E1C04 09/14/2010
14:7:44 Page 149
Example 4.12
Determine the number of measurements required to reduce the 95% confidence interval of
the mean value of a variable to within Æ1 unit, if the variance of the population is estimated to be
64 units.
KNOWN
CI ¼ Æ1 units ¼ 2 units
P ¼ 95%
d ¼ 1
s
2
¼ 64 units
ASSUMPTIONS
s
2
% s
2
x
FIND N required
SOLUTION Equation 4.47 has two unknowns in N and t
N %
t v;95 s x
d
2
95%
ð
Þ
We iterate by using a trial-and-error approach until convergence. Suppose we begin by guessing that
N ¼ 61 so that t n;95 ¼ 2:00 Then, we solve Equation 4.47 with t n;95 ¼ 2:00 and s x ¼ 8 units, to find
N ¼ 256. We now use N ¼ 256, so that n ¼ 255 and t n;95 ¼ 1:96. This gives the new estimate of N ¼
245. Repeat again with N ¼ 245, so that n ¼ 244, t n;95 ¼ 1:96. We again get N ¼ 245. The analysis is
converged at 245 measurements. Check the results after 245 measurements to ensure that the
variance level used was representative of the actual population.
COMMENT Since the confidence interval is reduced as N
1/2 , the procedure of increasing N to
decrease this interval becomes one of diminishing returns.
Example 4.13
From 21 preliminary measurements of a variable, the standard deviation of the data set is 160 units.
We want to reduce the 95% confidence interval in the mean value to Æ30 units. Estimate the total
number of measurements required.
KNOWN
S 1 ¼ 160 units
N 1 ¼ 21
d ¼ CI=2 ¼ 30 t 20;95 ¼ 2:093
ASSUMPTIONS s
2
% s
2
x
FIND N T
SOLUTION At 21 measurements, the confidence interval in the mean value is Æð2:093Þð160Þ=
ffiffiffiffiffi
21
p ¼ Æ73 units. We need to reduce this to Æ30 units. The total number of measurements required
is estimated by
N T %
t NÀ1;95 s 1
d
2
¼
2:093 Â 160
30
2
¼ 125 95%
ð
Þ
Thus, as a first guess, a total of 125 measurements are estimated to be necessary. Take an additional
104 measurements and then reanalyze to be certain that the constraint is met.
4.8 Number of Measurements Required 149
14:7:44 Page 149
Example 4.12
Determine the number of measurements required to reduce the 95% confidence interval of
the mean value of a variable to within Æ1 unit, if the variance of the population is estimated to be
64 units.
KNOWN
CI ¼ Æ1 units ¼ 2 units
P ¼ 95%
d ¼ 1
s
2
¼ 64 units
ASSUMPTIONS
s
2
% s
2
x
FIND N required
SOLUTION Equation 4.47 has two unknowns in N and t
N %
t v;95 s x
d
2
95%
ð
Þ
We iterate by using a trial-and-error approach until convergence. Suppose we begin by guessing that
N ¼ 61 so that t n;95 ¼ 2:00 Then, we solve Equation 4.47 with t n;95 ¼ 2:00 and s x ¼ 8 units, to find
N ¼ 256. We now use N ¼ 256, so that n ¼ 255 and t n;95 ¼ 1:96. This gives the new estimate of N ¼
245. Repeat again with N ¼ 245, so that n ¼ 244, t n;95 ¼ 1:96. We again get N ¼ 245. The analysis is
converged at 245 measurements. Check the results after 245 measurements to ensure that the
variance level used was representative of the actual population.
COMMENT Since the confidence interval is reduced as N
1/2 , the procedure of increasing N to
decrease this interval becomes one of diminishing returns.
Example 4.13
From 21 preliminary measurements of a variable, the standard deviation of the data set is 160 units.
We want to reduce the 95% confidence interval in the mean value to Æ30 units. Estimate the total
number of measurements required.
KNOWN
S 1 ¼ 160 units
N 1 ¼ 21
d ¼ CI=2 ¼ 30 t 20;95 ¼ 2:093
ASSUMPTIONS s
2
% s
2
x
FIND N T
SOLUTION At 21 measurements, the confidence interval in the mean value is Æð2:093Þð160Þ=
ffiffiffiffiffi
21
p ¼ Æ73 units. We need to reduce this to Æ30 units. The total number of measurements required
is estimated by
N T %
t NÀ1;95 s 1
d
2
¼
2:093 Â 160
30
2
¼ 125 95%
ð
Þ
Thus, as a first guess, a total of 125 measurements are estimated to be necessary. Take an additional
104 measurements and then reanalyze to be certain that the constraint is met.
4.8 Number of Measurements Required 149