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Normal Approximation to the Binomial Distribution

Normal Approximation to the Binomial Distribution. Objective. To understand and know how to calculate probabilities when asked to use a Normal approximation to the Binomial Distribution. To know the conditions under which this approximation can + should be used. Continuity Correction.

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Normal Approximation to the Binomial Distribution

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  1. Normal Approximation to the Binomial Distribution

  2. Objective • To understand and know how to calculate probabilities when asked to use a Normal approximation to the Binomial Distribution. • To know the conditions under which this approximation can + should be used.

  3. Continuity Correction • Because the binomial distribution is discrete and the normal distribution is continuous we need to apply a continuity correction. (last lesson)

  4. Conditions of Use • If X~Bi(n,p) then we can use the normal distribution if • n is large (>20) and • p is around 0.5 • This should lead to np>5 and nq>5 • Then X~N(np,npq) • The larger n is, the better approximation we calculate.

  5. What’s the point in approximating? • Calculations may be less tedious. • Calculations will be made easier and quicker. • Imagine, X~Bi(1000,0.45) find P(X≤342) • Tables are not big enough. • Lots of calculations.

  6. Example 1 • Find the probability of 4,5,6 or 7 heads when a dice is rolled 12 times, using: (i) the binomial distribution. (ii) the normal distribution.

  7. (i)

  8. (ii)

  9. Example 2: Exam Question • Drug manufacturer claims that a certain drug cures on average 80% of the time. He accepts the claim if k or more patients are cured. (i) State the distribution of X

  10. (ii) Find the probability that the claim will be accepted when 15 individuals are tested and k is set at 10.

  11. (iii) A more extensive trial is undertaken on 100 patients. The distribution may now be approximated by the Normal distribution. State the approximating distribution.

  12. (iv) Using the approximating distribution, estimate the probability that the claim will be rejected if k is set at 75.

  13. Work: Ex 2B pg 54 3,4,5,6

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