AP Statistics43/42 days until the AP Exam

Sampling Means

Sample Mean Formulas

Today’s Objectives

• I can set up and solve problems using the properties of the sampling distribution of means.

Quick Review of Vocab & Symbols

Parameter

Statistic

µ

These may be new:

p

x

p̂

REMEMBER!

It is very important to understand the difference between a parameter and a statistic.

Remember that a Statistic comes from a Sample and a Parameter comes from a Population.

Example Identify the number that appears in boldface type

as the value of either a parameter or a statistic and use the appropriate notation (i.e. ).

1) A department store reports that 84% of all customers who use the store’s credit plan pay their bills on time.

2) A sample of 100 students at a large university had a mean age of 24.1 years.

3) The Department of Motor Vehicles reports that 22% of all vehicles registered in a particular state are imports.

4) A hospital reports that based on the ten most recent cases, the mean length of stay for surgical patients is 6.4 days.

5) The Bureau of Labor Statistics last month interviewed 60,000

member of the U.S. labor force, of whom 7.2% were employed.

ˆ p 0.37

CUSS and BS (revisited)

• The sampling distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size from the same population.

• Remember, when describing a distribution, to CUSS and BS.

C-

U-

S-

S-

BS-

Sample Means Example:

Consider a small population consisting of the board of directors of a

day care center. Board member: Jay Carol Allison Teresa Anselmo Bob Roxy Vishal

# of children: 4 2 0 0 2 2 1 3

Find the average number of children for the entire group of eight:

This is a PARAMETER because it is derived from the ENTIRE population.

Question of the Day #1

• How is the parameter of the population related to a sampling distribution based on the population?

How to:

• To answer our question of the day, we will create a sampling distribution from our data using a sample size of 2.

• List all possible samples of size 2, and calculate the average for each of the samples.

Table:

The (official) sampling distribution (of a particular population) must:

• be based on a determined (fixed) sample size• be based on a determined (fixed) population• contain the values for all possible combinations

of the statistic

• In other words, a sampling distribution is the distribution of all possible values for a given sample size for a fixed population.

Example Continued

• Calculate the average value of the statistic( ) for the sampling distribution:

• What is the relationship between the population parameter and the mean of the sampling distribution?

• The “Mean of a Sampling Distribution” Formula:

x

x

Question of the Day #2

• What is the relationship between the population parameter and each sample statistic ?

x

Some more notation that we need to know:

Formulas!!!

• The mean of a sampling distribution??

• The standard deviation of a sampling distribution?

Note that we use the Normal Approximation to estimate probabilities for a sample mean.

nx

N(, n)

Example 2

• The average sales price of a single-family house in the United States is $243,756. Assume that the sales prices are normally distributed with a standard deviation of $44,000.

• Draw the normal distribution. Within what range would the middle 68% of the houses fall?

Continued

• Draw the sampling distribution for a sample size of 4 houses. Within what range would the middle 68% of the samples of size 4 houses fall?

• Draw the sampling distribution for a sample size of 16 houses. Within what range would the middle 68% of the samples of size 16 houses fall?

Continued

• Draw the sampling distribution for a sample size of 25 houses. Within what range would the middle 68% of the samples of size 25 houses fall?

What did you notice?

• I hope you noticed that, as the sample size n increased, the standard deviation of the sampling distribution decreased.

• Sampling distribution applet

Example 3

• The mean room and board expense per year at a four-year college is $5,850. You randomly select 9 four-year colleges. Assume that the room and board expenses are normally distributed with a standard deviation of $1125.

• a) Draw the sampling distribution for a sample size of 9 colleges.

• b) What is the probability that the mean room and board of the nine colleges is less than $6,180?

• c) What is the probability that the mean room and board of the nine colleges is more than $5,250?

Example 4• The Computer-Assisted Hypnosis Scale (CAHS) is designed to

measure a person’s susceptibility to hypnosis. In computer-assisted hypnosis, the computer serves as a facilitator of hypnosis by using digitized speech processing coupled with interactive involvement with the hypnotic subject. CAHS scores range from 0 to 12 (extremely high susceptibility). A study in Psychological assessment (Mar. 1995) reported as mean CAHS scores of 4.59 and a standard deviation of 2.95 for University of Tennessee undergraduates. Suppose CAHS scores are normally distributed and a psychologist uses CAHS to test a random sample of 50 subjects.

• Give the mean and standard deviation of the sampling distribution of .

• Find )4( xPx

Example 5• The College Student Journal (Dec. 1992) investigated differences in

traditional and nontraditional students, where nontraditional students are generally defined as those 25 year old or older. Based on the study results, we can assume that the population mean and standard deviation for the GPA of all nontraditional students is and . Suppose that a random sample of 100 nontraditional students is selected from the population of all nontraditional students that is normally distributed.

• a) Give the mean and standard deviation of the sampling distribution of .

• b) What is the approximate probability that the nontraditional student sample has a mean GPA between 3.40 and 3.65?

• c) What is the approximate probability that the sample of 100 nontraditional students has a mean GPA that exceeds 3.62?

• d) How would the sampling distribution of change if the sample size were doubled?

x

x

35. .5

Rule of Thumb

• An unbiased statistic falls sometimes above and sometimes below the actual mean, it shows no tendency to over or underestimate.

• As long as the population is much larger than the sample (rule of thumb, 10 times larger), the spread of the sampling distribution is approximately the same for any size population.

Central Limit Theorem

• Take a random sample of size n from any population with mean and standard deviation . When n is large, the sampling distribution of the sample mean is close to the normal distribution.

• How large a sample size is needed depends on the shape of the population distribution.

Uniform distribution

Sample size 1 Sample size 2

Sample size 3 Sample size 4

Uniform Distribution (2)

• Notice that, given a large enough sample size, the uniform distribution becomes normal.Sample size 8

Sample size 16

Law of Large Numbers

• As the sampling standard deviation continually decreases, what conclusion can we make regarding each individual sample mean with respect to the population mean ?

• As the sample size increases, the mean of the observed sample gets closer and closer to .