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.

Math

Answer 1

Mean = 5.85, SD = 0.24. Compute z-scores.

Part A
5.61 is 0.24 below the mean: z = (5.61−5.85)/0.24 = −1.00.
6.09 is 0.24 above the mean: z = (6.09−5.85)/0.24 = +1.00.
P(X < 5.61) = P(Z < −1) ≈ 0.1587 and P(X > 6.09) = P(Z > 1) ≈ 0.1587.
Yes — they are equal because the normal distribution is symmetric about the mean and the two values are equal distances from the mean.

Part B
P(X < 5.61) ≈ 0.1587 (as above).
P(X > 5.61) = 1 − P(X ≤ 5.61) ≈ 1 − 0.1587 = 0.8413.
No — these are not equal because 5.61 is below the mean, so the probability of being greater than 5.61 is much larger than the probability of being less than 5.61.