HP 50g User Manual page 591

The parameter
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 . The 100(1- ) % [i.e., 99%, 95%,
90%, etc.], central, two-sided confidence interval for the population mean
( X z / n , X+z
/2
is exceeded with a probability of
mean, X, is / n.
The one-sided upper and lower 100(1- ) % confidence limits for the population
mean are, respectively, X+z
sided, confidence interval is defined as (- , X+z
sided, confidence interval as (X z
intervals we use the value z , rather than z
In general, the value z
value of z whose probability of exceedence is k, i.e., Pr[Z>z
= 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 . The 100 (1
95%, 90%, etc.] central two-sided confidence interval for the population mean
is ( X t
n-1, /2
variate with = n-1 degrees of freedom and probability /2 of exceedence.
The one-sided upper and lower 100 (1- ) % confidence limits for the
population mean
is known as the significance level. Typical values of
/ n ), where z
/2
/ n , and X z
in the standard normal distribution is defined as that
k
S / n , X+ t
n-1, /2
are, respectively,
X + t S/ n , and X t
n-1, /2
is a standard normal variate that
/2
/2. The standard error of the sample
/ n . Thus, a lower, one-
/ n), and an upper, one-
/ n,+ ). Notice that in these last two
.
S/ n ), where t
n-1, /2
] = k, or Pr[Z<z
k
) % [i.e., 99%,
is Student's t
n-1, /2
S / n.
Page 18-24
are
is
]
k