Examine the computation formula for r, the sample correlation coefficient.
(a) In the formula for r, if we exchange the symbols x and y, do we get a different result or do we get the same (equivalent) result? Explain your answer.
The result is different because the formula is dependent on the symbols.The result is the same because the formula is dependent on the symbols. The result is the same because the formula is not dependent on the symbols.The result is different because the formula is not dependent on the symbols.
(b) If we have a set of x and y data values and
we exchange corresponding x and y values to get a
new data set, should the sample correlation coefficient be the same
for both sets of data? Explain your answer.
The result is different because the formula is dependent on which values are the x values and which values are the y values.The result is the same because the formula is dependent on which values are the x values and which values are the y values. The result is the same because the formula is not dependent on which values are the x values and which values are the y values.The result is different because the formula is not dependent on which values are the x values and which values are the y values.
(c) Compute the sample correlation coefficient r for each
of the following data sets and show that r is the same for
both. (Use 3 decimal places.)
(i) | x | 6 | 5 | 9 |
y | 2 | 4 | 5 |
(ii) | x | 2 | 4 | 5 |
y | 6 | 5 | 9 |
r | |
(i) | |
(ii) |
a) The result is the same because the formula is not dependent on the symbols.
b) The result is the same because the formula is not dependent on which values are the x values and which values are the y values
c)
i)
S.No | X | Y | (x-x̅)^{2} | (y-y̅)^{2} | (x-x̅)(y-y̅) |
1 | 6 | 2 | 0.4444 | 2.7778 | 1.1111 |
2 | 5 | 4 | 2.7778 | 0.1111 | -0.5556 |
3 | 9 | 5 | 5.4444 | 1.7778 | 3.1111 |
Total | 20 | 11 | 8.6667 | 4.6667 | 3.6667 |
Mean | 6.667 | 3.667 | SSX | SSY | SXY |
correlation coefficient r= | S_{xy}/(√S_{xx}*S_{yy}) = | 0.577 |
ii)
correlation coefficient r= | S_{xy}/(√S_{xx}*S_{yy}) = | 0.577 |
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