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Using the Tables for the standard normal distribution

Using the Tables for the standard normal distribution. Tables have been posted for the standard normal distribution. Namely. The values of z ranging from -3.5 to 3.5. If X has a normal distribution with mean m and standard deviation s then. has a standard normal distribution. Hence.

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Using the Tables for the standard normal distribution

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  1. Using the Tables for the standard normal distribution

  2. Tables have been posted for the standard normal distribution. Namely The values of z ranging from -3.5 to 3.5

  3. If X has a normal distribution with mean mand standard deviation sthen has a standard normal distribution. Hence

  4. Example: Suppose X has a normal distribution with mean m=160 and standard deviation s=15 then find:

  5. This also can be explained by making a change of variable Make the substitution when and Thus

  6. The Normal Approximation to the Binomial

  7. The Central Limit theorem If x1, x2, …, xn is a sample from a distribution with mean m, and standard deviations s, Let Then the distribution of approaches the standard normal distribution as

  8. Hence the distribution ofapproaches the Normal distribution with or the distribution of approaches the normal distribution with

  9. Thus The Central Limit theorem states That sums and averages of independent R.Vs tend to have approximately a normal distribution for large n. Suppose that X has a binomial distribution with parameters n and p. Then where are independent Bernoulli R.V.’s

  10. Thusfor large n the Central limit Theorem states that has approximately a normal distribution with Thus for large n where X has a binomial (n,p) distribution and Y has a normal distribution with

  11. The binomial distribution

  12. The normal distribution m = np, s2 = npq

  13. Approximating Normal distribution Binomial distribution Binomial distribution n = 20, p = 0.70

  14. Normal Approximation to the Binomial distribution • X has a Binomial distribution with parameters n and p • Y has a Normal distribution

  15. Approximating Normal distribution P[X = a] Binomial distribution

  16. P[X = a]

  17. Example • X has a Binomial distribution with parameters n = 20 and p = 0.70

  18. Using the Normal approximation to the Binomial distribution Where Y has a Normal distribution with:

  19. Hence = 0.4052 - 0.2327 = 0.1725 Compare with 0.1643

  20. Normal Approximation to the Binomial distribution • X has a Binomial distribution with parameters n and p • Y has a Normal distribution

  21. Example • X has a Binomial distribution with parameters n = 20 and p = 0.70

  22. Using the Normal approximation to the Binomial distribution Where Y has a Normal distribution with:

  23. Hence = 0.5948 - 0.0436 = 0.5512 Compare with 0.5357

  24. Comment: • The accuracy of the normal appoximation to the binomial increases with increasing values of n

  25. Example • The success rate for an Eye operation is 85% • The operation is performed n = 2000 times • Find • The number of successful operations is between 1650 and 1750. • The number of successful operations is at most 1800.

  26. Solution • X has a Binomial distribution with parameters n = 2000 and p = 0.85 where Y has a Normal distribution with:

  27. = 0.9004 - 0.0436 = 0.8008

  28. Solution – part 2. = 1.000

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