Quick 6-4 Supplement! Sampling Distribution: The distribution of sample

statistics with all samples having the same size, and the procedure repeated indefinitely.

Unbiased Estimators: Sample statistics that target the population parameter.• The mean of the sampling distribution will approach the

real parameter value. (the sample proportion)

Biased Estimators: Don’t target parameters.• Median, Range, s

Section 6.5The Central Limit Theorem

Given

1. The random variable x has a distribution (which may or may not be normal) with mean and standard deviation .

2. Simple random samples all of the same size n are selected from the population.

Conclusions

1. The distribution of sample means will, as the sample size increases, approach a normal distribution.

2. The mean of all sample means is the population mean .

3. The standard deviation of all sample means is .

Practical Rules Commonly Used

1. For a population with any distribution, if n > 30, then the sample means have an approximately normal distribution.

2. If n ≤ 30 and the original population has a normal distribution, then the sample means have an approximately normal distribution.

3. If n ≤ 30 and the original distribution does not have a normal distribution then the methods of this section do not apply.

The Central Limit

Theorem:In PicturesAs the sample size increases,

the distribution of sample means approaches a

normal distribution.

Important Points to Note• As the sample size increases,

the distribution of sample means

tends to approach a normal

distribution. • The mean of the sample means is the same as the

mean of the original

population.

Important Points to Note

• As the sample size increases,

the width of the graph becomes

narrower, showing that the

standard deviation of the sample mean

becomes smaller.

Cool … So Why Do We Care?

Think of what happens when we have a normal distribution. What can we do?• Table A2• Calculate any area, therefore any

probability. The central limit theorem shows us

that most samples can fit a normal distribution.

So our calculation power (and therefore understanding) is endless!!!

Water Taxi Safety Previously, we talked about the Baltimore

water taxi that sank because of old weight limit standards. The water taxi assumed

that the average person weight was 140lbs.

Given that today the weights of men are normally distributed with a mean of 172 lbs and a standard deviation of 29 lbs. Find the probability that if an individual

man is random selected, his weight will be greater than 175lbs.

Water Taxi Safety Previously, we talked about the Baltimore

water taxi that sank because of old weight limit standards. The water taxi assumed that the average person weighs 140lbs.

Given that today the weights of men are normally distributed with a mean of 172 lbs and a standard deviation of 29 lbs. Find the probability that 20 randomly selected men

will have a mean weight that is greater than 175lbs (so that their total weight exceeds

the current safe capacity of 3500 lbs).

Reminder!!!

The central limit theorem works if the sample size is greater than 30, or if the original population is

normally distributed.

The Fundamental Difference

INDIVIDUAL VALUE SAMPLE OF VALUES

When working with an individual value from a normally distributed population, use the methods of Section 6.3.

When working with a mean for some sample be sure to use the value of for

the standard deviation of the sample mean.

Water Taxi Safety Previously, we talked about the Baltimore

water taxi that sank because of old weight limit standards. The water taxi assumed that the average person weighs 140lbs.

Given that today the weights of men are normally distributed with a mean of 172 lbs and a standard deviation of 29 lbs. Find the probability that 20 randomly selected men

will have a mean weight that is greater than 175lbs (so that their total weight exceeds

the current safe capacity of 3500 lbs).

Coke Cans Coke cans are labeled to have 12 oz.

Recently, however, it was found that in a sample of 36 cans, the sample

mean was 12.19 oz. Assume that Coke cans are filled to

have a mean of 12.00 oz. and a standard deviation of 0.11 oz. What is

the probability that 36 cans would produce a sample mean of 12.19 oz.

or greater?

Homework Pg. 295-296

#2, 5-7