Normal Approximation to the Binomial Distribution:
Normal Approximation to the Binomial Distribution:
Consider the binomial distribution with:
n = number of trials
r = number of successes
p = probability of success
q = probability of failure = 1 – p
If np > 5 and nq > 5, then r has a binomial distribution that is approximated by a normal distribution with
µ = np and σ = √npq
*** View Example 4 and 5 on text p. 423 – 426.
Converting r Values to x Values
Remember that when using the normal distribution to approximate the binomial, we are computing the areas under bars. The bar over r goes from r – 0.5 to r + 0.5. If r is a left endpoint of an interval, we subtract 0.5 to get the corresponding normal variable x. If r is a right endpoint of an interval, we add o.5 to get the corresponding variable x.
For example, P(6 < x < 10) where r is a binomial variable is approximated by P(5.5 < x < 10.5) where x is the corresponding normal variable.
Both the binomial and Poisson distributions are for discrete random variables. Therefore, adding or subtracting 0.5 to r was not necessary when we approximated the binomial distribution by the Poisson distribution. However, the normal distribution is for a continuous random variable. In this case, adding or subtracting 0.5 (as appropriate) to r will improve the approximation of the normal to the binomial distribution.
*** View Guided Exercise 4 on text p. 426 – 427.
Complete text p. 427 – 430.
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