Higher Engineering Mathematics

SAMPLING AND ESTIMATION THEORIES 585

J

(6) is used, i.e. the confidence limits of the estimated
mean of the population areμx±zcσx.
For an 90% confidence level,zc= 1 .645 (from
Table 61.1), thus,

μx±zcσx= 16. 50 ±(1.645)(1.4)

= 16. 50 ± 2 .30 m.

Thus,the 90% confidence level of the mean time
to failure is from 14.20 minutes to 18.80 minutes.

Problem 8. The sampling distribution of ran-

dom samples of capacitors drawn from a large

batch is found to have a standard error of

the standard deviations of 0.12μF. Determine

the 92% confidence interval for the estimate

of the standard deviation of the whole batch,

if in a particular sample, the standard devi-

ation is 0.60μF. It can be assumed that the

values of capacitance of the batch are normally

distributed.

For the sample: the standard deviation,s= 0. 60 μF.
For the sampling distribution: the standard error of
the standard deviations,

σs= 0. 12 μF

When the confidence level is 92%, then by using
Table 58.1 of partial areas under the standardized
normal curve,

area=

0. 9200

2

= 0 .4600,

giving zc as ±1.751 standard deviations (by
interpolation).
Since the population is normally distributed,
the confidence limits of the standard deviation of
the population may be estimated by using expres-
sion (7), i.e.s±zcσs= 0. 60 ±(1.751)(0.12)=
0. 60 ± 0. 21 μF.
Thus,the 92% confidence interval for the esti-
mate of the standard deviation for the batch is
from 0.39μF to 0.81μF.

Now try the following exercise.

Exercise 221 Further problems on the

estimation of population parameters based

on a large sample size

1. Measurements are made on a random sample
   of 100 components drawn from a popula-
   tion of size 1546 and having a standard

deviation of 2.93 mm. The mean measure-

ment of the components in the sample is

67.45 mm. Determine the 95% and 99% con-

fidence limits for an estimate of the mean of

the population. [

66 .89 and 68.01 mm,

66 .72 and 68.18 mm

]

2. The standard deviation of the masses of
   500 blocks is 150 kg. A random sample of
   40 blocks has a mean mass of 2.40 Mg.

(a) Determine the 95% and 99% confidence

intervals for estimating the mean mass of

the remaining 460 blocks.

(b) With what degree of confidence can it be

said that the mean mass of the remaining

460 blocks is 2.40⎡ ±0.035 Mg?

⎣

(a) 2.355 Mg to 2.445 Mg;

2 .341 Mg to 2.459 Mg

(b) 86%

⎤

⎦

3. In order to estimate the thermal expansion
   of a metal, measurements of the change of
   length for a known change of temperature
   are taken by a group of students. The sam-
   pling distribution of the results has a mean
   of 12. 81 × 10 −^4 m◦C−^1 and a standard error
   of the means of 0. 04 × 10 −^4 m◦C−^1. Deter-
   mine the 95% confidence interval for an
   estimate of the true value of the thermal
   expansion of the metal, correct to two decimal
   places.

[

12. 73 × 10 −^4 m◦C−^1 to

12. 89 × 10 −^4 m◦C−^1

]

4. The standard deviation of the time to failure
   of an electronic component is estimated as
   100 hours. Determine how large a sample of
   these components must be, in order to be 90%
   confident that the error in the estimated time
   to failure will not exceed (a) 20 hours and
   (b) 10 hours.
   [(a) at least 68 (b) at least 271]


5. A sample of 60 slings of a certain diameter,
   used for lifting purposes, are tested to destruc-
   tion (that is, loaded until they snapped). The
   mean and standard deviation of the break-
   ing loads are 11.09 tonnes and 0.73 tonnes
   respectively. Find the 95% confidence inter-
   val for the mean of the snapping loads of all