Page 172 - Statistics for Dummies

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To find the normal approximation to the binomial distribution when n is
large, use the following steps:
1. Verify whether n is large enough to use the normal approximation by
checking the two appropriate conditions.
For the coin-flipping question, the conditions are met because n ∗ p =
100 ∗ 0.50 = 50, and n ∗ (1 – p) = 100 ∗ (1 – 0.50) = 50, both of which are at
least 10. So go ahead with the normal approximation.
2. Translate the problem into a probability statement about X.
For the coin-flipping example, you need to find p(X > 60).
3. Standardize the x-value to a z-value, using the z-formula:
For the mean of the normal distribution, use
binomial), and for the standard deviation , use
Part III: Distributions and the Central Limit Theorem (the mean of the
(the standard
deviation of the binomial; see Chapter 8).
For the coin-flipping example, use and
= . Then put these values into
the z-formula to get . To solve the problem, you
need to find p(Z > 2).
On an exam, you won’t see μ and σ in the problem when you have a
binomial distribution. However, you know the formulas that allow you to
calculate both of them using n and p (both of which will be given in the
problem). Just remember you have to do that extra step to calculate the
μ and σ needed for the z-formula.
4. Proceed as you usually would for any normal distribution. That is, do
Steps 4 and 5 described in the earlier section “Finding Probabilities
for a Normal Distribution.”
Continuing the example, p(Z > 2.00) = 1 – 0.9772 = 0.0228 from the Z-table
(appendix). So the chance of getting more than 60 heads in 100 flips of a
coin is only about 2.28 percent. (I wouldn’t bet on it.)
When using the normal approximation to find a binomial probability, your
answer is an approximation (not exact) — be sure to state that. Also show that
you checked both necessary conditions for using the normal approximation.

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