Question 1185839: Dear sir/madam,
How to find easily higher factorial values ?In this example do we need to approximate the mean as 2.718?And then is it correct to get answer by 1- p( x=39)The example is here
When appointing computer operators a firm requires that the candidates pass a
written examination. The examination paper contains 100 multiple choice questions, each with a
choice of three answers of which only one is correct." A "pass is obtained by answering 40 or more
questions correctly. Use the Normal distribution as an approximation to the binomial distribution
to estimate the probability that a candidate who chooses the answer to each question randomly
will pass the examinations.
Answer by ikleyn(52817)   (Show Source):
You can put this solution on YOUR website! .
When appointing computer operators a firm requires that the candidates pass a
written examination. The examination paper contains 100 multiple choice questions, each with a
choice of three answers of which only one is correct. "A" pass is obtained by answering 40 or more
questions correctly. Use the Normal distribution as an approximation to the binomial distribution
to estimate the probability that a candidate who chooses the answer to each question randomly
will pass the examinations.
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The original problem is on Binomial distribution with n = 100 trials and p = 1/3 for individual
success (p= 1/3 to answer correctly and q = 1-p = 2/3 to answer incorrectly).
So, formally there are standard formulas for binomial distribution, but the Math does not
recommend to use it, because the formulas include high degree exponents and too great binomial
coefficients. Instead, there is another, very effective and standard approach on solving such problems.
This approach is to use an appropriate normal distribution as an approximation to the Binomial distribution.
The relevant normal distribution has the mean m = n*p =  = 33.33333
and the standard deviation S =  =  = 4.714.
This normal curve covers all 100 questions on the horizontal axis, and we should take the area under
this normal curve on the right of the threshold value 40.
A convenient tool to find the area of interest is to use the standard function normcdf
(which means Cumulative Distrifution Functionfor normal distribution).
This function is in any regular calculator like TI-83/84.
So, you write/print in your calculator
z1 z2 mean SD <<<---=== formatting pattern.
P = normcdf(39.5, 9999, 33.33333, 4.714)
Here z1 = 39.5 stands for 40 and presents the continuing correction; z2 stand for infinity on the horizontal axis,
mean is the mean and SD is the standard deviation.
Then the function will return the probability, which you are looking for: P = 0.0954.
It is the ANSWER to the problem's question P = 0.0954, obtained with the normal distribution approximation.
Solved, with explanations.
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If you want to learn more from the Internet about the normal distribution as an approximation for binomial distribution,
go to Google with these keywords "normal distribution as an approximation for binomial distribution" and learn from there.
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