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Confidence Intervals – Introduction

• A point estimate provides no information about the precision and reliability of estimation.

• For example, the sample mean is a point estimate of the population mean μ but because of sampling variability, it is virtually never the case that

• 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).

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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%.

• A confidence level of 100(1-α)% implies that 100(1-α)% of all 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,

• Proof:

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Example

• The National Student Loan Survey collected data about the amount

of money that borrowers owe. The survey selected a random sample

of 1280 borrowers who began repayment of their loans between four

to six months prior to the study. The mean debt for the selected

borrowers was $18,900 and the standard deviation was $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

define 100(1-α)% CI for the population mean μ.

1

/b

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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

two types of bound.

• A large sample upper confidence bound for μ is

• A large sample lower confidence bound for μ is

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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.

• 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

• A 100(1-α)% confidence interval for μ, from a large iid sample is

• 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-α)% CI for θ is….

 

 

• Example…

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