6.25 Help on statistic normal probability distribution?

An experimenter publishing in the Annals of Botany investigated whether the stem diameters of the dicot sunflower would change depending on whether the plant was left to sway freely in the wind or was artificially supported. Suppose that the unsupported stem diameters at the base of a particular species of sunflower plant have a normal distribution with an average diameter of 35 mm and a standard dev of 3 mm.


a. What is the probability that a sunflower plant will have a basal diameter of more than 40 mm?


b. If two sunflower plants are randomly selected, what is the probability that both plants will have a basal diameter of more than 40 mm?


c. Within what limits would you expect the basal diameters to lie, with probability .95?


d. What diameter represents the 90 percentile of the distribution of diameters?





Answers at the back of the book:


1. .0475 b. .00226 c. 29.12 to 40.88 d. 38.84





Please show how you derived at the answer step-by-step.

6.25 Help on statistic normal probability distribution?
D ~ Normal(μ = 35, σ = 3)


Z ~ Normal(0,1) (the standard normal)





to solve these problems you have to convert all the statements to standard units. My answers will be a little different because I did not round my answers to use the standard normal CDF tables that I bet your book is using. I used software, MINITAB, to get exact values.





a) P(D %26gt; 40) = P(Z %26gt; (40-35)/3) = P(Z %26gt; 5/3) = 0.04779





b) the two flowers are independent so you simply multiply the probabilities together. For my work that would be 0.04779 * 0.04779 = 0.00228


based on the answers you provided you have 0.0475 * 0.0475 = 0.00225625





c)


From the standard normal tables we know that:


P(-1.96 %26lt; Z %26lt; 1.96) = 0.95





so....


P(μ - 1.96σ %26lt; D %26lt; μ + 1.96σ) = 0.95


P(35 - 1.96 * 3 %26lt; D %26lt; 35 + 1.96*3) = 0.95


P(29.12 %26lt; D %26lt; 40.88) = 0.95





the interval is 29.12 %26lt; D %26lt; 40.88





d) the 90th percentile is P[ Z %26lt; 1.28 ] = 0.90


P[ D %26lt; 35 + 1.28 * 3] = P[D %26lt; 38.84]


the 90th percentile for the diameters is 38.84 mm