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

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

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

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:

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 0.3997. Value of Z corresponding to 0.3997 is -1.28. Next, convert Z to Z

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:

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”

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:

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,

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 )

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.

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.

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.

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.

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.

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

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