The lengths, \(L \mathrm {~mm}\), of housefly wings are normally distributed with \(L \sim \mathrm {~N} \left( 4.5,0.4 ^ { 2 } \right)\)
Find the probability that a randomly selected housefly has a wing length of less than 3.86 mm .
Find
the upper quartile ( \(Q _ { 3 }\) ) of \(L\)
the lower quartile ( \(Q _ { 1 }\) ) of \(L\)
A value that is greater than \(Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)\) or smaller than \(Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)\) is defined as an outlier.
Find these two outlier limits.
A housefly is selected at random.
Using standardisation, show that the probability that this housefly is not an outlier is 0.993 to 3 decimal places.
Given that this housefly is not an outlier,
showing your working, find the probability that the wing length of this housefly is greater than 5 mm .