Question

(a)
A citrus farmer who grows mandarin oranges finds that the diameters of mandarin oranges harvested on his farm follow a normal distribution with a mean diameter of 5.85 cm and a standard deviation of 0.24 cm.



Part A

A mandarin orange is harvested. Is the probability that the orange has a diameter less than 5.61 cm equal to the probability that the orange has a diameter greater than 6.09 cm? Explain your reasoning.

Math

Answer 1

Yes. For X ~ N(5.85, 0.24^2):

z for 5.61 = (5.61 − 5.85)/0.24 = −1.0, so P(X < 5.61) = P(Z < −1) ≈ 0.1587.
z for 6.09 = (6.09 − 5.85)/0.24 = +1.0, so P(X > 6.09) = P(Z > 1) = P(Z < −1) ≈ 0.1587.

Because the normal distribution is symmetric about its mean, points equally far below and above the mean have equal tail probabilities.