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# normal distribution calculation Watch

1. Can someone help me figure out the steps in between these two please?

Here's what I know:
on the RHS, it's 1-0.95 = 0.05. But which sign has that changed as a consequence? the >= outside the brackets or the <= inside- ie from P(Z<= ...) to P(Z>..)?
I know that it's divided by 2 since the normal dist. is symmetric about it's mean.
2. (Original post by zomgleh)
...
You may find it useful to sketch the areas on a diagram of the normal distribution, and hopefully all will be clear.

Last post for today.
3. (Original post by ghostwalker)
You may find it useful to sketch the areas on a diagram of the normal distribution, and hopefully all will be clear.

Last post for today.

Okay, done that and I think I get it. Just one more thing though, why has there been a change of sign from P(Z>z) < 0.025 and sqrt(n)/7.39 >=1.97 (the value for that prob)? Is it solely because the q. says ' find the smallest n such that'...?
4. (Original post by zomgleh)

Okay, done that and I think I get it. Just one more thing though, why has there been a change of sign from P(Z>z) < 0.025 and sqrt(n)/7.39 >=1.97 (the value for that prob)? Is it solely because the q. says ' find the smallest n such that'...?
If we rearrange to use the cumulative distribution, here are the details.

P(Z>z) < 0.025

Means 1-P(Z<=z) < 0.025

So P(Z<=z) >0.975

Hence z >= 1.97 (although it's 1.96 in my book, and that's the value I've seen elsewhere).

and sqrt(x)/7.39 >= 1.97
5. (Original post by ghostwalker)
If we rearrange to use the cumulative distribution, here are the details.

P(Z>z) < 0.025

Means 1-P(Z<=z) < 0.025

So P(Z<=z) >0.975

Hence z >= 1.97 (although it's 1.96 in my book, and that's the value I've seen elsewhere).

and sqrt(x)/7.39 >= 1.97
it makes sense this way, thanks, however we're only give the areas in the upper tail of the normal distribution. So P(Z>z) < .025
=z<1.97 so it doesn't make sense why the sign changes to z>... in the last bit? And yes, I got 1.96 too but I'm guessing it's because the question asks for the smallest z that satisfies P(Z>z)<.025 ? is that right do you think?
6. (Original post by ghostwalker)
If we rearrange to use the cumulative distribution, here are the details.

P(Z>z) < 0.025

Means 1-P(Z<=z) < 0.025

So P(Z<=z) >0.975

Hence z >= 1.97 (although it's 1.96 in my book, and that's the value I've seen elsewhere).

and sqrt(x)/7.39 >= 1.97
using another example-- why has the sign changed from P(Z>z) <= .025

Is it simply because the table I'm using gives the upper tail of the distribution?

to z>=1.96 ?
7. (Original post by zomgleh)
using another example-- why has the sign changed from P(Z>z) <= .025

Is it simply because the table I'm using gives the upper tail of the distribution?

to z>=1.96 ?
No, it has nothing to do with how you look up the tables - that's a later process.

This is purely a mathematical one.

Looking at my previously worked example, you need to think about what these things mean. Then it should be clear what's going on. I usually try and visualise for the normal distribution.

As to 1.97, rather than 1.96 - just seems wrong.

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Updated: April 3, 2013
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