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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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

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)

Summary of the Normal Approximation to the Binomial Distribution

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

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

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

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.

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.