Lecture 27Dr. MUMTAZ AHMED

MTH 161: Introduction To Statistics

Review of Previous Lecture

In last lecture we discussed:

Cumulative Distribution FunctionFinding Area under the Normal DistributionRelated examples

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Objectives of Current Lecture

In the current lecture:

Finding area under normal curve using MS-ExcelNormal Approximation to Binomial DistributionCentral Limit Theorem and its demonstrationRelated Examples

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Steps to find the X value for a known probability:

1. Find the Z value for the known probability2. Convert to X units using the formula:

ZσμX

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)

X μ Zσ=30+ 1.28 10 17.2sec

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:

Finding the X value for a Known Probability

(Inverse use of Normal Table)

Normal Distribution Approximation for Binomial

Distribution

Recall the binomial distribution:n independent trialsprobability 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”

nPμE(X)

P)nP(1-σVar(X) 2

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:

P)nP(1

npX

Var(X)

E(X)XZ

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,

P)nP(1

nPbZ

P)nP(1

nPaPb)XP(a

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) = 80Var(X) = σ2 = nP(1 – P) = 200(0.40)(1 – 0.40) = 48

( note: nP(1 – P) = 48 > 9 )

76 80 80 80P(76 X 80) P Z

48 48

P( 0.58 Z 0)

0.2190

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.

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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.

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CLT: An Example

Next: Consider the distribution of X1+X2 Bi(20,0.25)

Note: It looks more closer to normal.

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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.

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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.

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Review

Let’s review the main concepts:

Finding area under normal curve using MS-ExcelNormal Approximation to Binomial DistributionCentral Limit Theorem and its demonstrationRelated Examples

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Next LectureIn next lecture, we will study:

Joint DistributionsMoment Generating FunctionsCovarianceRelated Examples

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