Confidence intervals

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Confidence intervals
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Population mean

• Assumption sample from normal distribution

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1 Population variance ?2 is known
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Given X1, X2, , XN independent, normally
distributed, what is the distribution of
?
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Confidence interval
95 of data
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With probability 0.95
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Equivalently
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2 Population variance ?2 is not known
Use estimate
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Not true. Intuition tells us that the interval
should be wider
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E.G., N7
95 CI looks like this
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Student t distribution
With N degrees of freedom (d.f.)
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Gamma function
L. Euler, XXVIII century
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Normal samples and t distribution
X1, X2, XN independent normal
Has t distribution with N-1 d.f.
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Linear regression and t - distribution
U normally distributed disturbance xi -
controlled variable, fixed values
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(No Transcript)
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Assumption
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Leads to
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Residual mean square estimates variance of
normal disturbance U
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Follow t distribution with n-2 degrees of freedom
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Non normal samples

• For large n, estimates

can still be used due to Central Limit Thorem