Question

The heights of trees in a large grove are normally distributed with a mean of 85 feet and a standard deviation of 10 feet. Explain your answers and provide a drawing of the probability (area under the curve) for each situation.

Answer 1

A. What is the probability that a tree selected at random is more 100 feet tall?

Answer 2

B. What is the percentile rank of a tree that is 80 ft. tall?

Answer 3

C. What is the minimum height of a tree that is in the top 10% of heights?

Answer 4

HELLLLLLLLPPPPP PLEASE!!!!!!!!!!!!!!!

Answer 5

I DONT WANT THE ANSWER I WANT GUIDANCE!

Answer 6

is it too hard?

Answer 7

A. What is the probability that a tree selected at random is more 100 feet tall?
The population mean is 85 feet and the population standard deviation is 10 feet.

You'll need to find the z-score for 100 feet. Remember how to do that?

\[z=\frac{ x-\mu }{ \sigma}\]
Substituting x=100, mu=85, and sigma=10, we get \[z=\frac{ 100-85 }{10 }=1.5\]

We're interested in finding the probability that a given tree is 100 feet tall or taller.
Thus, we need to find the probability that z is 1.5 or greater.
The easiest way to do this is to find the probability that z is 1.5 or less, and then to subtract this result from 1.00.

You can find P(z<1.5) either from a table of standard normal probabilities or through using a calculator such as the TI-84 Plus.


Using my TI-83, I found that P(z<1.5) is 0.9331. Thus, P(z>1.5) is 1.0000-0.9331.

Now, g.u.y., you understand that I have no idea where you're coming from. Please read over this work and ask questions about anything that is not clear to you, before we move on to the next question.

Answer 8

yes, this makes sense, but how did you get the probability that z is .9331 algebraically, calcs confuse me!?...

Answer 9

Do you have a TI-84 calculator?

Answer 10

and why do we have to subtract 1.00?

Answer 11

yes. i have a ti 84

Answer 12

Great. have you used the normalcdf( function before?
Alternatively, have you a table of standard normal probabilities?

Answer 13

yes.

Answer 14

first I'll just demonstrate what you need to type in, and then I'll answer questions if what I've typed is insufficiently clear for you. We want the area under the standard normal curve to the left of z=1.5.

so type in normalcdf(-100, 1.5) and press enter. Want to try that right now?

Answer 15

Your result should be .93319. How are you doing?

Answer 16

oh, i got .933

Answer 17

yeah, thats what i got. this is called the gaussian thing right? i remember.

Answer 18

now explain why we have to subtract 1.00, please and thank you.

Answer 19

I wouldn't use the adjective "gaussian" myself. Basically, you're finding probabilities by finding areas under a standard normal curve between one limit and another.

Now you wanted to know why we subtract that result from 1.0000.

We want to know the probability that a given tree is taller than 100 feet. This probability is represented by the area to the RIGHT of x=100. Your 0.9331 is the area to the LEFT of x=100. So, subtracting .9331 from 1.0000 gives you the area to the RIGHT of x=100, and therefore the probability that the tree you pick is taller than 100 feet.

Answer 20

P(tree is taller than 100 feet) = 1.0000 - P(tree is shorter than 100 feet).
P(tree is taller than 100 feet) = 1.0000 - 0.9331.

Answer 21

OOOOh, yeah. probabilites cant exceed 1.00. now i remember.

Answer 22

so the probability of a tree being higher than 100 ft is .067. right? i feel so happy right now.

Answer 23

that looks very reasonable! Each of us has to develop a sense for whether a given "answer" to a problem makes sense or not.

Have you considered the next question yet?

Answer 24

glad you're feeling happy!!

Answer 25

for the next question (part B) we would have to find the z score again right?

Answer 26

oh, and and for part A, why did we use -100 for normalcdf(-100, 1.5)????

Answer 27

Yes, you'll need to find the z-score corresponding to tree height 80 feet.
Then you'll need to find the area to the left of this z-score.

Answer 28

so the z score -.5? becausse 80-85/10=-.5

Answer 29

I'm glad you asked. We want the area under the std norm. curve from 'way out to z=1.5.
The area under the curve to the left of z=-3 or z=-4 is very small, but not zero. That's why using a left limit that's 'way out, like z=-100, will result in an accurate answer.
Drop the -100 and substitute -5; you'll see what I mean.

Answer 30

that's right: your latest z-score is -0.5. What is the area under the std norm. curve to the left of z = -0.5?

Answer 31

dow e have to use the z chart?

Answer 32

You can find the area in question either way: Use the "z chart" (which is the table of standard normal probabilities), or use your calculator in exactly the same way as before.

Answer 33

but i got error when i used the calc.