> 
> 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 $40$ 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 $90$th percentile of the
distribution of diameters?

$\mathbf{\text{MY ATTEMPT:}}$

a-) $P(X >40)=P(Z>1.66)=0.0475$

b-) for this part , i tried to use the formula for selection more than one elements such that $$P\bigg(Z > \frac{40-35}{3 / \sqrt{2}}\bigg)=P\bigg(Z > \frac{5}{2.12132}\bigg) =P\bigg(Z >2.357\bigg)=0.00921$$

However , the answer is $0.00226$ and it has such a solution : $$P(Z>1.66) \times P(Z>1.66)=(0.0475)^2 $$

Why didn't my formula work ? I always use it to find normal dist. probabilities when we choose more than one object.

c-) I could not do this with mathematical formula , the only way i could do is to seach for values to satisfy this condition in z table , so i am looking for a nice answer for it. By the way , the given answer is $29.12$ to $40.88$

d-) I have never solved a percentile problem , answer says that $P(Z \geq -z) =0.9$ and answr is $38.84$. I could not understand why they used "-z" insteaf of "+z" and why we used $"\geq"$ instead of $>$.

Thanks in advance..