Question

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 is 0.24 below the mean and 6.09 is 0.24 above the mean, so their z-scores are −1 and +1. For a normal distribution symmetric about the mean,
P(X < 5.61) = P(Z < −1) = 0.1587 and P(X > 6.09) = P(Z > 1) = 0.1587.

Part B — No.
P(X < 5.61) = 0.1587 (as above) while P(X > 5.61) = 1 − 0.1587 = 0.8413. For a continuous distribution these two probabilities sum to 1 and are equal only when the cutoff equals the mean.