CHP 83
Confidence interval for the population mean (µ) is:
(Std deviation)
mean ('xhat')
±
√
n
Must ask ourselves, “where did std deviation come from in the equation?

If it came from same sample as the mean, came from the t table (
s
– sample stan dev.)

If it came from the
Z
table, it is the population standard deviation (σ)
The sample mean, 'x bar', is the estimate of
µ and is also called a
point estimate
, because it consists of a single number.
Population mean
– accuracy depends on sample size. We measure the precision of this estimate by constructing a
confidence interval. When defining a confidence interval for µ, we can define 99%, 95%, 90% confidence interval, or
whatever. The specific percentage represents the confidence level. The
higher
the confidence level, the
wider
the confidence
interval.
8.4
If σ is unknown, it is impossible to determine a confidence interval for µ. This is no longer a standard normal random
variable, Z. However, it does follow another identifiable distribution, the t distribution. Degrees of Freedom = df =
n
– 1
A more accurate confidence interval is always obtained using the t table when the sample standard deviation (
s
) is used in
the construction of this interval.
8.5
Necessary sample size equation is:
Must estimate std deviation
A carefully chosen large sample generally provides a better representation of the population than does a smaller sample. To
obtain a rough approximation of
σ, you could ask a person familiar with the data to be collected two Qs: 1.) What do you
think will be the highest value in the sample (H) or 2.) What will be the lowest value (L)
H  L
a rough approximation of σ is the anticipated range (H – L) divided by 4
so.
.. σ
=
4
If the pop stand dev (σ) is known, the stan normal table is used to derive the confidence interval. If σ is unknown, the t table
is used to derive the confidence interval. For situations where is σ unknown and the sample size is greater than 30,
approximate confidence intervals for the population mean can be constructed using the standard normal table.
CHP 9
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 Spring '10
 Impson
 Normal Distribution, Valuation

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