Transcript mean and standard deviation for the sampling distribution of a

9.2 Objectives
• Describe the sampling distribution of a sample proportion.
(Remember that “describe” means to write about the shape,
center, and spread.)
• Compute the mean and standard deviation for the sampling
distribution of p-hat.
• Identify the “rule of thumb” that justifies the use of the recipe
for the standard deviation of p-hat.
• Identify the conditions necessary to use a Normal
approximation to the sampling distribution of p-hat.
• Use a Normal approximation to the sampling distribution of
p-hat to solve probability problems involving p-hat.
• Use a Normal approximation to the sampling distribution of
p-hat to solve probability problems involving p-hat.
9.3 Objectives
• Give the mean and standard deviation of a population,
calculate the mean and standard deviation for the
sampling distribution of a sample mean.
• Identify the shape of the sampling distribution of a
sample mean drawn from a population that has a
Normal distribution.
• State the central limit theorem.
• Use the central limit theorem to solve probability
problems for the sampling distribution of a sample
mean.
In 9.2, we found that p-hat is
approximately Normal under the right
conditions. What were those?
Wouldn’t it be nice if we could say
something similar about the sampling
distribution x-bar?
• Categorical variables  sampling
proportions
• Quantitative variables  sampling
distribution stats such as median, mean and
standard deviation.
Fig. 9.15 (p592)
(a) The distribution of returns
for a NYSE common stocks in
1987. (b) The distributions of
returns for portfolios of 5 stocks
in 1987.
The Figures emphasize a
principle that will be made
precise in this section:
• Means of random samples are
less variable than individual
observations.
• Means of random samples are
more Normal than individual
observations.
In this section, we are still
considering distributions of
sample statistics, but we are
shifting our attention to x-bar.
The behavior of x-bar in repeated samples is similar to that of
sample proportion p-hat.
• The sample mean x-bar is an unbiased estimator of the
population mean mu.
• The values of x-bar are less spread out for larger sample. Their
standard deviation decreases at the rate sqrt(n). You will need
to take a sample four times as large in order to half the stdev.
• Use sigma/sqrt(n) for the stdev of x-bar only when the
population is at least 10 times the sample. (This is almost
always the case.)
FYI
“Describe”
• Describing the behavior of ANY
distribution means to talk about
– SHAPE
– CENTER and
– SPREAD
Fig 9.16 (p 595)
Practice:
P 595
31 & 33
Central Limit Theorem
• Watch the videos as homework
• CLT discusses the SHAPE (& only the
shape) of the sampling distribution of x-bar
when the sample is sufficiently large. If n is
not large enough, the shape of the sampling
distribution of x-bar more closely resembles
the shape of the original population.
Thus there are 3 situations to consider when
discussing the shape of the sampling distribution
• The population has a Normal distribution—shape
of sampling distribution: Normal, regardless of
sample size
• Any population shape, small n—shape of
sampling distribution: similar to shape of parent
population.
• Any population shape, large n—shape of
sampling distribution: close to Normal (CLT)
Practice:
35, 37, 38 & 47