The normal approximation
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The Normal Approximation. To the Binomial Distribution. Recall: Binomial Distributions have: 1. Fixed number of trials 2. Independent Outcomes 3. Exactly two outcomes 4. Same probabilities for each trial. Binomial Distributions.

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The Normal Approximation

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The normal approximation

The Normal Approximation

To the Binomial Distribution


Binomial distributions

  • Recall: Binomial Distributions have:

  • 1. Fixed number of trials

  • 2. Independent Outcomes

  • 3. Exactly two outcomes

  • 4. Same probabilities for each trial

Binomial Distributions


The normal approximation

  • We can use the Normal Distribution to find probabilities for Binomial distributions that have large n, or when the calculations become difficult.

  • We can only use then when n * p > 5 AND n * q >5


The normal approximation

A correction for continuity may be used .

-a correction employed when a continuous distribution is used to approximate a discrete distribution.

-Ex. P(X) = 8 would imply having to find P(7.5<X<8.5)


The normal approximation

  • Summary of the Normal Approximation to the Binomial Distribution

  • Table 6-2 on p. 342 summarizes the corrections for continuity required……


Steps used to approximate binomial distributions

  • 1. Check to make sure np> 5 and nq > 5

  • 2. Find the mean and the standard deviation

  • 3. Write the problem in probability notation, using X.

  • 4. Rewrite the problem by using the continuity correction factor and show the area under the normal distribution.

  • 5. Find the corresponding z-values.

  • 6. Find the solution.

Steps used to approximate Binomial Distributions


Example

  • 6% of American Drivers read the newspaper while driving. If 300 drivers are selected at random, find the probability that exactly 25 read the newspaper while driving.

  • 1. (300*.06) = 18 (300 * .94) = 282

  • 2.

  • 3. What P(X) are we trying to find?? P(X = 25)

  • 4. Use the continuity correction…. P(24.5<X<25.5)

  • 5. Find both z-values.

  • = 1.58

  • 6. The area between the two z-values are:

    0.9656 – 0.9429 = 0.0227 So 2.27% read the paper and drive

Example


The normal approximation

  • 10% of members at a golf club are widowed. If 200 club members are selected at random, find the probability that exactly 10 will be widowed.


The normal approximation

  • If a baseball player’s batting average is 0.32, find the probability that the player will get at most 26 hits in 100 times at bat.


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