The times, \(t\) seconds, taken to swim 100 m were recorded for a group of 9 swimmers and were found to be as follows.
95, 126, 117, 135, 120, 125, 114, 119, 136
- Find the values of \(\Sigma(t - 120)\) and \(\Sigma(t - 120)^2\).
- Using your values found in part (i), calculate the variance of \(t\).
Solution
(i) Calculate \(\Sigma(t - 120)\):
\(\Sigma(t - 120) = (95 - 120) + (126 - 120) + (117 - 120) + (135 - 120) + (120 - 120) + (125 - 120) + (114 - 120) + (119 - 120) + (136 - 120)\)
\(= -25 + 6 - 3 + 15 + 0 + 5 - 6 - 1 + 16 = 7\)
Calculate \(\Sigma(t - 120)^2\):
\(\Sigma(t - 120)^2 = (-25)^2 + 6^2 + (-3)^2 + 15^2 + 0^2 + 5^2 + (-6)^2 + (-1)^2 + 16^2\)
\(= 625 + 36 + 9 + 225 + 0 + 25 + 36 + 1 + 256 = 1213\)
(ii) Calculate the variance of \(t\):
Variance \(= \frac{\Sigma(t - 120)^2}{9} - \left(\frac{\Sigma(t - 120)}{9}\right)^2\)
\(= \frac{1213}{9} - \left(\frac{7}{9}\right)^2\)
\(= 134.2\)
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