Question

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 millimeters (mm) and a standard deviation of 3 mm.

(a) What is the probability that a sunflower plant will have a basal diameter of more than 38 mm? (Round your answer to four decimal places.)

(b) If two sunflower plants are randomly selected, what is the probability that both plants will have a basal diameter of more than 38 mm? (Round your answer to four decimal places.)

(c) Within what limits (in mm) would you expect the basal diameters to lie, with probability 0.95? (Round your answers to two decimal places.)

lower limit mm:

upper limit mm:

(d) What diameter (in mm) represents the 90th percentile of the distribution of diameters? (Round your answer to two decimal places.)

mm:

You may need to use the appropriate appendix table or technology to answer this question.

Answer 1

Here by the problem ,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. Here we suppose that the unsupported stem diameters at the base of a particular species of sunflower plant (say X for any randomly selected stem) have a normal distribution with an average diameter of millimeters (mm) and a standard deviation of mm.

(a) Then the probability that a sunflower plant will have a basal diameter of more than 38 mm

(using R command)

(b) Now if two sunflower plants are randomly selected, then the probability that both plants will have a basal diameter of more than 38 mm (assuming they are selected independently)

=P(first plant have a basal diameter of more than 38 mm and the second plant have a basal diameter of more than 38 mm)

=P(first plant have a basal diameter of more than 38 mm). P(second plant have a basal diameter of more than 38 mm)

(c) As we know a random variable which follows normal distribution is distributed symmetrically around the mean hence we can assume within the limits we can expect the basal diameters to lie, with probability 0.95 where a is such that,

since Z is symmetric around 0 hence P(Z<a)=P(Z>-a)=1-P(Z<-a)

From Z-table we get,

Hence putting the values we get,

(d) Now let d be the diameter (in mm) that represents the 90th percentile of the distribution of diameters

Then,

From standard normal table we get,

Hence the value

Hence the answer.................

Thank you.............