Question

If the values of random variables \(X,Y\) are (121,360), (242,364), (363,362), the correlation coefficient (rounded to one decimal place) is \(\underline{\hspace{1cm}}\).

Hint:

Correlation uses covariance divided by product of standard deviations.

Answer 1

Correct Answer: 0.5

Three data pairs: \((121,360)\), \((242,364)\), \((363,362)\).
Compute means:
\[ \bar{X} = \frac{121 + 242 + 363}{3} = 242 \]
\[ \bar{Y} = \frac{360 + 364 + 362}{3} = 362 \]
Compute covariance numerator:
\[ (121-242)(360-362) + (242-242)(364-362) + (363-242)(362-362) \]
\[ = (-121)(-2) + (0)(2) + (121)(0) = 242 \]
Compute denominator:
\[ \sqrt{ \sum (X-\bar{X})^2 } \cdot \sqrt{ \sum (Y-\bar{Y})^2 } \]
\[ \sum (X-\bar{X})^2 = 121^2 + 0^2 + 121^2 = 29282 \]
\[ \sum (Y-\bar{Y})^2 = (-2)^2 + 2^2 + 0^2 = 8 \]
Thus:
\[ r = \frac{242}{\sqrt{29282 \cdot 8}} = \frac{242}{\sqrt{234256}} = \frac{242}{484} \approx 0.5 \]