Chapter 4
ECE 3800
John Stahl
Western Michigan University
Vocab
Population (N): a set of data being studied.
Population mean(𝐗): the average within the
population.
Sample (n): a subset of data taken from the
population.
Sample Mean(𝐗): the average within the sample
True Mean(m or 𝑿): Population mean.
Unbiased Estimator: when the sample mean is
equal to the true mean.
Vocab
True Variance: The actual variance of the population.
Sample Variance (S2): The variance of the sample
population.
Sampling with Replacement: When samples taken from
the population after measurements are returned to the
population.
Sampling without Replacement: The samples taken from
the population are not returned after measurement.
Example 4-2.1
An endless production line is turning out solidstate diodes and every 100th diode is tested for
its reverse current (IR) and its forward current
(IF) by applying ±1v.
a. IR has a true mean of 10-6 and a variance of
10-12. How many diodes must be tested to
obtain a sample mean (𝐗) whose standard
deviation is 5% of the true mean?
Part 4-2.1.a
Standard Deviation
(± 𝜎 2 ) will relate to the %
difference around the
mean. Since we want 5%
around the true mean,
than SD = 5% ⋅ 𝑋.
𝑣𝑎𝑟 𝑋 = 0.05 ⋅ 𝑋
𝑣𝑎𝑟 𝑋 = 0.05 ⋅ 10−6
2
𝑣𝑎𝑟 𝑋 = 0.25 × 10−12
𝜎2
𝑣𝑎𝑟 𝑋 =
𝑛
0.25 ×
The sample variance can
be related to the true
variance.
2
10−12
10−12
=
𝑛
10−12
𝑛=
= 40
0.25 × 10−12
𝑛 =40
Part 4-2.1.b
IF has a true mean (𝑿) of 0.1 and a variance of 0.0025. How many diodes
must be tested to obtain a sample mean (𝐗) whose standard deviation is 2%
of the true mean.
𝑣𝑎𝑟 𝑋 = 0.02 ⋅ 𝑋
2
𝑣𝑎𝑟 𝑋 = 0.02 ⋅ 0.1
2
𝑣𝑎𝑟 𝑋 = 4 × 10−6
𝜎2
𝑣𝑎𝑟 𝑋 =
𝑛
4 × 10−6
2.5 × 10−3
=
𝑛
2.5 × 10−3
𝑛=
= 625
4 × 10−6
Part 4-2.1.c
Using the large sample size what is the
standard deviations of the sample mean for
each test?
We use 625 samples since we
need to capture the 2% SD for
test performed in part b).
Part a.
Part b.
𝜎2
𝑣𝑎𝑟 𝑋 =
𝑛
𝜎2
𝑣𝑎𝑟 𝑋 =
𝑛
10−12
𝑣𝑎𝑟 𝑋 =
625
𝑆𝐷 =
𝑣𝑎𝑟 𝑋 =
𝑆𝐷 = 4 × 10−8
1.6 × 10−15
0.0025
𝑣𝑎𝑟 𝑋 =
625
𝑆𝐷 =
𝑣𝑎𝑟 𝑋 =
𝑆𝐷 = 2 × 10−3
4 × 10−6