sample variance
1
1
1
(
)
(in List MF26)
1
1
(
)
1
(
)
1
n
s
n
x
x
x
x
n
n
n
x
a
x
a
n
n
Example 8.8 :
A manufacturer claims that the average life span of his electric light bulbs is 2000 hours. A random
sample of 64 bulbs is tested and the life span
x
in hours recorded. The results obtained are as follows:
2
127808,
(
)
9694.6
x
x
x
Is there sufficient evidence, at 2% level of significance, that the manufacturer is over-estimating the
lifetime of his light bulbs? Assume that the distribution of the life span of light bulb is normal.
Solution :
Unbiased estimates for population mean and population
variance,
127808
1997
64
x
,
2
2
1
1
(
)
153.8825
1
n
i
i
s
x
x
n
Let
denote the mean lifetime of an electric light bulb.
If
the
manufacturer
is
over-
estimating
the
lifetime,
the
population mean is lower than the
claimed value.
When the sample size is large (
50
n
) and distribution of
X
is
not
normally distributed, the
distribution of
X
is approximately normal by Central Limit Theorem.
Hence (i)
2
~ N
,
X
n
if population variance is known, or
(ii)
2
~ N
,
s
X
n
approximately, if population variance is unknown.
The test statistic is
N(0,1
~
)
/
X
Z
n
or
N(0,1
~
)
/
X
Z
s
n
approximately

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