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 m and standard deviation s then
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:

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

7. The Normal Approximation to the Binomial

8. 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

9. Hence the distribution of approaches
the Normal distribution with
or the distribution of
approaches the normal distribution with

10. 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

11. Thus for 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

12. The binomial distribution

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

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

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

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

18. P[X = a]

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

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

21. Hence
= =
Compare with

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

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

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

27. Hence
= =
Compare with

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

29. 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.

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

31. = =

32. Solution – part 2.
= 1.000