Question

A botanist wondered if the stem diameters of sunflowers would change depending on whether the plant was left to sway freely in the wind or artificially supported. Suppose that the unsupported stem diameters of a particular species of sunflower plant have a normal distribution with an average diameter of 35 millimeters and a standard deviation of 3 millimeters. Within what limits would you expect the diameters to lie, with probability 80%?

I know the answer is 31.16 to 38.84 but could someone write out the solution to this problem step by step? im having a hard time understanding this problem.

Answer 1

Here we have = 35, = 3

We need to find limits within which we expect the diameter to lie with 80% probability.

p ( a < x < b ) = 0.80

p ( z₁ < Z < z₂ ) = 0.80

So middle are is 80 % . So area below z1 + area above z2 = 0.20

So, p ( Z < z₁ ) = 0.10 and p ( Z > z₂ ) = 0.10

So we use normal probability table to find z table.

p ( Z < z₁ ) = 0.10

We look in to probability table section for probability 0.10 and look across z value.  

So we get z₁ = -1.28 .

p ( Z > z₂ ) = 0.10  

1 - p ( Z < z₂ ) = 0.10

p ( Z < z₂ ) = 0.90

We look in to probability table section for probability 0.90 and look across z value.  

z₂ = 1.28 .

a = 31.16

And

b = 38.84

So answer is 31.16 and 38.84.