Download Confidence Intervals – Introduction

Confidence Intervals – Introduction
• A point estimate provides no information about the precision and
reliabilityy of estimation.
• For example, the sample mean X is a point estimate of the
population mean μ but because of sampling variability, it is virtually
never the case that x = μ .
• A point estimate says nothing about how close it might be to μ.
• An alternative to reporting a single sensible value for the parameter
being estimated it to calculate and report an entire interval of
plausible values – a confidence interval (CI).
(CI)
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Confidence level
• A confidence level is a measure of the degree of reliability of a
confidence interval. It is denoted as 100(1-α)%.
• The most frequently used confidence levels are 90%, 95% and 99%.
( ) implies
p
that 100(1-α)%
( ) of all
• A confidence level of 100(1-α)%
samples would include the true value of the parameter estimated.
• The higher the confidence level, the more strongly we believe that
the true value of the parameter being estimated lies within the
interval.
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CI for μ When σ is Known
• Suppose X1, X2,…,Xn are random sample from N(μ, σ2) where μ is
unknown and σ is known.
• A 100(1-α)% confidence interval for μ is,
x ± zα ⋅
2
σ
n
• Proof:
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Example
• The National Student Loan Survey collected data about the amount
y selected a random sample
p
of moneyy that borrowers owe. The survey
of 1280 borrowers who began repayment of their loans between four
to six months prior to the study. The mean debt for the selected
b
borrowers
was $18
$18,900
900 andd the
h standard
d d deviation
d i i was $49,000.
$49 000
Find a 95% for the mean debt for all borrowers.
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Width and Precision of CI
• The precision of an interval is conveyed by the width of the interval.
• If the confidence level is high and the resulting interval is quite
narrow, the interval is more precise, i.e., our knowledge of the value
of the parameter is reasonably precise.
• A very wide CI implies that there is a great deal of uncertainty
concerning the value of the parameter we are estimating.
• The width of the CI for μ is ….
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Important Comment
• Confidence intervals do not need to be central, any a and b that
solve
⎛
⎞
X −μ
P⎜⎜ a <
< b ⎟⎟ = 1 − α
σ/ n
⎝
⎠
define 100(1-α)% CI for the population mean μ.
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One Sided CI
• CI gives both lower and upper bounds for the parameter being
estimated.
• In some circumstances, an investigator will want only one of these
yp of bound.
two types
• A large sample upper confidence bound for μ is
σ
μ < x + zα ⋅
n
• A large sample lower confidence bound for μ is
μ > x − zα ⋅
σ
n
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CI for μ When σ is Unknown
• Suppose X1, X2,…,Xn are random sample from N(μ, σ2) where both
μ and σ are unknown.
unknown
• If σ2 is unknown we can estimate it using s2 and use the tn-1 distribution.
•
A 100(1-α)% confidence interval for μ in this case, is …
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Large Sample CI for μ
• Recall: if the sample size is large, then the CLT applies and we have
X −μ
σ/ n
d
⎯
⎯→
Z ~ N (0,1).
• A 100(1-α)% confidence interval for μ, from a large iid sample is
σ
x ± zα ⋅
2
n
• If σ2 is not known we estimate it with s2.
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Example – Binomial Distribution
• Suppose X1, X2,…,Xn are random sample from Bernoulli(θ)
distribution A 100(1
distribution.
100(1-α)%
α)% CI for θ is….
is
• Example…
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