7.1 Basic Properties of Confidence Intervals

Transcription

1 7.1 Basic Properties of Confidence Intervals

2 What s Missing in a Point Just a single estimate What we need: how reliable it is Estimate? No idea how reliable this estimate is some measure of the variability

3 Ideas behind CI Realizing that a single estimate is insufficient Try to use an interval to cover our target Crucial aspects: how meaningful is this interval? - width how reliable is this interval? - confidence level

4 Introducing Confidence Intervals for population mean Start with the sample mean We have If X is normal, X = X 1 + X X n n E[ X] = µ, V [ X] = σ2 n P X µ σ/ n < z α/2 = 1 α

5 Interval Derivation X µ σ/ n < z α/2 z α/2 < X µ σ/ n < z α/2 z α/2 σ n < X µ < z α/2 σ n µ z α/2 σ n < X < µ + z α/2 σ n X z α/2 σ n < µ < X + z α/2 σ n

6 Notice only Interpreting CI Interval is random X is random, not µ The probability that this random interval covers µ is 1 α If we repeat the experiment many many times, we should have 100(1 α)% of the intervals covering Typical 1 α values: 90%, 95%, 99%... µ

7 Common Misconceptions Say 95%, z = 1.96, one interval (1.5,1.8) obtained from a sample False claims: there is a 95% chance that is in this interval 95% of the observations are in this interval both are wrong! µ

8 Calculating the Bounds Determine the needed See if σ α Check the normal distribution requirement is available (most likely not, so approximation is needed) x ± z α/2 σ n x z α/2 σ n, x + z α/2 σ n

9 Examples x = 58.3, n = 50, σ = 3 (a). 95%:, z = 1.96 z σ n = = (b). 99%: (57.47, 59.13) z = 2.57, z σ n = = 1.09 (57.2, 59.39) more confidence, wider interval, less useful!

10 Role of Sample Size n Everything else being equal n increased by 4 times - interval narrowed by half n decreased by 4 times - interval enlarged by double if a particular w (width) is required n = 2z α/2 σ 2 w

11 7.2 Large-Sample CI for a Population Mean and Proportion

12 Conditions for Previous CI Discussion Focused on population mean Require random samples σ 2 S 2 Assume the population distribution to be normal Population variance is assumed to be known Approximation of by is often used, assuming large sample size n µ (X 1,X 2,...,X n ) σ 2

13 Getting around the normal No reason to assume a normal distribution in practice (example, the proportion of successes) Often the estimator can be approximately normal if the sample size is large enough (CLT at work!) Formulas are similar, but the implications can be subtle

14 Large-Sample Interval for µ Estimator Approximately normal if n is large Standardize Large-Sample CI for X Z = X µ σ/ n P z α/2 <Z<z α/2 =1 α µ x ± z α/2 s n

15 Issues need to be resolved How large is large enough? Typical rule of thumb: Replacing σ with s n>40 Sample variance itself is random Choosing n to have a desired width? highly dependent on the sample variance

16 From µ to more general parameters Consider more general Do we still have an estimator that is approximately normal? Require ˆθ approximately normal unbiased an expression for σˆθ θ

17 Constructing Large-Sample CI If ˆθ satisfies these requirements, we have P z α/2 < ˆθ θ Still need to find an interval for θ! does σˆθ involve θ? σˆθ <z α/2 1 α or another parameter? available approximation?

18 CI for Population Proportion Parameter in question: population proportion Natural estimator: Expression for σˆθ Solve for the CI: θ = p ˆp = X/n σˆp = p(1 p)/n p = ˆp + z 2 α/2 2n ± z α/2 ˆpˆq n + z2 α/2 4n 2 1+(z 2 α/2 )/n

19 Two Different CI s for p More accurate More simplified p = ˆp + z 2 α/2 2n ± z α/2 Advantages of the first: (1) closer to the the intended confidence level; (2) even works with sample sizes not so large! 1+(z 2 α/2 )/n p =ˆp ± z α/2 ˆpˆq/n ˆpˆq n + z2 α/2 4n 2

20 One-Sided CI (Confidence Bounds) large-sample upper CB µ< x + z α large-sample lower CB µ> x z α s n s n

21 7.3 Intervals Based on Normal Population Distribution

22 Key Assumptions Normal population distribution Population standard dev σ NOT known, so sample standard dev S is used Sample size not large! Problem: is NOT normal! X µ S/ n

23 A New Distribution What do we know about this distribution? more variability than the normal - more spread additional parameter: the number of degree of freedom (df) = n-1, where n is the sample size Student s t-distribution, or just the t- distribution

24 Properties of t Distribution Denote t ν the density function curve for ν df Each t ν curve is bell-shaped, centered at 0 Each t ν normal curve As ν curve is more spread out than the standard increases, the spread decreases As ν, the sequence of t ν curves approaches the standard normal curve

25 Standardize the rv We use the following in place of Z T = X µ S/ n T has the t distribution with df n-1 where n is the sample size As n becomes large, T can be replaced by Z

26 Probability Statement Replacing t α,ν with is the location on measurement axis for which the area under the curve to the right of t α,ν is α t α,ν z α t α,ν is called a t critical value Different df leads to different critical values

27 One-Sample t CI 100(1 α) µ confidence interval for is x t α/2,n 1 s n, x + t α/2,n 1 Upper confidence bound lower confidence bound s n x + t α,n 1 x t α,n 1 s n s n Can we find the n to give a desired interval width?

28 Prediction Interval (PI) Already have a sample of size n ( X 1, X 2,..., X n ) About to pick up the next one ( X n+1 ) What can we predict? one shot: the sample mean better prediction: an interval Prediction interval (PI) for a single observation x ± t α/2,n 1 s n

29 Tolerance Intervals Want to find an interval that captures certain percentage of the values in a normal distribution Allow some uncertain factor (confidence level) Tolerance interval for capturing at least k% of the values, with a confidence level 95% x ± (tolerance critical value) s Critical values found in Appendix Table A.6

30 7.4 CIs for Variance and Standard Dev of a Normal Population

31 Estimating Variance σ always needed in estimation of other parameters A point estimate often suffices, but sometimes we would like to have a CI Distribution of S usually more complicated than that of X

32 Normal Population Assumption We need to start from something familiar Population with normal distribution X is normal, but S is not chi-square χ 2 distribution Another parameter involved (df)

33 A Theorem Let X 1, X 2,..., X n be a random sample from a normal distribution N(µ, σ 2 ) the rv (n 1)S 2 σ 2 = (Xi X) 2 σ 2 has a chi-square distribution with n-1 df

34 chi-squared critical values similar ideas as χ 2 α,ν z α, t α,ν additional parameter - df (n-1 in our applications) denotes the number of the measurement axis such that α of the area under the curve with ν df lies to the right of χ 2 α,ν

35 What s different? χ 2 not symmetric, because it is positive, but unbounded in the positive direction Two ends: lower - χ 2 1 α/2,n 1, upper - Interval for σ 2, confidence level (n 1)S 2 χ 2 α/2,n 1 < σ 2 < Formulas for one-sided: χ 2 α/2,n 1 (n 1)S2 χ 2 1 α/2,n 1 α/2 α 100(1 α)%