For the accompanying data set, (a) draw a scatter diagram of the data, (b) by hand, compute the correlation coefficient, and (c) determine whether there is a linear relation between x and y.
x y
2 4
4 8
6 12
6 12
7 20
Data set
x 
22 
44 
66 
66 
77 


y 
44 
88 
1212 
1212 
2020 
Critical Values for Correlation Coefficient
n 


3 
0.997 
4 
0.950 
5 
0.878 
6 
0.811 
7 
0.754 
8 
0.707 
9 
0.666 
10 
0.632 
11 
0.602 
12 
0.576 
13 
0.553 
14 
0.532 
15 
0.514 
16 
0.497 
17 
0.482 
18 
0.468 
19 
0.456 
20 
0.444 
21 
0.433 
22 
0.423 
23 
0.413 
24 
0.404 
25 
0.396 
26 
0.388 
27 
0.381 
28 
0.374 
29 
0.367 
30 
0.361 
n 
(b) By hand, compute the correlation coefficient.
The correlation coefficient is
r=__?__.
(Round to three decimal places as needed.)
(c) Determine whether there is a linear relation between x and y.
Because the correlation coefficient is
▼
positive
negative
and the absolute value of the correlation coefficient, __?__,
is
▼
greater
not greater
than the critical value for this data set, __?__,
▼
a positive
a negative
no linear relation exists between x and y.
(Round to three decimal places as needed.)
a.
b.
X Values
∑ = 25
Mean = 5
∑(X  M_{x})^{2} = SS_{x} = 16
Y Values
∑ = 56
Mean = 11.2
∑(Y  M_{y})^{2} = SS_{y} = 140.8
X and Y Combined
N = 5
∑(X  M_{x})(Y  M_{y}) = 44
R Calculation
r = ∑((X  M_{y})(Y  M_{x})) /
√((SS_{x})(SS_{y}))
r = 44 / √((16)(140.8)) = 0.927
c. Here df=n2=3
So critical value is 0.997
Because the correlation coefficient is positive and the absolute value of the correlation coefficient, r=0.927,
is not greater than the critical value for this data set, 0.997, a positive
linear relation exists between x and y.
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