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The Normal Approximation

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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  1. The Normal Approximation To the Binomial Distribution

  2. Recall: Binomial Distributions have: • 1. Fixed number of trials • 2. Independent Outcomes • 3. Exactly two outcomes • 4. Same probabilities for each trial Binomial Distributions

  3. 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

  4. 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)

  5. Summary of the Normal Approximation to the Binomial Distribution • Table 6-2 on p. 342 summarizes the corrections for continuity required……

  6. 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

  7. 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

  8. 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.

  9. 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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