Transcript Chapter 9 - Humble ISD

Chapter 9
Indentify and describe
sampling distributions.
Vocabulary
Parameter
A number that describes the population
(p)
Statistic
A number that describes a sample. A
statistic is used to estimate an unknown
parameter. ( p̂)
Mean of population is μ
Mean of sample is x
Pg 489 #1 a. state whether it is a
parameter or a statistic and b. use
appropriate notation to describe
each number; for ex p = 0.65
A carload lot of ball bearings has
mean diameter 2.5003 centimeters
(cm). This is within the specifications
for acceptance of the lot by the
purchaser. By chance, an inspector
chooses 100 bearings from the lot
that have mean diameter 2.5009 cm.
Because this is outside the specified
limits, the lot is mistakenly rejected.
Can the statistic be used to
represent the parameter?
Sampling Variability – the value of the
statistic varies in repeated random
sampling
Sampling Distribution – the
distribution of values taken by the
statistic in all possible samples of the
same size from the same population.
The distribution of the sample proportion p hat
from SRSs of size 100 drawn from a pop with
p = .7. The results of 1000 samples taken.
Describing sampling distributions
Overall shape
Center of distribution
Spread (variability)
Outliers
Proportion of sample who watched
Survivor II in samples of n = 100
Center – very
close to the
true value
p=.37, in fact
the mean of
the 1000 ‘p
hat’s is .372
and their
median is
exactly .37
Overall shape
of distribution is
symmetric and
approx normal
There are no
outliers or other
important
deviations from
the overall pattern.
Values of
‘p hat’
have a
large
spread.
Range
from .22 to
.54, stand.
dev is .05
The approx sampling dist. of the
sample from SRSs of size 1000
On the
same scale
as the SRSs
of size 100
from
previous
slide
N = 1000, redrawn with expanded
scale to better display shape.
Spread, is
much less
from .321 to
.421, stand
dev is .016
Center is
again close
to .37,
mean is
.3697, and
median is
.37
Shape again
close to
normal.
Unbiased Statistic – a statistic used to
estimate a parameter is unbiased if the
mean of its sampling distribution is equal to
the true value of the parameter being
estimated.
Variability of a Statistic – described by the
spread of its sampling distribution.
This spread is determined by the sampling
design and size of the sample.
Larger samples size smaller spreads.
Rule of Thumb
Population at least 10 times as large
as sample. The spread of the
sampling distribution is approximately
the same for any population size.
High/Low Bias
High/Low Variability
2,3,4,10,11,17
In class 1 & 8
Section 9.2 – Sampling Proportions
Sampling Distribution of a Sample
Proportion – choose a SRS of size n from a
large population with population parameter
p, have some characteristic of interest. Let
p̂ be the proportion of the sample having
that characteristic. Then
The mean of the sampling distribution is exactly
p so…… p̂  p
The standard deviation of the sampling
distribution is  pˆ  p(1  p)
n
p̂ is less variable with larger samples
 p̂ decreases as n increases.
Rule of Thumb 1
Use the recipe for the standard
deviation of p̂ only when the
population is at least 10 times as
large as the sample
Rule of Thumb 2
We will use the normal approximation
to the sampling distribution of p̂ for
values of n and p that satisfy np ≥10,
and n(1-p)≥10
Example 9.7 applying to college
SRS of size
college
n=1500
SRS of 1500 first year
students whether they applied for
Population
has
proportion p=.35,
admission to anyso
other
college.
sampling distribution of ‘ ’ has
p̂ mean
  .35
35% of all first
–
year
students
applied
So standard deviation?
From first Rule
of one they are
to colleges besides
the
Thumb, pop must
attending. contain at least
10(1500) = 15,000
What is probability
people. Therethat
are 1.7random
million first-yr
college will give a
sample of 1500
students
students so…
result within 2 percentage points of
this true value?
pˆ
 pˆ 
p(1  p)
.35(1  .35)

 0.0123
n
1500
Second rule of thumb:
np=1500(.35)=525,
n(1-p)= 1500(.65) = 975
Both are MUCH larger
than 10 so YES, normal
approx is accurate.
Can we use normal distribution to
approximate the sampling distribution
of ‘p hat’?
What is probability that ‘p hat’ falls
between .33 and .37?
Area of shaded
region is
0.33≤’p hat’≤0.37
20, 21, 22, 25, 27, 30
In class 19
9.3 Sample Means
Mean and Standard deviation of a
sample mean suppose that x is the
mean of an SRS of size n drawn from
a large population with mean μ and
standard deviation σ.
Then the mean of the sampling
distribution of x is  x  

The standard deviation is  x  n
The sample mean ‘x bar’ is an
unbiased estimator of the population
mean μ.
The values of ‘x bar’ are less spread
out for larger samples
Their standard deviation decreases at
the rate n

Use  x 
when ROT#1 is met.
n
Sampling Distribution of a Sample
Mean from a Normal Population
Draw a SRS of size n from a
population that has the normal
distribution with mean μ and standard
deviation σ. Then the sample mean

N
(

,
)
‘x bar’ has the normal distribution
n
with mean μ and standard deviation 
Z
x  x
x
n
In class 31, 35,
Homework 32, 33, 34, 36, 39, 40, 42