Cofidece Itervals Itroductio A poit estimate provides o iformatio about the precisio ad reliability of estimatio. For example, the sample mea X is a poit estimate of the populatio mea μ but because of samplig variability, it is virtually ever the case that x = μ. A poit estimate says othig about how close it might be to μ. A alterative to reportig a sigle sesible value for the parameter beig estimated it to calculate ad report a etire iterval of plausible values a cofidece iterval (CI). week 5 1
Cofidece level A cofidece level is a measure of the degree of reliability of a cofidece iterval. It is deoted as 100(1-α)%. The most frequetly used cofidece levels are 90%, 95% ad 99%. A cofidece level of 100(1-α)% implies that 100(1-α)% of all samples would iclude the true value of the parameter estimated. The higher the cofidece level, the more strogly we believe that the true value of the parameter beig estimated lies withi the iterval. week 5 2
Large Sample CI for μ Recall: a poit estimate of the populatio mea μ is the sample mea. If the sample size is large, the the CLT applies ad we have X σ / μ d Z ~ N ( 0,1). A 100(1-α)% cofidece iterval for μ, from a large iid sample is x z σ α ± 2 This iterval is ot radom; it either does, or does ot cotai μ. If we make repeated CI s the 100(1-α)% will cotai μ ad 100 α% will ot. If σ 2 is ot kow we estimate it with s 2. week 5 3
Example The Natioal Studet Loa Survey collected data about the amout of moey that borrowers owe. The survey selected a radom sample of 1280 borrowers who bega repaymet of their loas betwee four to six moths prior to the study. The mea debt for the selected borrowers was $18,900 ad the stadard deviatio was $49,000. Fid a 95% for the mea debt for all borrowers. week 5 4
Width ad Precisio of CI The precisio of a iterval is coveyed by the width of the iterval. If the cofidece level is high ad the resultig iterval is quite arrow, the iterval is more precise, i.e., our kowledge of the value of the parameter is reasoably precise. A very wide CI implies that there is a great deal of ucertaity cocerig the value of the parameter we are estimatig. The width of the CI for μ is. week 5 5
Importat Commet Cofidece itervals do ot eed to be cetral, ay a ad b that solve X μ P a < < b = 1 α σ / defie 100(1-α)% CI for the populatio mea μ. week 5 6
Oe Sided CI CI gives both lower ad upper bouds for the parameter beig estimated. I some circumstaces, a ivestigator will wat oly oe of these two types of boud. A large sample upper cofidece boud for μ is σ μ < + A large sample lower cofidece boud for μ is σ μ > x z x z α α week 5 7
Choice of Sample Size Sample size ca be determied if we kow (i) the width (W=2B) of the desired CI (ii) a estimate of σ ad (iii) the cofidece level The sample size for a 100(1-α)% CI for μ with a desired width 2B is 2 ˆ α / 2 σ z B week 5 8
Example You wat to ret a ufurished oe-bedroom apartmet for ext semester. How large a sample of oe-bedroom apartmets would be eeded to estimate the mea µ withi ±$20 with 99% cofidece? week 5 9
Cofidece iterval for Populatio Proportio A large sample cofidece iterval for populatio proportio, p, is pˆ ± z α 2 The sample size for a 100(1-α)% CI for p with a desired width 2B is zα / 2 p *( 1 p *) B where p* is a guessed value for the proportio of successes i a future sample. Ca use the sample proportio from a give sample as the value of p* or ay other value i which the ivestigator strogly believe. The most coservative approach is to choose p* = 0.5. Why? 2 pq ˆ ˆ week 5 10
Example I a sample of 400 computer memory chips made at Digital Devices, Ic., 40 were foud to be defective. Give a 95% cofidece iterval for the proportio of defective chips i the populatio from which the sample was take? What sample size is ecessary if the 90% CI for the proportio of defective chips, p, is to have width of at most 0.1? week 5 11