Download CONFIDENCE INTERVAL for the mean ̅ ( ) √ ̅ ( ) √

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Transcript
CONFIDENCE INTERVAL
for the mean
unknown mean of the population
known standard deviation
size of a random sample
mean of the random sample
̅
Compute an interval with high probability of containing .
 Choose a number , such as 0.01, 0.05, or 0.1, the interval will have
100(
)% chance of containing , such as 99%, 95% or 90%.
⁄ corresponding to the
 In the standard normal table look up the z-value,
⁄ . If you also have a table of upper percentages you may find
probability
⁄ , corresponding to ⁄ in this table. See the figure below for
the z-value
a visual description:
area
area
area
0
As ̅ , , , and
containing is:
are known, the confidence interval with probability of
̅
( )
√
̅
( )
√
of
CONFIDENCE INTERVAL
for a difference of means
One way to compare populations would be to look at the difference of their means.
Let
̅
̅
mean of random sample of size
mean of random sample of size
from one population, and
from another population.
The means of the populations, and are unknown so it is necessary to calculate a
confidence interval for their difference
.
 Choose small , so that the confidence interval will have a probability
containing
.
of
 Find
If the standard deviation,
1
and
2, of
the two populations are known, then
The confidence interval with the probability
̅
̅
( )√
̅
of containing
̅
is:
( )√
If the standard deviations, and , of the two populations are unknown and the sample
sizes are large enough, the calculated variances and can be substituted for
and
, respectively.