Download Analytical Methods I

Chapter 9: Sampling Distributions
1
Activity 9A, pp. 486-487
2
We’ve just begun a sampling
distribution!

Strictly speaking, a sampling distribution is:


A theoretical distribution of the values of a statistic
(in our case, the mean) in all possible samples of
the same size (n=100 here) from the same
population.
Sampling Variability:


The value of a statistic varies from sample-tosample in repeated random sampling.
We do not expect to get the same exact value for
the statistic for each sample!
3
Sampling Distribution

The sampling distribution answers the
question, “What would happen if we
repeated the sampling or experiment
many times?”

Formal statistical inference is based on the
sampling distribution of statistics.
4
Definitions

Parameter:




A number that describes the population of interest.
Rarely do we know its value, because we do not (normally)
have all values of all individuals from a population.
We use µ and σ for the mean and standard deviation of a
population.
Statistic:


A number that describes a sample. We often use a statistic
to estimate an unknown parameter.
We use x-bar and s for the mean and standard deviation of
a sample.
5
Problems 9.1-9.4, p. 489:
Parameter or Statistic?
6
Example 9.4, p. 491

Compare Figures 9.2 and 9.4
7
Probability Distribution of Random Digits
8
All possible samples of size n=2
9
Sampling Distribution of the Mean
10
Exercise 9.7, p. 494
11
What happens to a sampling distribution
when we increase our sample size (n)?
Example 9.5, pp. 494-496
12
Results of 1000 SRSs of size n=100
13
Results of 1000 SRSs of size n=1000
14
Expanded scale of previous slide
15
Statistic Bias

If the mean of the sampling distribution is
equal to the population parameter, the
statistic is said to be unbiased.

Now, be careful—the sample mean you actually
get may in fact be “off” the parameter mean.
However, there is no systematic tendency, on
repeated samplings, to overestimate or
underestimate the parameter.
16
Variability of Statistic (pp. 498-499)

“Properly chosen statistics computed
from random samples of sufficient size
will have low bias and low variability.”
17
Figure 9.9, p. 500
18
Spread of a sampling distribution

As long as N>10n, the spread of the sampling
distribution does not depend on the size of
the population.



National poll (300,000,000): need approx.
n=1,100 for ±3% margin of error.
Asheville city poll (70,000): need approx. n=1,100
for ±3% margin of error.
See p. 498 for discussion.
19
Homework



Read through p. 504
9.10, p. 501
9.15 and 9.17, p. 503
20
9.2 Sample Proportions

We use p^ as an estimate of p (the parameter). What
does the sampling distribution of p^ look like?


Knowing the center, shape, and variability of the sampling
distribution will give us an idea of how confident we can be
in using p^ as an estimate of p.
If the population is at least 10X larger than the
sample, we can use binomial distribution facts to
develop equations for the mean and standard
deviation of a sampling distribution for p^ :
 p

p
 

p
p(1  p)
n
21
Sampling distribution for proportion
22
Using the Normal Approximation for p^

Example 9.5 showed us that for large samples, the
sampling distribution of p^ is approximately normal
(pp. 495-496).

Following the convention of this text, we will use the normal
approximation for the sampling distribution of p^ as long as
the following conditions are satisfied:
np  10 and n(1  p)  10

Using the normal approximation is quite accurate if
the above conditions are met, plus we can take
advantage of the useful standard normal probability
calculations.
23
Exercises

Read over Example 9.7, p. 507


Be sure to read Example 9.8 tonight.
Exercise 9.19, p. 511
24
Homework

Problems:





9.22, p. 511
9.30, p. 514
Reading through p. 514
Quiz, 9.1-9.2 Wednesday
Chapter 9 Test on Tuesday
25
9.3 Sample Means

In 9.2 we were dealing with a sample
proportion.


This statistic is used when we are
interested in some categorical variable.
In 9.3 we switch to looking at the
sample mean.

Used for quantitative variables.
26
Sampling Distribution for
a Sample Mean

See bulleted list on p. 516:



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.
Box on p. 517


The text tells us that if we draw a SRS of size n
from a normal distribution, the sampling distribution
will also be normal.
But what about drawing samples from a population
whose distribution is not normal?
27
The Central Limit Theorem (p. 521)


One of the more important ideas of statistics.
If we draw a sample that is large enough …

…the sampling distribution is approximately
normal no matter what the shape of the
underlying distribution!

How large the sample must be to get close to
a normal distribution depends on the shape
of the underlying distribution, but samples of
size n=25 to n=30 generally suffice.
28
Example 9.12, p. 521
(exponential distribution)
29
Exercises


9.31, p. 518
9.35, p. 524
30
Exercise 9.31

Important ideas:


Averages are less
variable than
individual
observations.
Averages are more
normal than
individual
observations.
31
Homework


Exercises 9.39 through 9.42, pp. 525-526
Test on Tuesday
32