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I am looking through an example (found here https://answers.yahoo.com/question/index?qid=20080303223718AA2T5qI) on the central limit theorem. I understand most of the best answer and am able to follow how they derive most of the numbers. What I am confused on is where the numbers for this line (= 0.8286574 - 0.1713426) come from? How are they derived. I am guessing this is basic stat principle that I am not grasping.

I have pasted a part of the solution given from the best answer on the webpost below

In this question we have 
X ~ Normal( μx = 0.49 , σx² = 0.0009996 ) 
X ~ Normal( μx = 0.49 , σx = 0.03161645 ) 

Find P( 0.46 < X < 0.52 )
= P( ( 0.46 - 0.49 ) / 0.03161645 < ( X - μ ) / σ < ( 0.52 - 0.49 ) / 0.03161645 )
= P( -0.948873 < Z < 0.948873 ) 
= P( Z < 0.948873 ) - P( Z < -0.948873 ) 
= 0.8286574 - 0.1713426    <------HOW ARE THESE NUMBERS DERIVED???
= 0.6573148
soccergal_66
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  • Using math typesetting will attract more readers and answers. Some more information: https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference – Sycorax Mar 25 '17 at 03:04
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    They're straight out of normal tables (or equivalently straight out of normal cdf functions of computer packages). See the examples [here](http://stats.stackexchange.com/questions/252645/calculate-p-value-for-a-negative-z) and [here](http://stats.stackexchange.com/questions/75056/how-to-determine-the-x-where-px-x-67-from-a-normal-distribution/75058#75058) which solve slightly different problems but illustrate the use of tables. – Glen_b Mar 25 '17 at 05:13
  • @Glen_b Reading through the comments you had provided on your 2nd example, do I understand this correctly, the (Z < 0.948873) is a z-value and 0.8286 is the corresponding value that goes with the z-value=0.948873? Looking at the z-table [here](http://stats.stackexchange.com/questions/75056/how-to-determine-the-x-where-px-x-67-from-a-normal-distribution/75058#75058) in your post, I see that a z-value of 0.94 is 0.8264 so how did they get 0.8286? – soccergal_66 Mar 25 '17 at 20:01
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    1. The "value that goes with 0.948873" is a probability, specifically the probability that Z is less than 0.948873 (it's also an *area*, since in the density function, area represents probability). 2. That table only has two decimal places for Z so under ideal conditions you can get about two decimal place accuracy back (more if you use interpolation). However, notice that 0.948873 is a *lot* closer to 0.95 than 0.94, so if you're not doing interpolation, look up 0.95 not 0.94. ... ctd – Glen_b Mar 26 '17 at 00:07
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    ctd... My guess is what they've actually done is call a function on a computer (stats packages can do it, as can essentially any general purpose spreadsheet program). If I do that in R I get exactly the same value they do. If I use tables but with linear interpolation (how to do that is discussed in detail [here](http://stats.stackexchange.com/questions/64538/how-do-i-find-values-not-given-in-interpolate-in-statistical-tables/64539)) it would give 0.828656, which is very close to the exact 0.82865739... (linear interpolation often works pretty well in normal tables) – Glen_b Mar 26 '17 at 00:07
  • @Glen_b thanks that is an incredibly helpful link on linear interpolation and understandable. Out of curiosity how do you calculate it in R as I am also in the process of learning R? – soccergal_66 Mar 26 '17 at 00:43
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    In R ... `pnorm(0.948873)` gives `[1] 0.8286574` ... by comparison, in Excel or LibreOffice, `=NORMSDIST(0.948873)` should give `0.828657396`. Similar commands for the standard normal cdf are easy to find in other packages. – Glen_b Mar 26 '17 at 01:05

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