Download Chapter 10: Sampling Distributions

Chapter 10: Sampling Distributions
Some Vocabulary

Parameter – any number that describes a population, unknown in
statistical practice because we cannot examine the entire population

Statistic – number that can be computed from sample data without
making use of any unknown parameters – often used to estimate the
value of a population parameter
Example: µ is the population mean and x is the sample mean
10.1 Your local newspaper contains a large number of advertisements for
unfurnished one-bedroom apartments. You choose 10 at random and
calculate that their mean monthly rent is $540 and that the standard
deviation of their rents is $80.
10.2 Voter registration records show that 68% of all voters in Indianapolis
are registered as Republicans. To test a random-digit dialing device,
you use the device to call 150 randomly chosen residential telephones
in Indianapolis. Of the registered voters contacted, 73% are registered
Republicans.
10.3 A carload lot of ball bearings has a mean diameter of 2.5003 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 a
mean diameter of 2.5009 cm. Because this is outside the specified
limits, the lot is mistakenly rejected.
Statistical Estimation and The Law of Large Numbers
Draw observations at random from any population with finite mean µ. As
the number of observations drawn increases, the mean x of the
observed values gets closer and closer to the mean of the population.
10.5 The idea of insurance is that we all face risks that are unlikely but carry
high cost. Think of a fire destroying your home. Insurance spreads
the risk: we all pay a small amount, and the insurance policy pays a
large amount to those few whose homes burn down. An insurance
company looks at the records for millions of homeowners and sees
that the mean loss from fire in a year is $250 per person. The
company plans to sell fire insurance for $250 plus enough to cover its
costs and profit. Explain clearly why it would be unwise to sell only 12
policies, Then explain why selling thousands of such policies is safe
business.
What is the Fire Threat?
Number of Residential Fires in 1999
Number of Fire Deaths in the Home in 1999
Cost of Residential Fires in 1997
371,000
2,895
$4,565,000,000
Source: Fire loss in the United States during 1999, National Fire Protection Association (NFPA)
Sampling Distributions
- the distribution of values taken by the statistic in all possible samples
of the same size from the same population
Sampling Distribution of x
- Suppose that x is the mean of a SRS of size n drawn from a large
population with mean µ and standard deviation σ. Then the mean of
the sampling distribution is µ and the standard deviation is σ/√n
- If individual observations have the N(µ,σ) distribution, then the sample
mean x of n independent observations has the N(µ, 
) distribution.
n
Example: Suppose the heights of American women are distributed N(64, 2.7).
A. What is the probability that if 1 woman is selected at random, her
average height will be more than 66 inches?
B. What is the probability that if 9 women are selected at random, their
average height will be more than 66 inches?
C. What is the probability that if 25 women are selected at random, their
average height will be more than 66 inches?
Central Limit Theorem
Draw an SRS of size n from any population with mean µ and finite standard
deviation σ. When n is large, the sampling distribution of the sample mean is
approximately normal: N(µ, 
).
n
For sufficiently large samples, the sampling distribution of X n will be
approximately Normal.

Typically, a sample size of 25 or 30 is sufficiently large

The amazing and counter-intuitive thing about the central limit theorem
is that no matter what the shape of the original distribution, the sampling
distribution of the mean approaches a normal distribution

The necessary sample size depends on the normality or skewness of the
distribution of the population

The larger the sample size, the better the normality


Averages are less variable than individual observations

The Central Limit Theorem holds when the distribution of the population
X n ~ N  x , x 
is either unknown or non-Normal

If the population is Normal, then the sampling distribution of X n will be
Normal regardless of the sample size

If we can claim that the sampling distribution is Normal, we can then
make statements concerning the probability of obtaining certain values
for the sample mean
In a Nutshell
1. The mean of the sampling distribution of means is equal to the mean of the
population from which the samples were drawn.
2. The variance of the sampling distribution of means is equal to the variance of the
population from which the samples were drawn divided by the size of the samples.
3. If the original population is distributed normally, the sampling distribution of means
will also be normal. If the original population is not normally distributed, the sampling
distribution of means will increasingly approximate a normal distribution as sample
size increases.
Example: The number of flaws per square yard in a type of carpet material
varies with a mean of 1.6 flaws per square yard and standard deviation of 1.2
flaws per square yard. The population distribution cannot be Normal, because
a count takes only whole-number values. An inspector samples 200 square
yards of the material, records the number of flaws found in each square yard,
and calculates x , the mean number of flaws per square yard inspected. Use the
central limit theorem to find the approximate probability that the mean number
of flaws exceeds 2 per square yard.