AP Statistics9.3 Sample Means

Know the mean and standard deviation for a sampling distribution of means

Understand the central limit theorem and how it applies to sampling distributions

Know the difference between the CLT and the law of large numbers

Use the characteristics of sampling distributions of sample means to solve statistical problems

Learning Objectives

Parametersthe mean and standard deviation (μ

and σ) of the POPULATION!!!

Statisticsthe mean and standard deviation (

) of the SAMPLE!!!

Suppose that (x bar)is the mean of an SRS of size n drawn from a large population with mean μ and the standard deviation σ .

Then the mean of the sampling distribution

of is: μ

The standard deviation is:

**** The behavior of in repeated samples is very similar to that of the sample proportion .

1. is an unbiased estimator of the population

mean µ

2. The values of are less spread for larger samples. Their standard deviation decreases at a rate 1/√n.

3. Only use for the standard deviation of when the population is at least 10 times as large as the sample

Ex: The height of young women varies approximately according to N(64.5, 2.5) distribution. If we measure the height of an SRS of 10 young women, the sampling distribution of their sample mean height will have a mean μ= and standard deviation s=___.

  μ= 64.5 s= (2.5/√10)=0.79 inches

9.26) µ=-3.5% s= 26%/√5= 11.628%

9.27) µ=18.6 s= 5.9/√76= 0.6768

9.28) µ=40.125 s= 0.001

Complete ex. 9.26- 9.28

Draw an SRS of size n from a population that has the normal distribution with mean _μ_ and standard deviation _σ_. Then the sample mean _μ_ has the normal distribution

N(μ , σ/√n )

Applet: http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html

Sampling Distribution of a Sample Mean:

Central Limit Theorem (CLT) Draw an SRS of size n from any

population with mean µ and standard deviation σ. When n large, the sampling distribution of the sample mean (x bar) is close to the normal distribution N(μ , σ/√n ) with mean µ and standard deviation σ/√n.

 

Ex: The time that a technician requires to perform preventive maintenance on an air-conditioning unit is governed by the exponential distribution whose density curve is approximately normal. The mean time is µ=1hour and the standard deviation is σ=1 hour.

Your company decides to operate 70 units. What is the probability that their average maintenance time exceeds 50 minutes?

Complete ex. 9.30-9.32

Draw observations at random from any population with mean µ. As the number of observations drawn increases, the mean _____ of the observed values gets closer and closer to μ.

Law of Large Numbers

Complete ex. 9.35

How does the CLT differ from the Law of Large Numbers?

-Law of large numbers: Larger sample size, closer your sample mean gets to your true mean.

-CLT: Larger sample size, closer your sample gets to N(μ , σ/√n ) .