E1C04 09/14/2010
14:7:44 Page 148
SOLUTION Based on the 10 data points, the sample mean and sample standard deviation are
found from Equations 4.14a and c to be x ¼ 27 psi with an s x ¼ 3.8. But the tire pressure should not
vary much between two readings beyond the precision capabilities of the measurement system and
technique, so data point 9 (x ¼ 18) is a potential outlier.
Apply Chauvenet’s criterion to this data point. For N ¼ 10, Equation 4.44 gives 1 À 2 Â Pðz 0 Þ
ð
Þ
< 1=2N ¼ 1=20 ¼ 0:050. So the criterion identifies the data point as an outlier if it lies outside 1 À
0.050 ¼ 0.950, which for 10 data points is the 95% probability spread of the data. For x i ¼ 18, z 0 ¼
2.368. Then P(z 0 ) ¼ P(2.368) ¼ 0.4910, so that 1 À 2 Â 0:4910
ð
Þ ¼ 0:018. As 0.018 < 0.050, this test
identifies the data point x ¼ 18 as a potential outlier.
4.8 NUMBER OF MEASUREMENTS REQUIRED
We can use the previous discussions to assist in the design and planning of a test program. For
example, how many measurements, N, are required to reduce the estimated value for random error in
the sample mean to an acceptable level? To answer this question, begin with Equation 4.19:
x
0
¼ x Æ t v;P s x P%
ð Þ
ð4:19Þ
Let CI be the confidence interval in Equation 4.18, that is,
CI ¼ Æt v;P s x ¼ Æt v;P
s x
ffiffiffiffi
N
p P%
ð Þ
ð4:45Þ
This interval is two sided about the mean, defining a range from Àt v;P
s x ffiffi ffi
N
p to þt v;P
s x ffiffi ffi
N
p . We introduce
the one-sided precision value d as
d ¼
CI
2
¼ t v;P
s x
ffiffiffiffi
N
p
ð4:46Þ
For example, if the confidence interval is Æ1 units, an interval of width 2 units, then d ¼ 1 unit. It
follows that the required number of measurements is estimated by
N %
t v;P s x
d
2
P%
ð Þ
ð4:47Þ
Because the degrees of freedom in t depends on N, solving Equation 4.47 requires iteration.
Equation 4.47 provides a first estimate for the number of measurements needed. How closely it
reduces the range of random error to the constraint depends on how well the assumed value of s x
approximates the population s.
A shortcoming of this method is the need to estimate s x , which could be based on experience or
other knowledge of the population. One approach is to make a preliminary number of measurements, N 1 , to obtain an estimate of the sample variance, s
2
1 , to be expected. Then use s 1 to estimate
the number of measurements required. The total number of measurements, N T , will be estimated by
N T %
t NÀ1;95 s 1
d
2
95%
ð
Þ
ð4:48Þ
This establishes that N T À N 1 additional measurements are required.
148 Chapter 4 Probability and Statistics
14:7:44 Page 148
SOLUTION Based on the 10 data points, the sample mean and sample standard deviation are
found from Equations 4.14a and c to be x ¼ 27 psi with an s x ¼ 3.8. But the tire pressure should not
vary much between two readings beyond the precision capabilities of the measurement system and
technique, so data point 9 (x ¼ 18) is a potential outlier.
Apply Chauvenet’s criterion to this data point. For N ¼ 10, Equation 4.44 gives 1 À 2 Â Pðz 0 Þ
ð
Þ
< 1=2N ¼ 1=20 ¼ 0:050. So the criterion identifies the data point as an outlier if it lies outside 1 À
0.050 ¼ 0.950, which for 10 data points is the 95% probability spread of the data. For x i ¼ 18, z 0 ¼
2.368. Then P(z 0 ) ¼ P(2.368) ¼ 0.4910, so that 1 À 2 Â 0:4910
ð
Þ ¼ 0:018. As 0.018 < 0.050, this test
identifies the data point x ¼ 18 as a potential outlier.
4.8 NUMBER OF MEASUREMENTS REQUIRED
We can use the previous discussions to assist in the design and planning of a test program. For
example, how many measurements, N, are required to reduce the estimated value for random error in
the sample mean to an acceptable level? To answer this question, begin with Equation 4.19:
x
0
¼ x Æ t v;P s x P%
ð Þ
ð4:19Þ
Let CI be the confidence interval in Equation 4.18, that is,
CI ¼ Æt v;P s x ¼ Æt v;P
s x
ffiffiffiffi
N
p P%
ð Þ
ð4:45Þ
This interval is two sided about the mean, defining a range from Àt v;P
s x ffiffi ffi
N
p to þt v;P
s x ffiffi ffi
N
p . We introduce
the one-sided precision value d as
d ¼
CI
2
¼ t v;P
s x
ffiffiffiffi
N
p
ð4:46Þ
For example, if the confidence interval is Æ1 units, an interval of width 2 units, then d ¼ 1 unit. It
follows that the required number of measurements is estimated by
N %
t v;P s x
d
2
P%
ð Þ
ð4:47Þ
Because the degrees of freedom in t depends on N, solving Equation 4.47 requires iteration.
Equation 4.47 provides a first estimate for the number of measurements needed. How closely it
reduces the range of random error to the constraint depends on how well the assumed value of s x
approximates the population s.
A shortcoming of this method is the need to estimate s x , which could be based on experience or
other knowledge of the population. One approach is to make a preliminary number of measurements, N 1 , to obtain an estimate of the sample variance, s
2
1 , to be expected. Then use s 1 to estimate
the number of measurements required. The total number of measurements, N T , will be estimated by
N T %
t NÀ1;95 s 1
d
2
95%
ð
Þ
ð4:48Þ
This establishes that N T À N 1 additional measurements are required.
148 Chapter 4 Probability and Statistics