (in inches) on the Snake River watershed in Wyoming for 1919 to 1935.

x: 23.1 32.8 31.8 32.0 30.4 24.0 39.5 24.2 52.5 37.9 30.5 25.1 12.4 35.1 31.5 21.1 27.6

y: 10.5 16.7 18.2 17.0 16.3 10.5 23.1 12.4 24.9 22.8 14.1 12.9 8.8 17.4 14.9 10.5 16.1

We can construct the following table

n  $\sum x$     $\sum y$     $\sum x^2$       $\sum y^2$        $\sum xy$

______________________________________________

1    23.1    10.5   533.61     110.25     242.55

2   32.8   16.7    1075.84   278.89   547.76

3   31.8    18.2    1011.24     331.24    578.76

4   32.0   17.0    1024         289        544

5   30.4   16.3    924.16      265.69   495.52

6   24.0   10.5    600.25     110.25     252

7   39.5    23.1    1560.25    533.61    912.45

8   24.2    12.4    585.64      153.76    300.48

9   52.5    24.9   2756.25    620.01   1307.25

10  37.9    22.8   1436.41      519.84    864.12

11   30.5    14.1     930.25      198.81     430.05

12   25.1    12.9    630.01       166.41     323.79

13   12.4     8.8     153.76       77.44      109.12

14   35.1     17.4    1232.01      302.76   610.74

15   31.5     14.9    992.25      222.01    469.35

16   21.1      10.5    445.21       110.25     221.55

17    27.6    16.1     761.76       259.21    444.36

____________________________________________

On this case n=17, $\sum x = 511.5$   $\sum y=267.1$   $\sum x^2 =16628.65$  $\sum y^2 =4549.43$   $\sum xy =8653.45$

And we can use the following formula to calculate the correlation coefficient:

$r=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^2-(\sum x)^2][n\sum y^2-(\sum y)^2]}}$

And replacing we have:

$r=\frac{17(8653.45)-(511.5)(267.1)}{\sqrt{[17(16628.65)-(511.5)^2][17(4549.43)-(267.1)^2]}}=0.933$