normal distribution

#1
Hi everyone!

I have a question about the answer of question 19a and e. question a: Find the probability that X is greater than 60. Why do they do 0,5 -0,3944 and not 1 - ...?
Question e was: the probability is 0,05 that X is in the symmetric interval about the mean between which two numbers? additional info: mean is 50 and sigma squared is 64. I don't really understand what I have to do here. Can anyone help me out?

Thanks in advance!
 

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Dason

Ambassador to the humans
#2
I'm guessing the z-table they use gives the area between 0 and whatever value you want. So if it asked for find P(0 < Z < some_value) then when you look up some_value in the table it will give you that probability directly. There are many different ways these tables get made so you just need to make sure you know what the particular table you're using actually does.

For your second question they're asking what two values are such that 1) they are symetric around the mean (so they have the same distance from the mean) and 2) the probability of being between those two values is .05. So essentially they asking you to solve for 'a' and 'b' in P(a < X < b) = 0.05 where a and b the same distance from the mean. Another way to put that is to solve for c in the following P(50 - c < X < 50 + c) = 0.05
 

obh

Active Member
#4
symmetric around the 0.5, so half probability to the right and half probability to the left, the total is 0.05

p(z<=Z1) - p(z<=Z2)=0.05

0.5+(0.05/2)=0.525
0.5-(0.05/2)=0.475

p(z<=Z1)=0.525 => Z1 = 0.0627061 => (x1-50)/8=~0.06 => x1=50.48
p(z<=Z2)=0.475 => Z2 = -0.0627061 => (x2-50)/8=~0.06 =>x2=49.52

Inv Z(0.525)= 0.0627061 =>
Inv Z(0.475)= 0.0627061 =>

P( x ≤ 50.501649 ) = 0.525
P( x ≤ 49.498351 ) = 0.475
 
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#5
Let me help with question (e)

The question says that the probability is 0.05 that X is in the symmetrical interval about the mean.

From the above statement, X's middle point is the mean meaning that they have the same distance from the mean. We are therefore safe to say that from the mean(0) to the left the probability is 0.025 and that from the mean(0) to the right the probability is 0.025. 0.025+0.025=0.05.

0.5-0.025= 0.475
When you check on your z-table, 0.475 is in row 0.0 and column 0.06. our z is therefore 0.06.
Since it's in the symmetrical interval about the mean, it's a reflection. We thus have -0.06.
P(-0.06<z<0.06) we then now unstandardize it.

x-50(divided by √64) < 0.06
x-50 < 0.48
x < 50.48

-0.06 < x-50(divided by √64)
-0.48 < x-50
49.52 < x

P(49.52 < x < 50.48)=0.05
 

Dason

Ambassador to the humans
#6
Thanks! I hope you stick around to help with more questions. I wouldn't be offended if the OP doesn't respond to a response to a year and a half old question though. But we still appreciate the input.