Download Chapter 9 PPT Vocabulary

UNIT FOUR/CHAPTER NINE
“SAMPLING DISTRIBUTIONS”
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(1) “Sampling Distribution of Sample Means”
> When we take repeated samples and calculate x
from each one, these sample means vary. The shape,
mean, and standard deviation of their graph is called the
sampling distribution of sample means. This applies to
having taken many, many samples – not just one or two.
(2) When the population of individual values in a population is
Normally distributed, then the sampling distribution of
x
is also Normally distributed.
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(3) The sampling distribution of x is less variable than that
of the entire population of individual values. (For example,
the yellow Post-Its on the wall form a graph that has a smaller
standard deviation than the red curve of the population of
individuals.)
(4) The sampling distribution of sample means has two
parameters with special symbols and values:
>> on the next page!
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(4) (continued)
x  
In words, this says that the mean of the
sampling distribution of sample means is the same as the mean
of the population of individuals. (For example, the yellow curve
of Post-Its – each of these is a sample mean – has the same
mean as the red curve of individuals.)
x 

n In words, this says that the standard deviation of
the sampling distribution of sample means is much smaller
than the standard deviation of the population of individuals.
Both of these formulas are true regardless of whether the
population of individuals is Normal or not
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(5) Section 9-3:
The Central Limit Theorem says this:
> IF “n” is large (generally 30 or larger),
> THEN the sampling distribution of sample means is
approximately Normal, regardless of whether the
population of individual values is normal, skewed, or
anything else.
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Section 9-2: Similarly, suppose that we are gathering data, but
the thing we are recording is not a sample mean, but a sample
proportion – a p-hat. Naturally, these would vary, since p-hats
are statistics. If we made a graph of these p-hats over time –
taking many, many samples and recording the sample
proportion each time, then these would form a shape with
certain values also.
(6)
 phat  p
The mean of these p-hats would equal the
population percentage – the true value of “p”.
 p hat 
p (1  p )
The standard deviation of these p-hats is
n
dependent on the true value of “p” and also the value of “n” –
the size of the samples from which the p-hats came.
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These p-hats – which vary, since they are statistics – can be
graphed, of course, and, IF certain conditions are met, then we
know what sort of shape they will make.
(7) IF all of these four conditions are met:
* the sample was an SRS,
* the size of the population is >10n,
* the value of np is >10, and
* the value of n(1-p) is >10,
THEN the sampling distribution of p-hat is
approximately Normal.
(This is similar to the Central Limit Theorem, but not called that.)
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