HW-pg. 577-578 (9.8 - 9.10)

9.1-9.2 Quiz Friday 1-18-13

www.westex.org HS, Teacher Website

1-15-13

Warm up—AP Stats

Let’s assume the population in our HS is 1000 students. We take an SRS of 10 students at our school and find the average amount of time spent on HW per night. We’d like to use this ______mean (______) to estimate the ______mean (______) for the population of students at our school. What if our SRS was 50? What differences might there be?

Name ______Date ______

AP Stats

9 Sampling Distributions

9.1 Sampling Distributions Day 2

Objectives

·  Explain how to describe a sampling distribution.

·  Define an unbiased statistic and an unbiased estimator.

·  Describe what is meant by the variability of a statistic.

·  Explain how bias and variability are related to estimating with a sample.

Describing Sampling Distributions

·  SHAPE-______

·  CENTER-______

·  SPREAD-______

·  OUTLIERS-______

What if our sample size changes from 100 to 1000? We’ll still take the same # of SRSs, 1000.

·  SHAPE-______

·  CENTER-______

·  SPREAD-______

·  OUTLIERS-______

______sampling is necessary to see the approximately normal appearance of a sampling distribution (or even a large number of SRSs of a population). Randomization of design allows us the opportunity to look at the trustworthiness of ______as estimators of parameters.

The Bias of a Statistic

Sampling distributions allow us to describe ______more precisely by speaking of the bias of a ______rather than bias in a ______method. Bias concerns the center of the sampling distribution. The center of an approximate sampling distribution is very close to true value of the population parameter. (think of our last example) The mean of an ACTUAL sampling distribution is ______equal to the mean of the population (parameter) mean. Why?

Unbiased Statistic/Estimator

A statistic used to estimate a parameter is ______if the mean of its sampling distribution is ______to the true value of the parameter being estimated. The statistic is then called an ______of the parameter.

***Remember, an ______statistic will sometimes be ______or ______the true value of the ______. The reason it is unbiased is because its ______is centered at the true value of the parameter. For this reason there is no systemic tendency to under or over estimate the parameter.***

Variability of a Statistic

The sample proportion, ___, from a random sample of any size is an unbiased estimator of the parameter, ___. Is there an advantage then of having a small or large sample size if they both are unbiased estimators?

Which is better and why?

Tomorrow we will give a formula for the standard deviation of the sampling distribution of ___. We will then see why the variability (______) of sample results decreases as sample size goes up. One seemingly surprising fact is that the spread of sampling distribution does not depend very much on the size of the ______.

Variability of a Statistic

***The variability of a statistic is described by the ______of its ______. This spread is determined by the sampling design and the size of the ______. Larger samples yield ______spreads. As long as the population is MUCH larger than the sample (at least 10 times larger), the spread of the sampling distribution is ______the same for ANY population size.***

Why does the population size have little influence on the behavior of statistics from random samples? Think about the bulk section of Whole Foods…

The fact that the variability of sample results is controlled by the size of the sample has important consequences for ______.

Bias and Variability

Think of a bull’s-eye on a target. The bull’s-eye is the true value of the parameter. If our sampling distribution doesn’t have the same mean as the population parameter then our samples will not cluster around the bull’s-eye. This is ______. If we use small sample size then our samples will have a lot of ______. In this case our dots will be scattered all around. We hope to have low _____ and low ______. It shouldn’t be hard to have unbiased statistics, but we have to make sure we are able to have large enough sample size so that we keep our variability low enough.

9.8 Bearing down-The table below contains the results of simulating on a computer 100 repetitions of drawing an SRS of size 200 from a large lot of ball bearings. Ten percent of the bearing in the lot do not conform to the specifications. That is, p = 0.10 for this population. The numbers in the table are the counts of nonconforming bearings in each sample of 200.

17 23 18 27 15 17 18 13 16 18 20 15 18 16 21

17 18 19 16 23 20 18 18 17 19 13 27 22 23 26

17 13 16 14 24 22 16 21 24 21 30 24 17 14 16

16 17 24 21 16 17 23 18 23 22 24 23 23 20 19

20 18 20 25 16 24 24 24 15 22 22 16 28 15 22

9 19 16 19 19 25 24 20 15 21 25 24 19 19 20

28 18 17 17 25 17 17 18 19 18

a) Make a table that shows how often each count occurs. For each count in your table, give the corresponding value of the sample proportion = count/200. Then draw a histogram for the values of the statistic .

b) Describe the shape of the distribution.

c) Find the mean of the 100 observations of . Mark the mean on your histogram to show its center. Does the statistic appear to have large or small bias as an estimate of the population proportion p?

d) The sampling distribution of is the distribution of the values of from all possible samples of size 200 from this population. What is the mean of this distribution?

e) If we repeatedly selected SRSs of size 1000 instead of 200 from this same population what would be the mean of the sampling distribution of the sample proportion ? Would the spread be larger, smaller, or about the same when compared with the spread of your histogram in a)?

Count / Frequency
9 / 1 / .045
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