A summary of the speeds, x kilometres per hour, of 22 cars passing a certain point gave the following information:
\(\Sigma(x - 50) = 81.4\) and \(\Sigma(x - 50)^2 = 671.0\).
Find the variance of the speeds and hence find the value of \(\Sigma x^2\).
Solution
First, find the mean of the speeds:
\(\bar{x} = 50 + \frac{81.4}{22} = 53.7\)
Next, calculate the variance using the formula:
\(\text{Var} = \frac{\Sigma(x - 50)^2}{22} - (\bar{x} - 50)^2\)
\(\text{Var} = \frac{671}{22} - 3.7^2 = 16.81\)
To find \(\Sigma x^2\), use the expanded equation:
\(\Sigma x^2 = \Sigma(x - 50)^2 + 2 \times 50 \times \Sigma(x - 50) + 22 \times 50^2\)
\(\Sigma x^2 = 671 + 2 \times 50 \times 81.4 + 22 \times 2500\)
\(\Sigma x^2 = 671 + 8140 + 55000 = 63811\)
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