Lecture13_570 统计笔记

H 1 : μ 5 ' 6"
Point Estimator, C.I., Test Statistical Inference Draw some conclusion on the population (parameters of interest) based on a random sample.
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5
2. The 90% CI, where ������ = ������. ������ and the corresponding ������������ = ������������.������������ = ������. ������������������
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3. The 99% CI, where ������ = ������. ������������ and the corresponding ������������ = ������������.������������������ = ������. ��������erties too.
<ii> Confidence Interval for - Intuitive approach (backwards derivation for the CI boundaries ������1 ������������������ ������2 ): ������(������1 ≤ ������ ≤ ������2 ) = 0.95 ������(−������1 ≥ −������ ≥ −������2 ) = 0.95 ������(������ − ������1 ≥ ������ − ������ ≥ ������ − ������2 ) = 0.95 ������ − ������1 ������/ ������ ������ − ������ ������/ ������ ������ − ������2 ������/ ������

n
) 1
P( X Z 2 P( X Z 2 P( X Z 2

n n
X Z 2

n
) 1

n
X Z 2 X Z 2

n
) 1 ) 1

n
∴ the 100(1-α)% C.I. for μ is [ X Z 2
n
)
ˆ ) E( X ) ˆ X is an unbiased estimator of E(
Intuitively, this means if you take “many” samples of size n from the population, then the mean of these samples means would be equal to if you take a large enough # of samples. X is also a maximum likelihood estimator(MLE) of . X is also a method of moment estimator (MOME) of .
and variance 2 . That is, X ~ N ( , 2 ), i 1,...,n .
iid.
-
For now, we assume that 2 is known.
ˆ X ~ N ( , <i> Point Estimator for :
2
That is, the estimated 95% confidence interval for the average height of all adult US male is 5’4”, 5’10” . … 5’4” and 5’10”. … This means that we are 95% sure the population mean μ would lie between
2
n
)
2
n
)
X

n
~ N (0,1)
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100(1-)% CI for , 0<<1 (e.g. =0.05 ⇒ 95% C.I.)
P(Z 2 Z Z 2 ) 1
P(Z 2
P( Z 2
X


n
n
Z 2 ) 1
X Z 2
Example 1.A random sample of 400 adult US male was taken and the sample mean was found to be ������ = 5’7” = 67 ������������������ℎ������������ . Based on past studies, it is believed that the population distribution of all adult US male is normal and the standard deviation is 30 inches. Please construct a 95% confidence interval for the average height of all adult US male based on this sample. Solution: The 95% CI for ������ is ������ − 1.96 ������ ������ , ������ + 1.96 ������ ������ = 67 − 1.96 30 400 , 67 + 1.96 30 400 ≈ 64, 70
∴Recall the 100(1-α)% symmetric C.I. for μ is [ X Z 2 *Please note that this CI is symmetric around ������ The length of this CI is:
Lsy 2 Z
1
1. The Exact Confidence Interval for μ when the population is normal &������������ is known
① Point estimator and confidence interval for
-

When the population is normal and the population variance is known. Let X 1 , X 2 ,, X n be a random sample for a normal population with mean
2

n
, X Z 2

n
]

n
Now we derive a non-symmetrical CI:
6
P( Z Z Z 2 ) 1
3 3
n
X
i 1
i
n
)
Other estimators: median, mode, trimmed mean, … <ii> Confidence Interval (C.I.) Eg) 95% C.I. for μ 99.9999% C.I. (‘6-9’ in the manufacture industry) <iii> Hypothesis Test Eg) H 0 : μ 5' 6"
������
≥
≥
= 0.95
Since we know
2
������ =
������ − ������ ������/ ������
~������(0,1)
We can compute the expressions for ������1 ������������������ ������2 . However, one question is that there areMANY ways to choose the C’s. Later you will see that for pivotal quantity with symmetric pdfs, the symmetric CIs are the optimal – in that they have the shortest lengths for the given confidence level100(1-)%.
3
Now we present a general approach to derive the CI’s. General approach for deriving CI’s : the Pivotal Quantity(P.Q.) approach
*Definition: A pivotal quantity is a function of the sample and the parameter of interest. Furthermore, its distribution is entirely known. 1. We start by looking at the point estimator of . X ~ N ( , * Is X a pivotal quantity for ? → X is not because is unknown. * function of X and : X ~ N (0, → Yes, it is pivotal quantity. * Another function of X and : Z → Yes, it is pivotal quantity. So, Pivotal Quantity is not unique. 2. Now that we have found the pivotal quantity Z, we shall start the derivation for the symmetrical CI’s for µ from the PDF of the pivotal quantity Z