Section 6-5 the Central Limit Theorem

Section 6-5 The Central Limit Theorem

I. Sampling Distribution of Sample Mean ( )

Example 1: Population Distribution Table

(a) Find the population mean and population standard deviation of the

population distribution table.

(b) Construct a probability histogram for x

Example 2: From the population distribution of example 1,

2 random variables are randomly selected.

(a) List out all possible combinations (sample space) and for each combination.

(b) Construct a probability distribution table for .

(d) Find the mean of the sampling distribution of .

(e) Find the standard deviation of the sampling distribution of .

(f) Compare with .

(g) Compare with .

Population parameterSample statistics

Mean

Standard deviation

Population Distribution Sampling Distribution

II. Central Limit Theorem

If the population distribution is normally distributed, the sampling distribution of will be normally distributed for n 30.

If the population distribution is not normally distributed, the sampling distribution of will be normally distributed for any size of n  30

Example 1: Population distribution

Given: = 50, = 10

(a) Find and for n = 4

(b) Is the sampling distribution normally distributed?

Example 2:(Ref: General Statistics by Chase/Bown, 4th Ed.)

A population has mean 325 and variance 144. Suppose the distribution of

sample means is generated by random samples of size 36.

(a)Find and

(b)Find

(c)Find

Example 3:

The average number of days spent in a North Carolina hospital for a coronary bypass in 1992 was 9 days and the standard deviation was 4 days (North Carolina Medical Database Commission, Consumer’s Guide to Hospitalization Charges in North Carolina Hospitals, August 1994). What is the probability that a random sample of 30 patients will have an average stay longer than 9.5 days?

Example 4:

Suppose the test scores for an exam are normally distributed with = 75, = 8

(a) What percent of the students has a score greater than 85?

(b) What is the probability that 4 randomly selected students will have a mean score

higher than 85?

Section 6 – 6 Normal Approximation to the Binomial Distribution

I.When to use a N dist. to approximate a Bi dist.?

Recall that a binomial distribution is determined by n and p. When p is

approximately 0.5, and as n increases, the shape of the binomial distribution

becomes similar to the normal distribution. (Ref: Elementary Statistics 3rd ed.

by Bluman). In order to use a normal distribution to approximate a binomial

distribution, n must be sufficiently large. It is known n will be sufficiently large

if np  5 and nq  5.

When using a normal distribution to approximate a binomial distribution, the

mean and standard deviation of the normal distribution is the same as the

binomial distribution. Now recall the formulas for finding the mean and

standard deviation of a binomial distribution .

II.Continuity Correction

In addition to the condition of np  5 and nq  5 , a correction for continuity is

used in employing a continuous distribution ( N dist.) to approximate a discrete

distribution ( Bi dist.).

Warning:The continuity correction should be used only when approximating

A binomial probability with a normal probability.

Don’t use the continuity correction with other normal probability problems.

Continuity correction  x  0.5

Example 1:

Use the continuity correction to rewrite each expression:

(a)Bi Dist.N Dist.(d)Bi Dist.N Dist.

P( x > 6) P( 1 < x < 7) 

(b)Bi Dist.N Dist.(e)Bi Dist.N Dist.

P( x  3) P ( 5  x  10) 

(c)Bi Dist.N Dist.(f)Bi Dist.N Dist.

P( x  9) P (4 < x  6) 

III.Using a Normal Distribution to approximate a Binomial Distribution

Step 1:Check whether the normal distribution can be used.

( np  5 and nq  5 )

Step 2:Find the mean and standard deviation .

Step 3:Write the problem in probability notation, using x.

Step 4:Rewrite the problem by using the continuity correction factor.

Continuity correction  x  0.5

Step 5:Find the corresponding z value(s).

Step 6:Use the z table to find the center area and adjust the center area to

answer the question.

Example 1:(Ref: General Statistics by Chase/Bown, 4th Ed.)

Assume that the experiment is a binomial experiment.

Find the probability of 10 or more successes,

where n = 13 and p = .4.

(a)Use the Bi table

(b)Using the normal approximation to the binomial.

Example 2:A dealer states that 90% of all automobiles sold have air

conditioning. If the dealer sells 250 cars, find the probability

that fewer than 5 of them will not have air conditioning.

Example 3:In a corporation, 30% of the people elect to enroll in the

financial investment program offered by the company.

Find the probability that of 800 randomly selected people,

between 260 and 300 inclusive have enrolled in the program.