HP 50G, Руководство пользователя

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Page 18-24
Θ
The parameter
α
is known as the significance level. Typical values of
α
are
0.01, 0.05, 0.1, corresponding to confidence levels of 0.99, 0.95, and
0.90, respectively.
Confidence intervals for the population mean when the
population variance is known
Let
⎯
X be the mean of a random sample of size n, drawn from an infinite
population with known standard deviation
σ
. The 100(1-
α
) % [i.e., 99%, 95%,
90%, etc.], central, two-sided confidence interval for the population mean
μ
is
(
⎯
X
−
z
α
/2
⋅σ
/
√
n ,
⎯
X+z
α
/2
⋅σ
/
√
n ), where z
α
/2
is a standard normal variate that
is exceeded with a probability of
α
/2. The standard error of the sample
mean,
⎯
X, is
⋅σ
/
√
n.
The one-sided upper and lower 100(1-
α
) % confidence limits for the population
mean
μ
are, respectively, X+z
α
⋅σ
/
√
n , and
⎯
X
−
z
α
⋅σ
/
√
n . Thus, a lower, one-
sided, confidence interval is defined as (-
∞
, X+z
α
⋅σ
/
√
n), and an upper, one-
sided, confidence interval as (X
−
z
α
⋅σ
/
√
n,+
∞
). Notice that in these last two
intervals we use the value z
α
, rather than z
α/2
.
In general, the value z
k
in the standard normal distribution is defined as that
value of z whose probability of exceedence is k, i.e., Pr[Z>z
k
] = k, or Pr[Z<z
k
]
= 1 – k. The normal distribution was described in Chapter 17.
Confidence intervals for the population mean when the
population variance is unknown
Let
⎯
X and S, respectively, be the mean and standard deviation of a random
sample of size n, drawn from an infinite population that follows the normal
distribution with unknown standard deviation
σ
. The 100
⋅
(1
−α
) % [i.e., 99%,
95%, 90%, etc.] central two-sided confidence interval for the population mean
μ,
is (
⎯
X
−
t
n-1,
α
/2
⋅
S /
√
n ,
⎯
X+ t
n-1,
α
/2
⋅
S/
√
n ), where t
n-1,
α
/2
is Student's t
variate with
ν
= n-1 degrees of freedom and probability
α
/2 of exceedence.
The one-sided upper and lower 100
⋅
(1-
α
) % confidence limits for the
population mean
μ
are, respectively,
X + t
n-1,
α
/2
⋅
S/
√
n , and
⎯
X
−
t
n-1,
α
/2
⋅
S /
√
n.