Applied Statistics I
Liang Zhang
Department of Mathematics, University of Utah
July 8, 2008
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
1 / 15
Distribution for Sample Mean
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
2 / 15
Distribution for Sample Mean
Example (Problem 54)
Suppose the sediment density (g/cm) of a randomly selected specimen
from a certain region is normally distributed with mean 2.65 and standard
deviation .85 (suggested in “Modeling Sediment and Water Column
Interactions for Hydrophobic Pollutants”, Water Research, 1984:
1169-1174).
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
2 / 15
Distribution for Sample Mean
Example (Problem 54)
Suppose the sediment density (g/cm) of a randomly selected specimen
from a certain region is normally distributed with mean 2.65 and standard
deviation .85 (suggested in “Modeling Sediment and Water Column
Interactions for Hydrophobic Pollutants”, Water Research, 1984:
1169-1174).
a. If a random sample of 25 specimens is selected, what is the
probability that the sample average sediment density is at most 3.00?
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
2 / 15
Distribution for Sample Mean
Example (Problem 54)
Suppose the sediment density (g/cm) of a randomly selected specimen
from a certain region is normally distributed with mean 2.65 and standard
deviation .85 (suggested in “Modeling Sediment and Water Column
Interactions for Hydrophobic Pollutants”, Water Research, 1984:
1169-1174).
a. If a random sample of 25 specimens is selected, what is the
probability that the sample average sediment density is at most 3.00?
b. How large a sample size would be required to ensure that the above
probability is at least .99?
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
2 / 15
Distribution for Sample Mean
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
3 / 15
Distribution for Sample Mean
The Central Limit Theorem (CLT)
Let X1 , X2 , . . . , Xn be a random sample from a distribution with mean
value µ and standard deviation σ. Then if n is sufficiently large, X has
approximately a normal distribution with mean value µ and standard
√
deviation σ/ n, and T0 also has approximately a normal distribution with
√
mean value nµ and standard deviation nσ. The larger the value of n, the
better the approximation.
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
3 / 15
Distribution for Sample Mean
The Central Limit Theorem (CLT)
Let X1 , X2 , . . . , Xn be a random sample from a distribution with mean
value µ and standard deviation σ. Then if n is sufficiently large, X has
approximately a normal distribution with mean value µ and standard
√
deviation σ/ n, and T0 also has approximately a normal distribution with
√
mean value nµ and standard deviation nσ. The larger the value of n, the
better the approximation.
Remark:
1. As long as n is sufficiently large, CLT is applicable no matter Xi ’s are
discrete random variables or continuous random variables.
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
3 / 15
Distribution for Sample Mean
The Central Limit Theorem (CLT)
Let X1 , X2 , . . . , Xn be a random sample from a distribution with mean
value µ and standard deviation σ. Then if n is sufficiently large, X has
approximately a normal distribution with mean value µ and standard
√
deviation σ/ n, and T0 also has approximately a normal distribution with
√
mean value nµ and standard deviation nσ. The larger the value of n, the
better the approximation.
Remark:
1. As long as n is sufficiently large, CLT is applicable no matter Xi ’s are
discrete random variables or continuous random variables.
2. How large should n be such that CLT is applicable?
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
3 / 15
Distribution for Sample Mean
The Central Limit Theorem (CLT)
Let X1 , X2 , . . . , Xn be a random sample from a distribution with mean
value µ and standard deviation σ. Then if n is sufficiently large, X has
approximately a normal distribution with mean value µ and standard
√
deviation σ/ n, and T0 also has approximately a normal distribution with
√
mean value nµ and standard deviation nσ. The larger the value of n, the
better the approximation.
Remark:
1. As long as n is sufficiently large, CLT is applicable no matter Xi ’s are
discrete random variables or continuous random variables.
2. How large should n be such that CLT is applicable?
Generally, if n > 30, CLT can be used.
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
3 / 15
Distribution for Sample Mean
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
4 / 15
Distribution for Sample Mean
Example (Problem 49)
There are 40 students in an elementary statistics class. On the basis of
years of experience, the instructor knows that the time needed to grade a
randomly chosen first examination paper is a random variable with an
expected value of 6 min and a standard deviation of 6 min.
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
4 / 15
Distribution for Sample Mean
Example (Problem 49)
There are 40 students in an elementary statistics class. On the basis of
years of experience, the instructor knows that the time needed to grade a
randomly chosen first examination paper is a random variable with an
expected value of 6 min and a standard deviation of 6 min.
a. If grading times are independent and the instructor begins grading at
6:50pm and grades continuously, what is the (approximate) probability
that he is through grading before the 11:00pm TV news begins?
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
4 / 15
Distribution for Sample Mean
Example (Problem 49)
There are 40 students in an elementary statistics class. On the basis of
years of experience, the instructor knows that the time needed to grade a
randomly chosen first examination paper is a random variable with an
expected value of 6 min and a standard deviation of 6 min.
a. If grading times are independent and the instructor begins grading at
6:50pm and grades continuously, what is the (approximate) probability
that he is through grading before the 11:00pm TV news begins?
b. If the sports report begins at 11:10pm, what is the probability that he
misses part of the report if he waits unitl grading is done before
turning on the TV?
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
4 / 15
Distribution for Sample Mean
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
5 / 15
Distribution for Sample Mean
The original version of CLT
The Central Limit Theorem (CLT)
Let X1 , X2 , . . . be a sequence of i.i.d. random variables from a
distribution with mean value µ and standard deviation σ. Define random
variables
Pn
Xi − nµ
Yn = i=1√
for n = 1, 2, . . .
nσ
Then as n → ∞, Yn has approximately a normal distribution.
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
5 / 15
Distribution for Sample Mean
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
6 / 15
Distribution for Sample Mean
Corollary
Let X1 , X2 , . . . , Xn be a random sample from a distribution for which only
positive values are possible [P(Xi > 0) = 1]. Then if n is sufficiently large,
the product Y = X1 X2 · · · Xn has approximately a lognormal distribution.
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
6 / 15
Distribution for Linear Combinations
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
7 / 15
Distribution for Linear Combinations
Proposition
Let X1 , X2 , . . . , Xn have mean values µ1 , µ2 , . . . , µn , respectively, and
variances σ12 , σ22 , . . . , σn2 , respectively.
1.Whether or not the Xi s are independent,
E (a1 X1 + a2 X2 + · · · + an Xn ) = a1 E (X1 ) + a2 E (X2 ) + · · · + an E (Xn )
= a1 µ1 + a2 µ2 + · · · + an µn
2. If X1 , X2 , . . . , Xn are independent,
V (a1 X1 + a2 X2 + · · · + an Xn ) = a12 V (X1 ) + a22 V (X2 ) + · · · + an2 V (Xn )
= a1 σ12 + a2 σ22 + · · · + an σn2
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
7 / 15
Distribution for Linear Combinations
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
8 / 15
Distribution for Linear Combinations
Proposition (Continued)
Let X1 , X2 , . . . , Xn have mean values µ1 , µ2 , . . . , µn , respectively, and
variances σ12 , σ22 , . . . , σn2 , respectively.
3. More generally, for any X1 , X2 , . . . , Xn
V (a1 X1 + a2 X2 + · · · + an Xn ) =
n X
n
X
ai aj Cov (Xi , Xj )
i=1 j=1
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
8 / 15
Distribution for Linear Combinations
Proposition (Continued)
Let X1 , X2 , . . . , Xn have mean values µ1 , µ2 , . . . , µn , respectively, and
variances σ12 , σ22 , . . . , σn2 , respectively.
3. More generally, for any X1 , X2 , . . . , Xn
V (a1 X1 + a2 X2 + · · · + an Xn ) =
n X
n
X
ai aj Cov (Xi , Xj )
i=1 j=1
We call a1 X1 + a2 X2 + · · · + an Xn a linear combination of the Xi ’s.
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
8 / 15
Distribution for Linear Combinations
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
9 / 15
Distribution for Linear Combinations
Example (Problem 64)
Suppose your waiting time for a bus in the morning is uniformly
distributed on [0,8], whereas waiting time in the evening is uniformly
distributed on [0,10] independent of morning waiting time.
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
9 / 15
Distribution for Linear Combinations
Example (Problem 64)
Suppose your waiting time for a bus in the morning is uniformly
distributed on [0,8], whereas waiting time in the evening is uniformly
distributed on [0,10] independent of morning waiting time.
a. If you take the bus each morning and evening for a week, what is your
total expected waiting time?
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
9 / 15
Distribution for Linear Combinations
Example (Problem 64)
Suppose your waiting time for a bus in the morning is uniformly
distributed on [0,8], whereas waiting time in the evening is uniformly
distributed on [0,10] independent of morning waiting time.
a. If you take the bus each morning and evening for a week, what is your
total expected waiting time?
b. What is the variance of your total waiting time?
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
9 / 15
Distribution for Linear Combinations
Example (Problem 64)
Suppose your waiting time for a bus in the morning is uniformly
distributed on [0,8], whereas waiting time in the evening is uniformly
distributed on [0,10] independent of morning waiting time.
a. If you take the bus each morning and evening for a week, what is your
total expected waiting time?
b. What is the variance of your total waiting time?
c. What are the expected value and variance of the difference between
morning and evening waiting times on a given day?
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
9 / 15
Distribution for Linear Combinations
Example (Problem 64)
Suppose your waiting time for a bus in the morning is uniformly
distributed on [0,8], whereas waiting time in the evening is uniformly
distributed on [0,10] independent of morning waiting time.
a. If you take the bus each morning and evening for a week, what is your
total expected waiting time?
b. What is the variance of your total waiting time?
c. What are the expected value and variance of the difference between
morning and evening waiting times on a given day?
d. What are the expected value and variance of the difference between
total morning waiting time and total evening waiting time on a
particular week?
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
9 / 15
Distribution for Linear Combinations
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
10 / 15
Distribution for Linear Combinations
Corollary
E (X1 − X2 ) = E (X1 ) − E (X2 ) and, if X1 and X2 are independent,
V (X1 − X2 ) = V (X1 ) + V (X2 ).
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
10 / 15
Distribution for Linear Combinations
Corollary
E (X1 − X2 ) = E (X1 ) − E (X2 ) and, if X1 and X2 are independent,
V (X1 − X2 ) = V (X1 ) + V (X2 ).
Proposition
If X1 , X2 , . . . , Xn are independent, normally distributed rv’s (with possibly
different means and/or variances), then any linear combination of the Xi s
also has a normal distribution. In particular, the difference X1 − X2
between two independent, normally distributed variables is itself normally
distributed.
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
10 / 15
Distribution for Linear Combinations
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
11 / 15
Distribution for Linear Combinations
Example (Problem 62)
Manufacture of a certain component requires three different maching
operations. Machining time for each operation has a normal distribution,
and the three times are independent of one another. The mean values are
15, 30, and 20min, respectively, and the standard deviations are 1, 2, and
1.5min, respectively.
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
11 / 15
Distribution for Linear Combinations
Example (Problem 62)
Manufacture of a certain component requires three different maching
operations. Machining time for each operation has a normal distribution,
and the three times are independent of one another. The mean values are
15, 30, and 20min, respectively, and the standard deviations are 1, 2, and
1.5min, respectively.
What is the probability that it takes at most 1 hour of machining time to
produce a randomly selected component?
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
11 / 15
Point Estimation
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
12 / 15
Point Estimation
Example (a variant of Problem 62, Ch5)
Manufacture of a certain component requires three different maching
operations. The total time for manufacturing one such component is
known to have a normal distribution. However, the mean µ and variance
σ 2 for the normal distribution are unknown. If we did an experiment in
which we manufactured 10 components and record the operation time,
and the sample time is given as following:
1
2
3
4
5
time 63.8 60.5 65.3 65.7 61.9
6
7
8
9
10
time 68.2 68.1 64.8 65.8 65.4
What can we say about the population mean µ and population variance
σ2?
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
12 / 15
Point Estimation
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
13 / 15
Point Estimation
Example (a variant of Problem 64, Ch5)
Suppose the waiting time for a certain bus in the morning is uniformly
distributed on [0, θ], where θ is unknown. If we record 10 waiting times as
follwos:
1
2
3
4
5
time 7.6 1.8 4.8 3.9 7.1
6
7
8
9
10
time 6.1 3.6 0.1 6.5 3.5
What can we say about the parameter θ?
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
13 / 15
Point Estimation
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
14 / 15
Point Estimation
Definition
A point estimate of a parameter θ is a single number that can be
regarded as a sensible value for θ. A point estimate is obtained by
selecting a suitable statistic and computing its value from the given sample
data. The selected statistic is called the point estimator of θ.
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
14 / 15
Point Estimation
Definition
A point estimate of a parameter θ is a single number that can be
regarded as a sensible value for θ. A point estimate is obtained by
selecting a suitable statistic and computing its value from the given sample
data. The selected statistic is called the point estimator of θ.
P
e.g. X = 10
i=1 Xi /10 is a point estimator for µ for the normal distribution
example.
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
14 / 15
Point Estimation
Definition
A point estimate of a parameter θ is a single number that can be
regarded as a sensible value for θ. A point estimate is obtained by
selecting a suitable statistic and computing its value from the given sample
data. The selected statistic is called the point estimator of θ.
P
e.g. X = 10
i=1 Xi /10 is a point estimator for µ for the normal distribution
example.
The largest sample data X10,10 is a point estimator for θ for the uniform
distribution example.
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
14 / 15
Methods of Point Estimation
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
15 / 15
Methods of Point Estimation
Definition
Let X1 , X2 , . . . , Xn be a random sample from a distribution with pmf or
pdf f (x). For k = 1, 2, 3, . . . , the kth population moment, or kth
moment
of the distribution f (x), is E (X k ). The kth sample moment
P
is n1 ni=1 Xik .
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
15 / 15
Methods of Point Estimation
Definition
Let X1 , X2 , . . . , Xn be a random sample from a distribution with pmf or
pdf f (x). For k = 1, 2, 3, . . . , the kth population moment, or kth
moment
of the distribution f (x), is E (X k ). The kth sample moment
P
is n1 ni=1 Xik .
Definition
Let X1 , X2 , . . . , Xn be a random sample from a distribution with pmf or
pdf f (x; θ1 , . . . , θm ), where θ1 , . . . , θm are parameters whose values are
unknown. Then the moment estimators θ̂1 , . . . , θ̂m are obtained by
equating the first m sample moments to the corresponding first m
population moments and solving for θ1 , . . . , θm .
Liang Zhang (UofU)
Applied Statistics I
July 8, 2008
15 / 15