Standard Error
Mathematics 47: Lecture 13
Dan Sloughter
Furman University
March 23, 2006
Dan Sloughter (Furman University)
Standard Error
March 23, 2006
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Root mean squared error
Definition
Suppose T is an estimator for a parameter θ. We call
q
p
r.m.s.e.(T ) = m.s.e.(T ) = E [(T − θ)2 ]
the root-mean square error of T .
Dan Sloughter (Furman University)
Standard Error
March 23, 2006
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Root mean squared error
Definition
Suppose T is an estimator for a parameter θ. We call
q
p
r.m.s.e.(T ) = m.s.e.(T ) = E [(T − θ)2 ]
the root-mean square error of T .
I
Note: if T is unbiased, r.m.s.e.(T ) = σT , the standard deviation of
T.
Dan Sloughter (Furman University)
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March 23, 2006
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Example
Dan Sloughter (Furman University)
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March 23, 2006
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Example
I
If X1 , X2 , . . . , Xn is a random sample from a distribution with mean
µ and variance σ 2 , then
σ
r.m.s.e.(X̄ ) = √ .
n
Dan Sloughter (Furman University)
Standard Error
March 23, 2006
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Example
I
If X1 , X2 , . . . , Xn is a random sample from a distribution with mean
µ and variance σ 2 , then
σ
r.m.s.e.(X̄ ) = √ .
n
I
In this case, we call
S
s.e.(X̄ ) = √
n
the standard error of X̄ .
Dan Sloughter (Furman University)
Standard Error
March 23, 2006
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Example
I
If X1 , X2 , . . . , Xn is a random sample from a distribution with mean
µ and variance σ 2 , then
σ
r.m.s.e.(X̄ ) = √ .
n
I
In this case, we call
S
s.e.(X̄ ) = √
n
the standard error of X̄ .
I
Note: usually we do not know σ, and so cannot compute r.m.s.e.(X̄ ),
but we can nevertheless estimate it with s.e.(X̄ ).
Dan Sloughter (Furman University)
Standard Error
March 23, 2006
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Example
Dan Sloughter (Furman University)
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Example
I
If X1 , X2 , . . . , Xn is a random sample from a Bernoulli distribution
with probability of success p, then
r
p(1 − p)
r.m.s.e.(p̂) =
.
n
Dan Sloughter (Furman University)
Standard Error
March 23, 2006
4/5
Example
I
If X1 , X2 , . . . , Xn is a random sample from a Bernoulli distribution
with probability of success p, then
r
p(1 − p)
r.m.s.e.(p̂) =
.
n
I
In this case, we call
r
s.e.(p̂) =
p̂(1 − p̂)
n
the standard error of p̂.
Dan Sloughter (Furman University)
Standard Error
March 23, 2006
4/5
Example
I
If X1 , X2 , . . . , Xn is a random sample from a Bernoulli distribution
with probability of success p, then
r
p(1 − p)
r.m.s.e.(p̂) =
.
n
I
In this case, we call
r
s.e.(p̂) =
p̂(1 − p̂)
n
the standard error of p̂.
I
Note: as noted before,
1
s.e.(p̂) ≤ √ .
2 n
Dan Sloughter (Furman University)
Standard Error
March 23, 2006
4/5
Example
Dan Sloughter (Furman University)
Standard Error
March 23, 2006
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Example
I
Suppose we wish to take a random sample of voters to estimate p,
the true proportion of voters who will vote for a certain candidate.
Dan Sloughter (Furman University)
Standard Error
March 23, 2006
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Example
I
Suppose we wish to take a random sample of voters to estimate p,
the true proportion of voters who will vote for a certain candidate.
I
Moreover, suppose we wish to ensure that s.e.(p̂) ≤ 0.01.
Dan Sloughter (Furman University)
Standard Error
March 23, 2006
5/5
Example
I
Suppose we wish to take a random sample of voters to estimate p,
the true proportion of voters who will vote for a certain candidate.
I
Moreover, suppose we wish to ensure that s.e.(p̂) ≤ 0.01.
I
Then we need to choose a sample size n so that
1
√ ≤ 0.01.
2 n
Dan Sloughter (Furman University)
Standard Error
March 23, 2006
5/5
Example
I
Suppose we wish to take a random sample of voters to estimate p,
the true proportion of voters who will vote for a certain candidate.
I
Moreover, suppose we wish to ensure that s.e.(p̂) ≤ 0.01.
I
Then we need to choose a sample size n so that
1
√ ≤ 0.01.
2 n
I
Hence we want
n≥
Dan Sloughter (Furman University)
1
10, 000
=
= 2500.
2
4(0.01)
4
Standard Error
March 23, 2006
5/5
Example
I
Suppose we wish to take a random sample of voters to estimate p,
the true proportion of voters who will vote for a certain candidate.
I
Moreover, suppose we wish to ensure that s.e.(p̂) ≤ 0.01.
I
Then we need to choose a sample size n so that
1
√ ≤ 0.01.
2 n
I
Hence we want
n≥
I
1
10, 000
=
= 2500.
2
4(0.01)
4
So we need to sample only 2500 voters.
Dan Sloughter (Furman University)
Standard Error
March 23, 2006
5/5
Example
I
Suppose we wish to take a random sample of voters to estimate p,
the true proportion of voters who will vote for a certain candidate.
I
Moreover, suppose we wish to ensure that s.e.(p̂) ≤ 0.01.
I
Then we need to choose a sample size n so that
1
√ ≤ 0.01.
2 n
I
Hence we want
n≥
1
10, 000
=
= 2500.
2
4(0.01)
4
I
So we need to sample only 2500 voters.
I
Why doesn’t this depend on the size of the voter population from
which we are sampling?
Dan Sloughter (Furman University)
Standard Error
March 23, 2006
5/5