The shape of the binomial distribution depends on the values of n and p.
For large n (say n > 20) and p not too near 0 or 1 (say 0.05 < p < 0.95) the distribution approximately follows the Normal distribution.
This can be used to find binomial probabilities.
If X ~ binomial (n,p) where n > 20 and 0.05 < p < 0.95 then approximately X has the Normal distribution with mean E(X) = np
so is approximately N(0,1).
MTB> set c1 DATA> 0:16 DATA> end MTB> pdf c1 c2; SUBC> binomial 16 0.5. MTB> name c2 'binomial' MTB> pdf c1 c3; SUBC> normal 8 2. *NOTE np = 8, npq = 4 * MTB> name c3 'normal' MTB> Gmplot c2 c1 c3 c1
Hence, if X has the binomial distribution ie. X~ binomial (n,p) and n is large, then X has approximately the Normal distribution with mean µ=np and standard deviation . This approximation is reasonably good when np>10 and n(1-p)>10.
In a particular faculty 60% of students are men and 40% are women. In a random sample of 50 students what is the probability that more than half are women?
Let RV X = number of women in the sample.
Assume X has the binomial distribution with
Then E(X) = np = 50 x 0.4 = 20
var(X) = npq = 50 x 0.4 x 0.6 = 12
so approximately X ~ N(20,12).
We need to find P(X > 25). Note - not P(X >= 25).
The exact answer calculated from binomial probabilities
is P(X>25) = P(X=26) + P(X=27) + ... + P(X=50) = 0.0573)
The approximate probability, using the continuity correction, is
(The value 25.5 was chosen as the outcome 25 was not to be included but the outcomes 26, 27, 50 were to be included in the calculation.)
Similarly, if the example required the probability that less than 18 students were women, the continuity correction would require the calculation
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