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The Normal Approximation to a Binomial Random Variable

Example 1: According to the Information Please almanac, 6% of the human population has blood type O-negative.  What is the probability that, in a simple random sample of 500, fewer than 25 have blood type O-negative?

In this binomial experiment, the random variable, X, is the number of individuals with blood type

O-negative, the probability, p, that an individual has type O-negative blood is .06 and the sample size, n, is 500.  We will approximate the probability that fewer than 25 individuals in the sample have type O-negative blood, that is, P(X<25) using the normal approximation to the binomial.

First, calculate the mean and standard deviation of this binomial random variable.  The mean, ,

equals n*p = 500*.06=30.  The standard deviation, , equals  = .

To approximate P(X<25) with a normal probability we calculate   (Note:  This adjustment from 25 to 24.5 is called a continuity correction).

Press 2^nd [DISTR], select 2:normalcdf( and type in  -1E99 , 24.5 , 30 , 5.31 ) and press ENTER.

You can compare this probability (.1505) that you obtained through a normal approximation to the actual probability obtained from the binomial distribution.

Press 2^nd [DISTR], select A:binomcdf( and type in  500 , .06 , 24) and press ENTER.

The actual P(X<25) is .1494.

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Example 2: According to Nielsen Media Research, 75% of all United States households have cable television.  Erica conducts a random sample of 1000 households in DuPage County and finds that 800 of them have cable.  What might Erica conclude?

In this binomial experiment, the random variable, X, is the number of households with cable TV, the probability, p, that a household has cable TV is .75 and the sample size, n, is 1000.  We will approximate the probability that at least 800 households in the sample have cable TV, that is

using the normal approximation to the binomial.

First, calculate the mean and standard deviation of this binomial random variable.  The mean, , equals n*p=1000*.75=750.  The standard deviation, , equals  = =

13.693.

To approximate  with a normal probability we calculate    (Note:  This adjustment from 800 to 799.5 is call a continuity correction).

Press 2^nd [DISTR], select 2:normalcdf( and type in 799.5 , 1E99 , 750 , 13.693) and press ENTER.

The answer is written in scientific notation.  To convert to standard notation, move the decimal point four places to the left.  The approximateis .00015, which is an extremely small probability.  This suggests that the percentage of households with cable TV is actually higher that 75%.

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