1. 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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2. 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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3. Large Sample CI for μ
Recall: a point estimate of the population mean μ is the sample
mean. 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
This interval is not random; it either does, or does not contain μ.
If we make repeated CI’s then 100(1-α)% will contain μ and 100∙α%
will not.
If σ2 is not known we estimate it with s2.
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4. 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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5. 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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6. 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.
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7. 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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8. Choice of Sample Size Sample size can be determined if we know
(i) the width (W=2B) of the desired CI
(ii) an estimate of σ and
(iii) the confidence level
The sample size for a 100(1-α)% CI for μ with a desired width 2B is
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9. Example You want to rent an unfurnished one-bedroom apartment for next
semester. How large a sample of one-bedroom apartments would be
needed to estimate the mean µ within ±$20 with 99% confidence?
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10. Confidence interval for Population Proportion
A large sample confidence interval for population proportion, p, is
The sample size for a 100(1-α)% CI for p with a desired width 2B is
where p* is a guessed value for the proportion of successes in a future sample.
Can use the sample proportion from a given sample as the value of
p* or any other value in which the investigator strongly believe.
The most conservative approach is to choose p* = 0.5. Why?
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11. Example
In a sample of 400 computer memory chips made at Digital Devices, Inc., 40 were found to be defective. Give a 95% confidence interval for the proportion of defective chips in the population from which the sample was taken?
What sample size is necessary if the 90% CI for the proportion of defective chips, p, is to have width of at most 0.1?
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