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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

- 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.

- 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

- 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.