Thread: Normal probability distributions

assume that men's weight are normaly distributed with mean 172 lb and a standard deviation 29 lb
If 100 men are randomly selected find the probability that weight of 10 of them is less than 150 lb and weight of
the remaining 90 of them is more or equal to 150 lb.
I'm lost.
I'l be very appreciate if somebody helps me.

Originally Posted by busyboy

assume that men's weight are normaly distributed with mean 172 lb and a standard deviation 29 lb
If 100 men are randomly selected find the probability that weight of 10 of them is less than 150 lb and weight of them is more or equal to 150 lb.
I'm lost.
I'l be very appreciate if somebody helps me.

The distribution of the mean of a sample of size n from a normal distribution
with mean mu and SD sigma, has mean m=mu and SD sm=sigma/sqrt(n).

So in this case the mean of the sample of 10 has mean 172 lb and SD 29/sqrt(10) ~= 9.17 lb.

Now the z score of a mean of 150 lb is:

z = (150 - m)/sm = (150 - 172)/9.17 ~= -2.40

Now we look this z-score up in a table of the cumulative standard normal distribution to
get the probability of observing a z-score this low or lower. Doing this we find the probability
is 0.0082 (0.82%), which is also the probability that the mean weight of 10 of the men is <= 150 lb.

The probability that the mean weight of a sample of 90 of the men is >150lb is derived
by observing for a sample of 90 the SD of the mean is sm = sigma/sqrt(90) = 29/sqrt(90) =3.06.
So the z-score is now:

z = (150 - m)/sm = (150 - 172)/3.06 ~= -7.19

and there is no need to look this up the required probability is so close to 1 that my tables
do not go that far.

RonL

Thank you very much for you help!
Why did you not use number 100 anywere ?
I am not sure,but I think that probability will depend from amount men which randomly selected .
Sincerely, Busyboy.

Originally Posted by busyboy

Thank you very much for you help!
Why did you not use number 100 anywere ?
I am not sure,but I think that probability will depend from amount men which randomly selected .
Sincerely, Busyboy.

Because that was just misdirection, the question was really about one sample
of 10 and another of 90.

RonL

I'm VERY appreciate you for your explanation!!!
Thanks a lot!!!!!
Busyboy.