Concept Byte: Approximating a Binomial Distribution

For Use With Lesson 11-9

In Lesson 11-8, you found probabilities for a binomial experiment using the binomial theorem. The process works well with a small number of probabilities. However, calculating a large number of probabilities can be difficult. Sometimes you can use a normal distribution to approximate the binomial distribution.

For any normally distributed data set, you can use the standard normal distribution to find probabilities. The standard normal distribution is a normal distribution centered on the y-axis. The mean of the standard normal curve is 0 and the standard deviation is 1.

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You can transform a normal distribution to a standard normal distribution using the formula z = X − μ σ , z equals . fraction x minus mu , over sigma end fraction . comma  where X is a value from the original normal distribution, μ mu  is the original mean, and σ sigma  is the original standard deviation. The z-score that you get for X is the number of standard deviations that X lies above or below the mean in the standard normal distribution.

Example 1

Biology In a given population, the weights of newborn humans are normally distributed about the mean, 3250 g. The standard deviation is 500 g. Find the probability that a baby chosen at random weighs from 2250 g to 4250 g.

• Step 1 Find the z-scores of the lower and upper limits.

  z -score = X − μ σ z , minusscore , equals . fraction x minus mu , over sigma end fraction
  z 1 = 2250 − 3250 500 = − 1000 500 = − 2 z sub 1 , equals . fraction 2250 , minus , 3250 , over 500 end fraction . equals . negative , 1000 over 500 . equals negative 2	z 2 = 4250 − 3250 500 = 1000 500 = 2 z sub 2 , equals . fraction 4250 , minus , 3250 , over 500 end fraction . equals . 1000 over 500 . equals 2

• Step 2 Use the z-scores to find the probability.

  A z-score of 2 means the original weight is 2 standard deviations above the mean. A z-score of − 2 negative 2  means the original weight is 2 standard deviations below the mean.

  The probability that the baby's weight is between 2250 g and 4250 g is 13.5% + 34% + 34% + 13.5% = 95%.

You can approximate a binomial distribution with a normal distribution if you know the number of trials and the probability of success on each trial.

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