Binomial approximation to normal

pighty

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Jul 10, 2014
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Hi, I have a problem in understanding the binomial approximation to normal.
Here is my exercise:
"For a normal distribution in which 1/4 of values are lower than 85 and 1/5 are lower than 75, calculate

[FONT=MathJax_Math-italic]σ
[FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math-italic]X[FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Math-italic]X[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Math-italic]z[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Math-italic]z[/FONT][FONT=MathJax_Main]2[/FONT][/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]75[FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]85[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]0.84[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]0.67[/FONT][/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]6.62

[/FONT]
[FONT=MathJax_Math-italic]μ[FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math-italic]X[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Math-italic]z[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Math-italic]σ[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]85[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]0.67[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]6.62[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]89.43[/FONT][/FONT]



[FONT=MathJax_Math-italic]X[FONT=MathJax_Main]20[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]75[/FONT][/FONT][FONT=MathJax_Math-italic]

[FONT=MathJax_Math-italic]X[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]−[/FONT]
[/FONT]


[FONT=MathJax_Math-italic]R[FONT=MathJax_Math-italic]I[/FONT][FONT=MathJax_Math-italic]Q[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math-italic]Q[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Math-italic]Q[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]1.34[/FONT][FONT=MathJax_Math-italic]σ[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]8.87[/FONT][/FONT]

[FONT=MathJax_Math-italic]X[FONT=MathJax_Main]97.5[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math-italic]μ[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]1.96[/FONT][FONT=MathJax_Math-italic]σ[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]102.4[/FONT][/FONT]

Calculate the probability of the following events:

1) Only 1 in 20 casually extracted values is between 75 and 85



[FONT=MathJax_Main]20[FONT=MathJax_Main]⋅[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]0.05[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]0.95[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]19[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]0.377[/FONT][/FONT]

2) At least 1 in 20 casually extracted values are between 75 and 85

[FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]([FONT=MathJax_Main]0.95[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]20[/FONT][/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]0.641[/FONT]


3) More than 25 in 35 casually extracted values are higher than 75

[FONT=MathJax_Math-italic]π[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]0.8[/FONT]
[FONT=MathJax_Math-italic]μ[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math-italic]n[/FONT][FONT=MathJax_Math-italic]π[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]28

[FONT=MathJax_Math-italic]σ[FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]n[/FONT][FONT=MathJax_Math-italic]π[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Math-italic]π[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Size2]√[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]2.36[/FONT][/FONT][/FONT]

[FONT=MathJax_Math-italic]z[FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]24.5[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]28[/FONT][FONT=MathJax_Main]2.36[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1.48[/FONT][/FONT]
[FONT=MathJax_Math-italic]p[FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]0.0694[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]0.930[/FONT][/FONT]

Those questions I think I could solve, but the following:
4) the mean of 15 casually extracted values is between 110 and 130 = ??
[FONT=MathJax_Math-italic]μ[FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math-italic]μ[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]84.93[/FONT][/FONT]

[FONT=MathJax_Math-italic]E[/FONT][FONT=MathJax_Math-italic]S[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math-italic]σ[FONT=MathJax_Math-italic]n[/FONT][FONT=MathJax_Main]√[/FONT][/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]6.62[FONT=MathJax_Main]15[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]√[/FONT][/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]1.71

[FONT=MathJax_Math-italic]z[FONT=MathJax_Main]=[/FONT][FONT=MathJax_Size3]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Math-italic]μ[/FONT][FONT=MathJax_Math-italic]E[/FONT][FONT=MathJax_Math-italic]S[/FONT][FONT=MathJax_Size3])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]130[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]84.93[/FONT][FONT=MathJax_Main]1.71[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]26.35[/FONT][/FONT][/FONT]


my z value is 26.65 and it's way to high!

5) the difference of 2 casually extracted values is higher than 45 = ??

[FONT=MathJax_Math-italic]s[FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]√[/FONT][FONT=MathJax_Math-italic]σ[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]9.36[/FONT][/FONT]

[FONT=MathJax_Math-italic]z[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]45[FONT=MathJax_Main]9.36[/FONT][/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]4.8
[/FONT]

again, my z is 4.8, really really high cos p = 8*10^(-7)
6) the third number extracted in 35 values is between 110 and 130
no clue here
7) Minimal values to extract to have at least 99% probability to have
- at least one value lower than 75 = ?
- at least one value between 75 and 85 = ?
8) Minimal number of pairs to extract to have at least 99% probabilty to have
- at least one pair of values whose difference is higher than 50
[/FONT]




Thank you!
 
Last edited:
Hi, I have a problem in understanding the binomial approximation to normal.
Here is my exercise:
"For a normal distribution in which 1/4 of central values are lower than 85 and 1/5 are lower than 75, calculate

I am not certain what you mean by "central values" but I assume you mean from probability 1/2- 1/8= 3/8 to 1/2+ 1/8= 5/8.
From a table of the standard normal distribution, I see that 1/4 of the values lie between -0.32 and .32 so that we have \(\displaystyle \frac{85- \mu}{\sigma}= .32\) and that 1/5 lie between -0.25 and 0.25 so that \(\displaystyle \frac{75- \mu}{\sigma}= .25\). Solve those two equations for \(\displaystyle \mu\) and \(\displaystyle \sigma\).

[FONT=MathJax_Math-italic]σ
[FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math-italic]X[FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Math-italic]X[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Math-italic]z[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Math-italic]z[/FONT][FONT=MathJax_Main]2[/FONT][/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]75[FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]85[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]0.84[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]0.67[/FONT][/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]6.62
You seem to be assuming that "1/4 of central values" is between 1/4 and 3/4, but that would be 3/4- 1/4= 1/2 of the whole. Instead use 3/8 and 5/8: 5/8- 3/8= 2/8= 1/4.

[/FONT][FONT=MathJax_Math-italic]μ[FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math-italic]X[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Math-italic]z[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Math-italic]σ[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]85[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]0.67[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]6.62[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]89.43[/FONT][/FONT]



[FONT=MathJax_Math-italic]X[FONT=MathJax_Main]20[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]75[/FONT][/FONT][FONT=MathJax_Math-italic]

[FONT=MathJax_Math-italic]X[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]−[/FONT]
[/FONT]


[FONT=MathJax_Math-italic]R[FONT=MathJax_Math-italic]I[/FONT][FONT=MathJax_Math-italic]Q[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math-italic]Q[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Math-italic]Q[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]1.34[/FONT][FONT=MathJax_Math-italic]σ[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]8.87[/FONT][/FONT]

[FONT=MathJax_Math-italic]X[FONT=MathJax_Main]97.5[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math-italic]μ[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]1.96[/FONT][FONT=MathJax_Math-italic]σ[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]102.4[/FONT][/FONT]

Calculate the probability of the following events:

1) Only 1 in 20 casually extracted values is between 75 and 85



[FONT=MathJax_Main]20[FONT=MathJax_Main]⋅[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]0.05[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]0.95[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]19[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]0.377[/FONT][/FONT]

2) At least 1 in 20 casually extracted values are between 75 and 85

[FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]([FONT=MathJax_Main]0.95[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]20[/FONT][/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]0.641[/FONT]


3) More than 25 in 35 casually extracted values are higher than 75

[FONT=MathJax_Math-italic]π[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]0.8[/FONT]
[FONT=MathJax_Math-italic]μ[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math-italic]n[/FONT][FONT=MathJax_Math-italic]π[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]28

[FONT=MathJax_Math-italic]σ[FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]n[/FONT][FONT=MathJax_Math-italic]π[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Math-italic]π[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Size2]√[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]2.36[/FONT][/FONT][/FONT]

[FONT=MathJax_Math-italic]z[FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]24.5[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]28[/FONT][FONT=MathJax_Main]2.36[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1.48[/FONT][/FONT]
[FONT=MathJax_Math-italic]p[FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]0.0694[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]0.930[/FONT][/FONT]

Those questions I think I could solve, but the following:
4) the mean of 15 casually extracted values is between 110 and 130 = ??
[FONT=MathJax_Math-italic]μ[FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math-italic]μ[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]84.93[/FONT][/FONT]

[FONT=MathJax_Math-italic]E[/FONT][FONT=MathJax_Math-italic]S[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math-italic]σ[FONT=MathJax_Math-italic]n[/FONT][FONT=MathJax_Main]√[/FONT][/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]6.62[FONT=MathJax_Main]15[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]√[/FONT][/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]1.71

[FONT=MathJax_Math-italic]z[FONT=MathJax_Main]=[/FONT][FONT=MathJax_Size3]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Math-italic]μ[/FONT][FONT=MathJax_Math-italic]E[/FONT][FONT=MathJax_Math-italic]S[/FONT][FONT=MathJax_Size3])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]130[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]84.93[/FONT][FONT=MathJax_Main]1.71[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]26.35[/FONT][/FONT][/FONT]


my z value is 26.65 and it's way to high!

5) the difference of 2 casually extracted values is higher than 45 = ??

[FONT=MathJax_Math-italic]s[FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]√[/FONT][FONT=MathJax_Math-italic]σ[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]9.36[/FONT][/FONT]

[FONT=MathJax_Math-italic]z[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]45[FONT=MathJax_Main]9.36[/FONT][/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]4.8
[/FONT]

again, my z is 4.8, really really high cos p = 8*10^(-7)
6) the third number extracted in 35 values is between 110 and 130
no clue here
7) Minimal values to extract to have at least 99% probability to have
- at least one value lower than 75 = ?
- at least one value between 75 and 85 = ?
8) Minimal number of pairs to extract to have at least 99% probabilty to have
- at least one pair of values whose difference is higher than 50
[/FONT]



Thank you!
 
Last edited:
I am not certain what you mean by "central values" but I assume you mean from probability 1/2- 1/8= 3/8 to 1/2+ 1/8= 5/8.
From a table of the standard normal distribution, I see that 1/4 of the values lie between -0.32 and .32 so that we have \(\displaystyle \frac{85- \mu}{\sigma}= .32\) and that 1/5 lie between -0.25 and 0.25 so that \(\displaystyle \frac{75- \mu}{\sigma}= .25\). Solve those two equations for \(\displaystyle \mu\) and \(\displaystyle \sigma\).


You seem to be assuming that "1/4 of central values" is between 1/4 and 3/4, but that would be 3/4- 1/4= 1/2 of the whole. Instead use 3/8 and 5/8: 5/8- 3/8= 2/8= 1/4.

I am sorry but I mistakenly reported the exercise! It said 1/4 of the VALUES. Sorry. Correcting
 
Last edited:
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