What is the probability that a tree’s height exceeds the mean if normally distributed.

Question

    The heights of trees in a grove are  normally distributed. (a) What is the probability that a tree’s height exceeds the mean? (b) Exceeds the mean by at least 1 standard deviation? (c) Exceeds the mean by at least 3 standard deviations ? (d) Is within 1.5 standard deviations? Use Normal distribution, not just the empirical rule

Answer 1

(a) 50% by definition of the mean as the central point of the distribution.

(b) For Z=1 the table gives the value 0.8413 which is 0.3413 above 0.5000 (mean). So 34.13% is the probability of being in the range from the mean and 1 SD from the mean. 0.5000-0.3413=1-0.8413=0.1587 (15.87%) is the probability of the height being at least 1 SD from the mean.

(c) 0.9987 is the probability for Z=3 SD from the mean, so 1-0.9987=0.0013 (0.13%) is the probability for the height in excess of 3 SD.

(d) 0.9332 is the probability for Z=1.5, so 0.4332 (43.32%) is the probability of being within 1.5 SD of the mean.