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Feb/Mar 2019 p62 q3
3130
The times taken, in minutes, for trains to travel between Alphaton and Beeton are normally distributed with mean 140 and standard deviation 12.
Find the probability that a randomly chosen train will take less than 132 minutes to travel between Alphaton and Beeton.
Solution
Let the random variable \(X\) represent the travel time in minutes. \(X\) is normally distributed with mean \(\mu = 140\) and standard deviation \(\sigma = 12\).
We need to find \(P(X < 132)\).
First, standardize the variable using the formula:
\(Z = \frac{X - \mu}{\sigma}\)
Substitute the values:
\(Z = \frac{132 - 140}{12} = -0.6667\)
Now, find \(P(Z < -0.6667)\).
Using standard normal distribution tables or a calculator, \(P(Z < -0.6667) = 0.252\).