week 51 confidence intervals – introduction a point estimate provides no information about the...
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week 5 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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week 5 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.
week 5 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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week 5 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.
week 5 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 ….
week 5 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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week 5 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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week 5 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 2
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week 5 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?
week 5 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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week 5 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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