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June 2012 p63 q6
3157
In another fish farm, the lengths of salmon, X cm, are normally distributed with mean 32.9 cm and standard deviation 2.4 cm.
Find the probability that a randomly chosen salmon is 34 cm long, correct to the nearest centimetre.
Solution
To find the probability that a randomly chosen salmon is 34 cm long, we need to consider the range from 33.5 cm to 34.5 cm due to rounding to the nearest centimetre.
We standardize the values using the formula for the standard normal distribution:
\(Z = \frac{X - \mu}{\sigma}\)
For 34.5 cm: \(Z = \frac{34.5 - 32.9}{2.4} = 0.667\)
For 33.5 cm: \(Z = \frac{33.5 - 32.9}{2.4} = 0.25\)
We find the probabilities using the standard normal distribution table:
\(\Phi(0.667) = 0.7477\)
\(\Phi(0.25) = 0.5987\)
The probability that a salmon is 34 cm long is the difference between these probabilities:
