1. MTH 161: Introduction To Statistics
Lecture 27
Dr. MUMTAZ AHMED

2. Review of Previous Lecture
In last lecture we discussed:
Cumulative Distribution Function
Finding Area under the Normal Distribution
Related examples

3. Objectives of Current Lecture
In the current lecture:
Finding area under normal curve using MS-Excel
Normal Approximation to Binomial Distribution
Central Limit Theorem and its demonstration
Related Examples

4. Finding the X value for a Known Probability (Inverse use of Normal Table)
Steps to find the X value for a known probability:
1. Find the Z value for the known probability
2. Convert to X units using the formula:

5. Finding the X value for a Known Probability (Inverse use of Normal Table)
Example: Time required to inject a shot of penicillin has been observed to be normally distributed with a mean of 30 seconds and a SD of 10 seconds. Find (a) 10th percentile.
Solution:
Lookup 0.4 in area table, it doesn’t appear
So take closest probability to 0.4 which is
Value of Z corresponding to is
Next, convert Z to Z

6. Finding the X value for a Known Probability (Inverse use of Normal Table)
Example: Time required to inject a shot of penicillin has been observed to be normally distributed with a mean of 30 seconds and a SD of 10 seconds. Find (b) 90th percentile.
Solution:

7. Normal Distribution Approximation for Binomial Distribution
Recall the binomial distribution:
n independent trials
probability of success on any given trial = P
Random variable X:
Xi =1 if the ith trial is “success”
Xi =0 if the ith trial is “failure”

8. Normal Distribution Approximation for Binomial Distribution
The shape of the binomial distribution is approximately normal if n is large
The normal is a good approximation to the binomial when nP(1 – P) > 9
Standardize to Z from a binomial distribution:

9. Normal Distribution Approximation for Binomial Distribution
Let X be the number of successes from n independent trials, each with probability of success P.
If nP(1 - P) > 9,

10. Binomial Approximation: An Example
40% of all voters support ballot. What is the probability that between 76 and 80 voters indicate support in a sample of n = 200 ?
E(X) = µ = nP = 200(0.40) = 80
Var(X) = σ2 = nP(1 – P) = 200(0.40)(1 – 0.40) = 48
( note: nP(1 – P) = 48 > 9 )

11. Central Limit Theorem: CLT
The central limit theorem says that sums of random variables tend to be approximately normal if you add large numbers of them together.
CENTRAL LIMIT THEOREM:
Let X1, X2, … Xn be random draws from any population.
Let S = X1 + X2 + … + Xn.
Then the standardization of S will have an approximately standard normal distribution if n is large.
Note:
Independence is required, but slight dependence is OK.
Each term in the sum should be small in relation to the SUM.

12. Consider the distribution of X  Bi(10,0.25)
CLT: An Example
we illustrate graphically the convergence of Binomial to a Normal distribution.
Consider the distribution of X  Bi(10,0.25)
Note: It does not look very normal.

13. Next: Consider the distribution of X1+X2  Bi(20,0.25)
CLT: An Example
Next: Consider the distribution of X1+X2  Bi(20,0.25)
Note: It looks more closer to normal.

14. Next: Consider the distribution of X1+X2+X3+X4  Bi(40,0.25)
CLT: An Example
Next: Consider the distribution of X1+X2+X3+X4  Bi(40,0.25)
Note: It looks even more closer to normal.
This just illustrates the Central Limit Theorem. As we add random variables, the distribution of the sum begins to look closer and closer to a normal distribution. If we standardize, then it looks like a standard normal.

15. CLT: An Example
Note: As we add random variables, the distribution of the sum begins to look closer and closer to a normal distribution.
If we standardize, then it looks like a standard normal.

16. Review Let’s review the main concepts:
Finding area under normal curve using MS-Excel
Normal Approximation to Binomial Distribution
Central Limit Theorem and its demonstration
Related Examples

17. Next Lecture In next lecture, we will study: Joint Distributions
Moment Generating Functions
Covariance
Related Examples