Download Chapter 9: Sampling Distributions

Chapter 9.1: Sampling
Distributions
Mr. Lynch
AP Statistics
The Heights of Women






The heights of women in the world follow: N(64.5, 2.5)
… Explain …
Let’s draw a sketch that helps illustrate this
MATH … PRB … 6:randNorm(64.5,2.5)
Stand up if your value is between [62, 67]
Stand up if your value is between [59.5, 69.5]
Stand up if your value is between [57, 72]
The Heights of Women








MATH … PRB … 6:randNorm(64.5,2.5, 100)
STO  L1
1-Var Stats: Mean? Median? S?
STAT PLOT 1: Histogram … L1, 1
WINDOW: X:[57,72, 2.5] …Y:[-10,60,10]
STAT PLOT 2: Boxplot … L1, 1
TRACE Histogram … Enter frequencies is chart
Repeat three times … fill out frequency chart as
shown
The Heights of Women
Interval Set #1 Set #2 Set #3 Total
57-59.5
3
2
3
8
%
2.7
59.5-62
62-64.5
64.5-67
67-69.5
8
39
37
11
15
33
38
10
11
33
32
18
34
105
107
39
11.3
35.0
95.0
70.7
99.4
35.7
13.0
69.5-72
1
2
2
5
1.7
Pooled Data Period 03 – January 2008
Interval
Row Row
1
2
Row 3
Lynch
Row
4
Row Row
5
6
Total
%
57 - 59.5
11
27
37
23
40
38
20
196
2.7%
59.5 – 62
40
158
208
120
180
155
126
987
13.7%
62 - 64.5
91
423
529
306
392
401
298
2440
33.9%
64.5 – 67
102
418
503
318
409
398
323
2471
34.3%
67 - 69.5
43
148
184
106
147
158
111
897
12.5%
69.5 – 72
13
25
39
26
32
50
19
204
2.8%
The Heights of Women





How did the “Empirical Rule” work out for you?
What do the Shape, Center, and Spread look
like?
Let’s look at the n = 7500 histogram!
How are we doing now?
Conclusion: This distribution is just a miniature
version of the population distribution with same
mean and standard deviation
The Heights of Women





Now, take 4 samples again … and one at a
time – Use 1-Var Stats to get the mean X .
Write that value on one of your post-it notes.
Repeat this 3 more times.
Place the notes upon the board CAREFULLY
in the correct slots to build a histogram!
Let’s record the values in L2.
The Heights of Women





How did the “Empirical Rule” work out here?
Compare a Boxplot for L2 in PLOT 3 – to the one we
did in PLOT 2 for the population.
What do the Shape, Center, and Spread look like for
THIS NEW distribution?
Let’s look at the new SAMPLING DISTRIBUTION of
Sample means of n = 100 histogram!
Conclusion: What is the relationship between the
mean of the population and the mean of the X bars?
What about the standard deviation of the population
and that of the X-bars?
Terminology

Population Parameter–
–
–
–
Numerical value that describes a population
A “mysterious” and essentially unknowable –
idealized value.
A theoretically fixed value
Ex: Population Mean, Population Standard
Deviation, Population Proportion,
Population Size
 ,  , p, N
Terminology

Sample Statistic
–
–
–
–
–
Numerical value that describes a sample (a subset of
a larger population)
An easily attainable and knowable value
Will vary from sample to sample
Used to estimate an unknown population parameter
Ex: Sample Mean, Sample Standard Deviation,
Sample Proportion, Sample Size
X , s, pˆ , n
Example and Exercises

EXAMPLE
9.1:
MAKING
MONEY
EXAMPLE
9.2:
DO
YOU BELIEVE
EXERCISE
9.4:
WELL-FED
RATS IN GHOSTS?
EXERCISE
9.2:
UNEMPLOYMENT
Sampling Variability


What would happen if we took many
samples?
EXAMPLE 9.3 BAGGAGE LUGGAGE
Sampling Variability


Sampling Distribution: of a statistic is the
distribution of values in ALL POSSIBLE
samples of the same size
EXAMPLE 9.4 RANDOM DIGITS
Describing Sampling Distributions

EXAMPLE 9.5: ARE YOU A SURVIVOR FAN?
1000 SRSs; n = 100; p = 0.37
1000 SRSs; n = 1000; p = 0.37
Using the
Using
same
a scale
x-axis
toscale
showas
shape!
to the left!
UNBIASED vs. BIASED


A Statistic is said to be UNBIASED if the
mean of the sampling distribution is equal to
the true parameter being estimated
When finding the value of a sampling
statistic, it is just as likely to fall above the
population parameter as it is to fall below it.
VARIABILITY of a STATISTIC


The larger the sample size, the less variability there
will be
EXAMPLE 9.6: THE STATISTICS HAVE SPOKEN
–
–
95% of the samples generated: Mean ± 2 Sd
With n = 100 …0.37 ± 2 (0.05) = 0.37 ± 2 (0.05)

–
With n = 1000 …0.37 ± 2 (0.01) = 0.37 ± 2 (0.01)


[0.32, 0.42]
[0.35, 0.39]
The N-size is irrelevant! Accuracy for n = 2500 is the
same for the entire 280M US, as it is for 775K in San Fran
BIAS & VARIABILITY (Revisited)
Precision
versus
Accuracy
BIAS & VARIABILITY (Revisited 2)
Homework Example

EXERCISE 9.9: BEARING DOWN

p = 0.1; 100 SRSs of size n = 200

Non-conforming ball bearings out of 200 are shown:
(e)
isa
repeated
this
exercise,
instead
(c)
Find
mean
of the
distribution
ofbut
p-hat;
markused
itofon
(d)What
Whatthe
iswe
the
mean
ofof
“the
sampling
distribution”
all
(b)
Describe
the
shape
thethe
distribution.
(a)
Make
table
that
shows
frequency
of each
count!
SRSs
of size
1000 instead
ofp-hat
200?values.
What would the
Draw
a
histogram
of
the
the
histogram.
Anyofevidence
of bias in the sample?
possible
samples
size
200?
mean
of this be?
Would
the spread
be larger, smaller
or about the same as the histogram from part (a)?