Let X be the length of a randomly selected housefly
By assumption, X is normal random variable with μ= 4.55 and σ=0.392
Your are asked to find P(4mm < X < 5mm). We can replace the strict inequalities by non-strict inequalities but for continuous distribution both will give you the same probability value.
The first thing you have to do is transform 4mm, X, and 5mm into a Z-scores using the Z-transform formula,
Z= (X- μ)/σ
This transformation allows you to use a Z-table (which is tabulated) to find the probability involving the normal distribution in the question (please note that the normal distribution in the question is not tabulated).
Using the Z-trasnform formula,
when X=4mm, we get z=(4-4.55)/0.392 = -1.403061
when X=X we get Z
when X= 5mm, we get z=(5-4.55)/0.392 = 1.147959
Thus, P(4mm < X < 5mm) = P( -1.403061<Z<1.147959). Now, we can use a table for the standard normal distribution (i.e. Z table) to find the probability in the question.
Looking up the Z-table, we find P( -1.403061<Z<1.147959) = 0.794208
Thus, P(4mm < X < 5mm) = 0.794208.
I hope this helps.
