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June 2002 p6 q4
3170
On another day the mean speed of cars on the motorway was found to be 107.6 km h-1 and the standard deviation was 13.8 km h-1. Assuming these speeds follow a normal distribution and that the speed limit is 110 km h-1, find what proportion of cars exceed the speed limit.
Solution
To find the proportion of cars exceeding the speed limit, we first standardize the speed limit using the z-score formula:
\(z = \frac{110 - 107.6}{13.8} = 0.174\)
Next, we find the probability that a car's speed is greater than 110 km h-1:
\(P(X > 110) = 1 - \Phi(0.174)\)
Using standard normal distribution tables, \(\Phi(0.174) \approx 0.5691\).