Confidence Intervals Introduction A point estimate provides no information about the precision and reliability 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. 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). STA 248 week 5 1
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. STA 248 week 5 2
Deriving a Confidence Interval Suppose X 1, X 2,,X n are random sample and we observed the data x 1, x 2,,x n which are realization of these random variables. We want a CI for some parameter θ. To derive this CI we need to find another random variable that is typically a function of the estimator of θ satisfying: 1) It depends on X 1, X 2,,X n and θ 2) Its probability distribution does not depend on θ or any other unknown parameter. Such a random variable is called a pivot. Example. STA 248 week 5 3
CI for μ When σ is Known Suppose X 1, X 2,,X n are random sample from N(μ, σ 2 ) where μ is unknown and σ is known. A 100(1-α)% confidence interval for μ is, Proof: x z 2 n STA 248 week 5 4
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. STA 248 week 5 5
Important Comment Confidence intervals do not need to be central, any a and b that solve X Pa b 1 / n define 100(1-α)% CI for the population mean μ. STA 248 week 5 6
One Sided CI for μ 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 x z x z n n STA 248 week 5 7
Large Sample CI for μ Recall: if the sample size is large, then the CLT applies and we have X d Z / n 0,1. A 100(1-α)% confidence interval for μ, from a large iid sample is x z 2 If σ 2 is not known we estimate it with s 2. ~ n N STA 248 week 5 8
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. STA 248 week 5 9
Example Binomial Distribution Suppose X 1, X 2,,X n are random sample from Bernoulli(θ) distribution. A 100(1-α)% CI for θ is. Example STA 248 week 5 10
One Sample Variance In many case we will be interested in making inference about the population variance. Suppose X 1, X 2,,X n are random sample from N(μ, σ 2 ) where both μ and σ are unknown. A CI for σ 2 is STA 248 week 5 11
Example STA 248 week 5 12
t distribution Suppose Z ~ N(0,1) independent of X ~ χ 2 (n). Then, T Z X / v ~ t. v Proof: week1 13
Claim Suppose X 1, X 2, X n are i.i.d normal random variables with mean μ and variance σ 2. Then, Proof: X ~ t S / n n1 week1 14
CI for μ When σ is Unknown Suppose X 1, X 2,,X n are random sample from N(μ, σ 2 ) where both μ and σ are unknown. If σ 2 is unknown we can estimate it using s 2 and use the t n-1 distribution. A 100(1-α)% confidence interval for μ in this case, is STA 248 week 5 15