SOLUTION: We have been using the normal distribution to approximate situations that are, in fact, binomial events. Create an example of a binomial event that can be approximated by a normal

Question 1194451: We have been using the normal distribution to approximate situations that are, in fact, binomial events. Create an example of a binomial event that can be approximated by a normal distribution and:
a)Demonstrate how accurate the approximation is by using both approaches to find the probability of the same event. Hint: Calculate the probability of the event as a binomial (sum of all binomial events) and calculate the probability of the approximated event using a normal distribution, and compare them to see how close the approximation is.
b)Describe the conditions under which the normal distribution would give a less accurate approximation.
Explain a situation in which the criteria for using the normal approximation would be met, i.e. np≥5 and n(1-p)≥5 , and yet you would decide not to use the normal distribution.
Answer by Boreal(15235) (Show Source): You can put this solution on YOUR website!
b and c first. Use a normal approximation for a skewed distribution binomial with small n and p=0.9, and it will have more error.
If a binomial is completely symmetrical, it may be approximated by a normal distribution quite well with small np or n(1-p).
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A baseball player is batting 0.400. What is the probability he will get no more than 10 hits in the next 50 at bats? The exact is 0.00076 or 0.0008 probability
look at 50C9*0.40^9*0.60^41=0.000527
for 8 hits 0.000169, and this for the others will converge to the exact.
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np=20 hits mean
np(1-p)=20*0.6=12 hits variance; SD is sqrt(V)=3.464 hits
probability of z < 9.5 is z(9.5-20)/3.464=-3.03
The probability of z < -3.03 is 0.0012, a decent approximation.
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