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


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(b)
Part B

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 5.61 cm? Explain your reasoning.

Answer 1

Part A — Yes.
5.61 = 5.85 − 0.24 and 6.09 = 5.85 + 0.24, so they are equally far from the mean in opposite directions. For a normal distribution P(X < μ − d) = P(X > μ + d). Numerically, z = (5.61−5.85)/0.24 = −1 and (6.09−5.85)/0.24 = +1, so both probabilities equal Φ(−1) ≈ 0.1587.

Part B — No.
P(X < 5.61) and P(X > 5.61) are complements (they sum to 1) for a continuous distribution, so they are equal only when each equals 0.5, i.e. when the cutoff is the mean. Since 5.61 ≠ 5.85, they are not equal: P(X < 5.61) ≈ 0.1587 while P(X > 5.61) ≈ 0.8413.