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**supastar****Member**- Registered: 2005-04-24
- Posts: 2

a large number of observations follow a normal distribution with a mean of 120 and a standard deviation of 80. What proportion of observations is:

i) below 100

ii) above 130

iii) between 100 and 130

iv) between 130 and 140

v) between 115 and 135??

anyone have an ideas because I certainly dont

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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,626

Arrggghhh indeed !

Let me put my thinking cap on here ...

You can read up on Mean and Standard Deviation if you want to, first.

Well, you said "normal" distribution. That is a special word that says it follows a certain well-defined pattern on a graph that looks like a bell. (Funny that "normal" means it is special ...)

You can look at the curve here

First step is to figure how many *standard deviations* (80 in your case) that each value is from the *mean* (120 in your case).

Let us work on your examples:

i) below 100 ==> 100 is 20 away from the mean of 120, so it is 20/80 = 0.25 standard deviations away.

ii) above 130 ==> 130 is 10 away from the mean of 120, so it is 10/80 = 0.125 standard deviations away.

The next step is to look up these values (0.25 and 0.375), which are called "z-scores", in a table. Yep, you gotta look em up, unless you have them as a special function on your calculator or software.

Looking up these values, I get:

0.25: 0.0987

0.12: 0.0478

That means that 0.0987, or 9.87%, of the population are between the mean (120) and 100

And that 0.0478, or 4.78% are between the mean and 130

The rest is just figuring what the question asked: above, below, between?

The first question says "below 100" - well, we know that 9.87% are between 100 and 120, and 50% must be greater than 120 (because 120 is the mean), so 59.87% are GREATER, and therefore 41.2% must be LESSER, or below!

Likewise "above 130" can be figured out to be 45.2%

All the others can be worked out in a similar way, and I leave that up to you ... !

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman

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**supastar****Member**- Registered: 2005-04-24
- Posts: 2

thanx a lot, much appreciated.

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