4-1
PROBLEM 4.1
KNOWN:
N > 1000;
x
= 9.2 units ; S
x
= 1.1 units
FIND:
Range of x in which 50% of all measurements
should fall.
ASSUMPTIONS:
Measurand follows a normal density
function
Data set sufficiently large such that
x
≈
x’
and
S
x
≈
σ
SOLUTION
By assuming that the data is sufficiently large such that its
population behaves as an infinite population, we can find the
interval defined by
x' - z
1
σ
≤
x
i
≤
x' + z
1
σ
as follows. We can find P(x' + z
1
σ
) from the one-sided
integral solution to
This solution is given in Table 4.3 for p(z
1
) = 0.25
(one half
of the 50% probability sought) as z
1
= 0.674. Then, we
d
β
e
]
[2
1
)
P(z
1
2
Z
0
/2
β
1/2
1
∫
−
π
=

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4-2
should expect that 50% of the x
i
values lie in the interval
given by
9.2 – 0.7425
≤
x
i
≤
9.2 + 0.7425
(50%)
COMMENT
We can see from Table 4.4 that as N becomes large the value
for t approaches a value given by z
1
.