Finding two unknown Z's...

#1
Hi! I have two problems in which I have to find the unknown Z or X... I've really done well with this section but somehow I just can't remember how to do this and am worried it'll be on the test. Can anyone point me in the right direction/give me the first step?

1. Find Z so that P(-z<Z<z)=0.3108
(I know that the bell shaped graph has the 0 mean in the middle, with a Z falling on each side... and I know the 0.3108 is the area between these 2 z's and that I have to work backwards to find them, I just can't figure out how to get started)

2. If X has a normal distribution with mean 65 and standard deviation σ, what is P(X<65)?
(I know this is finding the area/probability of the space that is less than the average of 65... again, just can't get started. One of the options is "cannot be determined from information given", so that could be it...)

Thank you in advance! :)
 

Dason

Ambassador to the humans
#3
Is the first problem as easy as dividing the number by 2 to get at the answer?
No. I mean most of the ways to solve the problem will have you dividing the .3108 by 2 and using that for something but the answer isn't just .3108/2
 
#4
Well the second one is certainly not "not enough info", in fact it's a pretty easy one. Your answer will depend on remembering two things about normal distributions:

-area under the entire curve when expressed as a proportion is 1.0
-the distribution is symmetrical (same amount to the left of the mean as to the right)


Also in regards to the previous posters' answers regarding #1...it doesn't say the "MIDDLE" .3108, so might this be a "not enough info" question? Without some phrase like that we can't necessarily assume that this ~32% chunk of the distribution is symmetric about the mean.

I doubt this is the case though because I doubt the instructor is trying to be this tricky. Also, because I just checked an online z-table and saw that .1554 is a precise proportion given on the table in the "between z and mean" column. :p
 

Dason

Ambassador to the humans
#5
Also in regards to the previous posters' answers regarding #1...it doesn't say the "MIDDLE" .3108, so might this be a "not enough info" question? Without some phrase like that we can't necessarily assume that this ~32% chunk of the distribution is symmetric about the mean.
I'm assuming their instructor is using the common convention that Z represents a standard normal random variable. In which case it would necessarily have to be the "middle" .3108.
 
#6
Maybe I'm missing something regarding the notation...is P(-z<Z<z) meant to imply that the two lowercase z's are the same (absolute) value? Because if not I'm still unsure why there aren't multiple correct answers.

For example, p(-1<Z<2) = .8185 ≈ p(-1.34<Z<1.34).
 

Dason

Ambassador to the humans
#7
Yes - I don't really see any other way you can interpret that. Of course there are multiple answers if it's not symmetric but the notation seems clear to me to imply the symmetry.
 

hlsmith

Less is more. Stay pure. Stay poor.
#8
No. I mean most of the ways to solve the problem will have you dividing the .3108 by 2 and using that for something but the answer isn't just .3108/2
Yes, that is why I said "to get at the answer" not "get the answer" : )

I agree that they are likely refering to a symmetrical region...
 
#9
I found the answers, but still have no idea how to actually work them out haha.
1. is 0.40 (I found that if I did invert the norm on the calculator--ti-84--and put in the 0.3108 in parens, it'd give me .49... close...but not the same. still stumped.)
2. is 0.5... so it's not the "not enough information" answer. Maybe I'll try putting it in the calculator and seeing what happens...