1. ## Normal distribution questions

Can someone check my answers for the first two questions, I have a feeling they're wrong. And I can't seem to get the right answer to question 2. a), I keep ending up with an 'impossible' z score.

Brain weights in a certain population of adult Swedish males are approximately normally distributed with a mean of 1400 g and a standard deviation of 100g.

1. a) For an individual chosen at random from this population calculate the probability that their brain weight is less than 1375 g or more than 1500 g.

=1-P(1375<X<1500)
=1-P(1375-1400/100 < Z < 1500-1400/100)
=1-P(-.25 < Z < 1)
=1-P(Z<1) - P(Z<-.25)
=1-.8413-.0062
=1-.8351
=.1649

b) For a random sample of 10 from this population calculate the probability that the mean brain weight is less than 1375 g or more than 1500 g.

=P(1375-1400/(100/square root of 10 < X < 1500-1400/(100/square root of 10)
=P(-.79 < Z < 3.16)
=P(Z<3.16) - P(Z<-.79)
=.9992-.2148 =.7844

2 a) How do you expect the sample mean brain weight to be distributed for random samples of size 20 from this population?
b) In such samples, how unusual would it be to find a mean brain weight in excess of 1470 g?

2. Originally Posted by mrtoilet
1. a) For an individual chosen at random from this population calculate the probability that their brain weight is less than 1375 g or more than 1500 g.

=1-P(1375<X<1500)
=1-P(1375-1400/100 < Z < 1500-1400/100)
=1-P(-.25 < Z < 1)
=1-P(Z<1) - P(Z<-.25)
=1-.8413-.0062
=1-.8351
=.1649
You are having notation problems. Be more careful and you will be more consistent and more accurate.

=1-P(1375-1400/100 < Z < 1500-1400/100)
Rewrite this to:
=1-P([1375-1400]/100 < Z < [1500-1400]/100)
You know what you mean by the original, but that is not what the notation says.

=1-P(Z<1) - P(Z<-.25)
Rewrite this to:
=1-[P(Z<-1) - P(Z<-.25)]
Again, you know what you mean by the original, but that is not what the notation says.

Also, you are missing the negative sign in the first probability. Your clue should have been that it was greater than 1/2. That is quite unlikely for a Normal Distribution and a value below the mean. Since this was the wrong value, they are now int he wrong order. You should have [P(Z<-0.25) - P(Z<-1.00)]

.0062
Is that supposed to be P(Z<-0.25)? You had better calculate that again. You simply must keep in mind some relative estimates. Keep the empirical rule in mind. Z = -0.25 is just a little below the mean. For P(Z < -0.25), you should get a value quite close to 1/2. For P(Z < -1), you should get a value in the neighborhood of 0.50 - 0.34 = 0.16

Good work. A little cleaner and a little more careful and you'll have it!!

3. Originally Posted by TKHunny
Also, you are missing the negative sign in the first probability. Your clue should have been that it was greater than 1/2. That is quite unlikely for a Normal Distribution and a value below the mean. Since this was the wrong value, they are now int he wrong order. You should have [P(Z<-0.25) - P(Z<-1.00)]
I'm confused. Why is the negative sign supposed to be there? I was following an example to help do this question and I can't work out where it's supposed to come from. Also, why are they in the wrong order?

Originally Posted by TKHunny
For P(Z < -0.25), you should get a value quite close to 1/2. For P(Z < -1), you should get a value in the neighborhood of 0.50 - 0.34 = 0.16
I'm using tables, so I don't know how much of a difference there is. For P(Z<-.25) I got .4013. For P(Z<-1) I got .1587. Is this about right?

Thanks.

4. That would be good (about 1/2 around 0.16) if only I were right.

I could have sworn both were below the mean on first reading. It should be +1 and you should get about 0.50 + 0.34 = 0.84 for p(Z < +1).

Show the whole problem again, using your best notation.

5. =1-P(1375<X<1500)
=1-P([1375-1400]/100 < Z < [1500-1400]/100)
=1-P(-.25<Z<1)
=1-[P(Z<1) - p(Z<-.25)]
=1-(.8413-.4013)
=1-.44
=.56

6. Now THAT is a work of art!

Moving on...
• Mean doesn't change. Excellent.
• Standard Deviation decreases by a factor of $\sqrt{10}$. Excellent.
• Notation. Needs a little work, but you hadn't had a chance to fix this one.
• Result? Well, except that you answered the wrong question, it's great. It's the same region as #1, right? Where's the "1 - "?

7. =1-P([1375-1400]/(100/square root of 10) < X < [1500-1400]/(100/square root of 10))
=1-P(-.79< Z<3.16)
=1-[P(Z<3.16) - P(Z<-.79)]
=1-(.9992-.2148)
=1-.7844
=.2156

?

8. Is this right? And do I do question 2a) the same way?