SAMPLING AND ESTIMATION THEORIES 581J
and (b) without replacement, correct to three
significant figures.
⎡⎢
⎣(a) μx= 1 .70 cm,
σx= 2. 91 × 10 −^3 cm
(b) μx= 1 .70 cm,
σx= 2. 89 × 10 −^3 cm⎤⎥
⎦A large batch of electric light bulbs have a
mean time to failure of 800 hours and the
standard deviation of the batch is 60 hours.
Use this data and also Table 58.1 on page 561
to solve Problems 4 to 6.- If a random sample of 64 light bulbs is drawn
from the batch, determine the probability that
the mean time to failure will be less than
785 hours, correct to three decimal places.
[0.023]- Determine the probability that the mean time
to failure of a random sample of 16 light bulbs
will be between 790 hours and 810 hours,
correct to three decimal places. [0.497] - For a random sample of 64 light bulbs, deter-
mine the probability that the mean time to
failure will exceed 820 hours, correct to two
significant figures. [0.0038] - The contents of a consignment of 1200 tins
of a product have a mean mass of 0.504 kg
and a standard deviation of 92 g. Deter-
mine the probability that a random sam-
ple of 40 tins drawn from the consignment
will have a combined mass of (a) less than
20.13 kg, (b) between 20.13 kg and 20.17 kg,
and (c) more than 20.17 kg, correct to three
significant figures.
[(a) 0.0179 (b) 0.740 (c) 0.242]61.4 The estimation of population
parameters based on a large
sample size
When a population is large, it is not practical to deter-
mine its mean and standard deviation by using the
basic formulae for these parameters. In fact, when a
population is infinite, it is impossible to determine
these values. For large and infinite populations the
values of the mean and standard deviation may be
estimated by using the data obtained from samples
drawn from the population.Point and interval estimatesAn estimate of a population parameter, such as mean
or standard deviation, based on a single number is
called apoint estimate. An estimate of a popula-
tion parameter given by two numbers between which
the parameter may be considered to lie is called an
interval estimate. Thus if an estimate is made of
the length of an object and the result is quoted as
150 cm, this is a point estimate. If the result is quoted
as 150±10 cm, this is an interval estimate and indi-
cates that the length lies between 140 and 160 cm.
Generally, a point estimate does not indicate how
close the value is to the true value of the quantity and
should be accompanied by additional information on
which its merits may be judged. A statement of the
error or the precision of an estimate is often called
itsreliability. In statistics, when estimates are made
of population parameters based on samples, usually
interval estimates are used. The word estimate does
not suggest that we adopt the approach ‘let’s guess
that the mean value is about...,’ but rather that a
value is carefully selected and the degree of confi-
dence which can be placed in the estimate is given
in addition.Confidence intervalsIt is stated in Section 61.3 that when samples are
taken from a population, the mean values of these
samples are approximately normally distributed, that
is, the mean values forming the sampling distribution
of means is approximately normally distributed. It is
also true that if the standard deviations of each of the
samples is found, then the standard deviations of all
the samples are approximately normally distributed,
that is, the standard deviations of the sampling dis-
tribution of standard deviations are approximately
normally distributed. Parameters such as the mean
or the standard deviation of a sampling distribu-
tion are calledsampling statistics,S. Letμsbe the
mean value of a sampling statistic of the sampling
distribution, that is, the mean value of the means
of the samples or the mean value of the standard
deviations of the samples. Also letσsbe the stan-
dard deviation of a sampling statistic of the sampling
distribution, that is, the standard deviation of the
means of the samples or the standard deviation of the
standard deviations of the samples. Because the sam-
pling distribution of the means and of the standard
deviations are normally distributed, it is possible to
predict the probability of the sampling statistic lying