Chapter 8 Sampling Variability and Sampling Distributions How can a sample be used to accurately represent the characteristics of a population?

Unit Activation:

Statistics and Sampling Variability 8. 1 Statistics and Sampling Variability 8.1 How are statistics and sampling variability related?

A statistic is any quantity computed from values in a sample Sampling distribution is the distribution of one statistic over many samples --we 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: the average of the samples approximates the pop ave. the average of the averages for a certain n approximates the pop ave the average of the Stand dev for a certain n approximates the pop stand dev.

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 four housemates. a) compute the mean of this population b) select a random sample of size 2 by separating the numbers on the bottom of the paper and selecting two. Compute the mean of the sample. c) Determine all the samples of size two d) Create a sampling distribution for the information—find E(x) 12.75 11, 14 ave= 12.5 Repeat 4 times. What is your average? 10,11 10.5 11, 10 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 10.5 12 12.5 13 13.5 15 1/6

Homework Homework page 441-442 1, 2, 3, 6, 9

The Sampling Distribution of a sample mean 8 The Sampling Distribution of a sample mean 8.2 What 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 likely to be variable)

Rules of the Sampling Distribution 1) (the average of all the sample average = pop average) 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 THEOREM—CLT the pennies and their distributions )

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 indicated in the formula above

(Note: always state the parameters as above) Example 8.5 page 448 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  (Note: always state the parameters as above) Z-chart

a) Determine the probability that your sample contains between 11 a) Determine the probability that your sample contains between 11.96 oz and 12.08 oz 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.

Homework HW Pg 450 to 452 10, 11, 13, 16, 19, 22

The Sampling Distribution of a sample proportion 8 The Sampling Distribution of a sample proportion 8.3 What is the sampling distribution of a sample proportion?

  p p p

   

Homework Hw pg 456-457 23, 24, 25, 28, 31

Review info

Chapter 8 Review Formulas: Distribution of = the population average = the sample average = the average of the several sample averages  

Chapter 8 Review Formulas: Distribution of Distribution of p If n is large the dist is The further π is from .5 Normal even if the pop isn’t the larger n must be for the n 30 distribution to be considered normal so: show the conservative or liberal test n p and n (1- p) 5 or 10  

Homework pg 459-460 32, 34, 35, 36 Test set up: 10 multiple choice—1 point each 4 computational—total of 45 points Test total 55 points

Review Homework pg 428 evens for review