Question 8.2: A study on breeding birds collects information such as the l......
A study on breeding birds collects information such as the length of their eggs (in mm). Assume that the length is normally distributed with μ = 42.1 mm and σ² = 20.8². What is the probability of
(a) finding an egg with a length greater than 50 mm?
(b) finding an egg between 30 and 40 mm in length?
Calculate the results both manually and by using R.
Given X ∼ N(42.1, 20.8²), we get:
(a)
P(X ≥ 50) = 1 − P(X ≤ 50) = 1 − \phi \left(\frac{x-\mu }{\sigma } \right)=1-\phi \left(\frac{50-42.1 }{20.8 } \right)
=1-\phi (0.37) ≈ 0.35.
We obtain the same results in R as follows:
1-pnorm(50,42.1,20.8)
(b)
P(30 ≤ X ≤ 40) = P(X ≤ 40) − P(X ≤ 30)
=\phi \left(\frac{40 − 42.1 }{20.8 } \right)- \phi \left(\frac{30 − 42.1}{20.8 } \right)
=\phi \left(−0.096\right)- \phi \left(−0.577\right)=1 − 0.538 − 1 + 0.718
≈ 18%.
We would have obtained the same results in R using:
pnorm(40,42.1,20.8)-pnorm(30,42.1,20.8)