Sampling Variability and Sampling Distributions

How can a sample be used to accurately represent the characteristics of a population?

Statistics and Sampling Variability 8.1

How are statistics and sampling variability related?

Statistic is any quantity computed from values in a sample

Sampling distribution the distribution of one statistic over many samples often look at histograms to determine patterns

Sampling variability is the recognition that the value of a statistic depends on the sample selected , the size of the sample and how many samples are evaluated

Since:

since the average of the samples approximates the pop ave.

x

x

nx

_

Consider a population consisting of the following four values {14, 16,10, 11}, which represent the number of video rentals during the

academic year for each of the housemates.

a) compute the mean of this population

b) select a random sample of size 2 by writing the numbers on slips of paper and selecting two. Compute the mean of the sample.

c) Select all the samples of size two

d) Create a sampling distribution for the information

12.75

11, 14 ave= 12.5

10,11 10.5 11, 10

10.5

10,14 12 11, 14

12.5

10,16 13 11,16 13.5

14, 10 12 16, 10

13

14, 11 12.5 16, 11

13.5

14,16 15 16, 14

15

10.5 12 12.5 13 13.5 15

1/6 1/6 1/6 1/6 1/6 1/6

Homework page 409-411 evens

For #12 the info is on pages 406 and 409

The Sampling Distribution of a sample mean 8.2What is the sampling distribution of a sample mean?

If is based on a large # of items (n) in the sample the is closer to µ than if n is small ( although does approximate µ regardless of the size of n)

If n is large σ is closer to Sx (meaning the data is less variable)

x xx

Rules of the Sampling Distribution1)

2) (correct as long as no more than 5% of the population is included

this is of more importance at higher stat levels and not stressed on

the AP exam)

3) when the population is normal, the sampling distribution of is also normal for any sample size of n

4) When n is sufficiently large (ie approx 30+) the sampling distribution of is approximated by a normal curve even when the population is not.

(known as the CENTRAL LIMIT THEOREMthink of the age of the pennies and their distribution )

x

x

x

nx

_

If n is large or the population distribution is normal we can use z-scores to answer questions

is the standard deviation of the sample averages and approximated as inicated in the formula above

nx

x xxz

x

Example 8.7 page 418

A soft-drink company claims that on average, cans contain 12 oz. of soda. Let x denote the actual volume of soda in a randomly selected can. Suppose that x is normally distributed with σ= .16 oz. Sixteen cans are to be selected and the soda volume determined for each one. Let denote the resulting sample average soda volume. Because the x distribution is normal, the sampling distribution is normal.

If the bottler’s claim is correct, the sample distribution of has a mean value and a standard deviation of

 

a) determine the probability that your sample contains between 11.96 oz and 12.08 oz

12_ x

04.16

16._

nx

Z-chart

x

b) Determine the probability that the manufacturer is correct if your sample of soda cans has an average of at least 12.1 oz in it.

HW Pg 420 to 422 8.14 to 8.22 even

The Sampling Distribution of a sample proportion 8.3What is the sampling distribution of a sample proportion?

The Sample proportion p =

RULES:1) µp = π

2) σp =

3) when n is large and π is not too near to 0 or 1 the sampling distribution of p is approximately normal

--the further π is from .5 the larger n must be for the sampling distribution to be approx normal.

Rule of thumb:conservative liberal

If nπ ≥ 10 and n(1-π) ≥10 nπ≥5 and n(1-π)≥5

Note: the book tends to use the conservative method and the AP test tends to use the liberal method according to the AP listserve and the AP grading rubrics it doesn’t make a difference just show your set up!!!!!

)( #

nsamplethein

successesofnumber

n

)1(

x

AP Statistics Example 8.11 page 426

The proportion of all blood recipients stricken with viral hepatitis was given as .07 in an article in the American Journal of medicine. Suppose a new treatment is believed to reduce the incidence rate of viral hepatitis. This treatment is given to 200 blood recipients. Only 6 of the 200 contracted hepatitis. It appears that the new treatment is better since p = 6/200 = .03.

Determine if the sampling distribution of p is normally distributed

Determine the probability that the sample distribution p could be less than .03 if the treatment were ineffective (i.e. it did not change --the original value of those contracting viral hepatitis under the old treatment)

Hw pg 427 8.28 to 8.32 even

Review info

Formulas:Distribution of

= the population average

= the sample average

= the average of the several

sample averages

x

x

x

Distribution of p

=the population probability

= the sample probability

= the average of the several sample probabilities

p

p

Formulas:Distribution of Distribution of p

If n is large the dist is The further π is from .5Normal even if the pop isn’t the larger n must be for the N 30

distribution to be considered normal so:

show the conservative orliberal testn and n (1- ) 5 or 10

x

x

n

)1(

1) µp = π

2) σp =

x

nx

_

Homework pg 428 evens

Test set up:10 multiple choice—1 point each4 computational—total of 45 points

Test total 55 points

pg 428 evens for review