Statistical Inference
: branch of statistics that has to do with using samples to guess/infer information about unknown populations, most notably, inferring
about parameters of unknown populations such as population means and population proportions
Point estimate
: a
single # calculated
from a sample to
guess a population
parameter ( for
example
x
is the
point estimator for
µ
A (1α)100% confidence interval
is a range of #’s characterized by a lower bound and upper bound that gives a range of likely values
for a population parameter. 90%=1.645=α:.10. 95%=1.96=α:.05. 98%=2.33=α:.02. 99%=2.58=α:.01
95% CI for µ is given by
± .
x
1 96sn
Sample Variance:
=


s2
xi1 xi2nn
1
Hypothesis test:
a means of deciding when two complimentary statements about a population parameter.
The null hypothesis (
Ho
) always states that the population parameter is equal to some hypothesized value (ie.
Ho
: µ=5)
The alternative hypothesis (
Ha
) always involves some form of inequality. ( <, ≠, >)
When
Ha
involves ≠, the test is 2tailed, otherwise the test is 1 tailed. If ≠, then the test is 2tailed, otherwise the test is 1tailed.
Type I error: rejecting the null hypothesis when it is really true. Type II error: not rejecting null hypothesis when null hypothesis is really false
Large sample
test about
population
mean
µ
For large sample tests,
our point estimates for the
population parameter will
have a normal
distribution. Usually, we
want to test whether the
population parameter (µ)
is equal to some
hypothesized value (
)
µo
1)
Ho:
2)
The
Zts
= 
x μosn
Notice that if
the sample mean (
x
)
is close to the
hypothesized value
(
)
μo
then
Zts
will be
close to 0. If
x
is far
from
μo
then
Zts
will
be far from 0.
3)
Next
Ie. For 1
tail test w/
Zts
= 3.12
For 2 tail
test w/
Steps for a small sample test about population mean
µ
When the sample size is small the test statistic no longer has a std. normal dist’n. instead it will have a t dist’n w/ some degrees of freedom
.
1)
Ho:
:
µ
=
μo
Ha= µ≠
μo
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 Spring '07
 Leong
 Statistics, Null hypothesis, Statistical hypothesis testing

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