Transcript Chapter09

Chapter 9
Sampling Distributions
9.1 Sampling Distributions
Key Term
• A parameter is a number that describes the
population. In statistical practice, the value of a
parameter is not known.
• A statistics is a number than can be computed
from the sample data without making use of any
unknown parameters. In practice, we often use a
statistic to estimate an unknown parameter.
• Refer to Example 9.1, page 564
Population and Sample Means
• We write µ for the mean of a population.
This is a fixed parameter that is unknown
when we use a sample for inference
• The mean of the sample is the sample is the
familiar x-bar, the average of the
observations in the sample.
Key Concept / Term
• Sampling Variability
– For example 9.2 we found a p-hat of
160/515 = 0.31 However, the next time we take
a random sample we choose different people
and get a different value of p-hat.
Key Term / Concept
• The sampling distribution of a statistic is
the distribution of values taken by the
statistic in all possible samples of the same
size from the same population
• View Figures 9.4, 9.5, 9.6, 9.7, 9.8
• Larger samples sizes are less variable
Two Digit Sample Mean
Sampling Distribution of x-bar for
Samples of size n = 2
Example 9.5: Proportions of samples who
watched Survivor : Guatemala in Sample size
of n = 100, 1000 samples
Figure 9.6: Sampling distribution of the
sample proportion p-hat from SRSs of size
1000 drawn from a population with
population proportion p = 0.37, 100 samples
Same Sample different scale
illustrated the normal distribution
Comparison of the two sample sizes
Key Term
• 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. See Figure 9.9,
page 576
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.
Homework
• Read 9.2
• Complete Exercises 1,2,4, 9 -12, 15 - 17
Chapter 9
Sampling Distributions
9.2 Sampling Proportions
Key Concept
• The variability of a statistic is described by the
spread of its sampling distribution. This spread is
determined by the sampling design and the size of
the sample. Larger samples give smaller spread.
• As long as the population is much larger than the
sample (at least 10 times as large), the spread of
the sample distribution is approximately the same
for any population size.
Key Term / Concept
• Retailers would like to know what
proportion of all adults find clothes
shopping frustrating and time-consuming.
This unknown population proportion is a
parameter p. A random sample of 2500
people found 1650 frustrated by clothes
shopping. The sample proportion p-hat =
1650/2500 = .66 is a statistic that we use to
gain information about the parameter p.
Sampling Distribution of a Sample
Proportion
Choose an SRS of size n from a large population with
population proportion p having some characteristic of
interest. Let p-hat be the proportion of the sample
having that characteristic. Then:
• The sampling distribution of p-hat is approximately
normal and is closer to a normal distribution when
the sample size n is large.
• The mean of the sampling distribution is exactly p.
• The standard deviation of the sampling distribution
is
p(1  p)
n
Rule of Thumb 1
• Use the recipe for the standard deviation of
p-hat only when the population is at least 10
times as large as the sample
• Example 9.7, page 584
Rule of Thumb 2
• We will use the normal approximation to
sample distribution of p-hat values of n and
p that satisfy:
np > 10 and n(1-p) > 10
• Example 9.5, page 476
Homework
• Read 9.3
• Complete Exercises #19, 20, 22, 24, 25, 28,
29, 30
Chapter 9
Sampling Distributions
9.3 Sample Means
Rate of Return (Individual Stocks)
Rate of Returns (Portfolios)
Key Terms
• The mean and standard deviation of a population
are parameters. We use Greek letters to write
these parameters: μ for the mean and σ for the
standard deviation.
• The mean and standard deviation calculated from
sample data are statistics. We write the sample
mean as x-bar and the sample standard deviation
as s.
Mean and Standard Deviation of a
Sample Mean
Suppose that x-bar 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-bar is μ and its
standard deviation is

n
Sampling Distribution of a Sample
Mean
• Draw an SRS of size n from a population that
has the normal distribution with mean μ and
the standard deviation σ. Then the sample
mean x-bar has the normal distribution N ( ,  )
n
with the mean μ and standard deviation

n
Central Limit Theorem
• Draw an SRS of size n from any population
whatsoever with mean μ and finite standard
deviation σ. When n is large, the sampling
distribution of the sample mean x-bar is close
to the normal distribution N (, n ) with mean
μ and standard deviation

n
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-bar of the observed values get
closer and closer to μ.
Homework
• Complete Exercises #35-46