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HW-pg. 577-578 (9.8 - 9.10)
9.1-9.2 Quiz Friday 1-17-14
www.westex.org HS, Teacher Website
1-14-14
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 40? What differences might there be?
Name _________________________
AP Stats
9 Sampling Distributions
9.1 Sampling Distributions Day 2
Date _______
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
17
17
16
20
9
28
23
18
13
17
18
19
18
18
19
16
24
20
16
17
27
16
14
21
25
19
17
15
23
24
16
16
19
25
17
20
22
17
24
25
17
18
18
16
23
24
24
17
13
18
21
18
24
20
18
16
17
24
23
15
15
19
18
19
21
22
22
21
18
20
13
30
24
22
25
15
27
24
23
16
24
18
22
17
23
28
19
16
23
14
20
15
19
21
26
16
19
22
20
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 p̂ = count/200. Then draw a histogram for the values of the statistic p̂ .
b) Describe the shape of the distribution.
c) Find the mean of the 100 observations of p̂ . Mark the mean on your histogram to show its center. Does the
statistic p̂ appear to have large or small bias as an estimate of the population proportion p?
d) The sampling distribution of p̂ is the distribution of the values of p̂ 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 p̂ ? Would the spread be larger, smaller, or about the same
when compared with the spread of your histogram in a)?
Count
9
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
30
Frequency
p̂
1
.045