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?