Confidence Intervals

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1 Cofidece Itervals Cofidece Itervals are a extesio of the cocept of Margi of Error which we met earlier i this course. Remember we saw: The sample proportio will differ from the populatio proportio by more tha the margi of error less tha 5% of the time. The Margi of Error for a sample of size is 1/ Suppose we wated to estimate a Populatio Mea rather tha a Populatio Proportio, ca we compute a Margi of Error for the Mea?

2 2 Cofidece Itervals provide us with the aswer. A Cofidece Iterval is a iterval of umbers cotaiig the most plausible values for our Populatio Parameter. The probability that this procedure produces a iterval that cotais the actual true parameter value is kow as the Cofidece Level ad is geerally chose to be 0.9, 0.95 or Cofidece Itervals take the form: Poit Estimate +/- Critical Value x Stadard Error

3 3 LARGE SAMPLE CONFIDENCE INTERVAL FOR A POPULATION MEAN GENERAL FORMULA x ± ( z critical value) σ The level of cofidece determies the z critical value. 99% % % Sice is large the ukow σ ca be replaced by the sample value s. x ± ( z critical value) s

4 4 Example A radom sample of st year statistics tutorials was selected from the past 5 years ad the umber of studets abset from each oe recorded. The results were x =11.6 ad s=4.1 abseces. Estimate the mea umber of abseces per tutorial over the past 5 years with 90% cofidece. 90% CI for μ is σ x ± ± = 116. ± = ( 1115., ) 225

5 5 INTERPRETATION: It is icorrect to say that there is a probability of 0.90 that μ is betwee ad I fact this probability is either 1 or 0 (μ either is or is ot i the iterval). The 90% refers to the percetage of all possible itervals that cotai μ i.e. to the estimatio process rather tha a particular iterval. It is also icorrect to say that 90% of all tutorials had betwee ad missig studets.

6 6 SMALL SAMPLE CONFIDENCE INTERVAL FOR A POPULATION MEAN Cosider oly samples from populatios which are (approx) ormal. x μ Use the distributio of s which is kow as the t distributio with (-1) degrees of freedom (df). Note: As the df icreases the t curve approaches the z curve. Areas uder the t curve are tabulated i tables 9 & 10 of NCST (ote ν=df) A small sample cofidece iterval for μ is x s ± ( tcritical value) with df = 1 NOTE: This cofidece iterval is appropriate for small samples ONLY whe the populatio distributio is ormal.

7 7 Example Sample of 15 test-tubes tested for umber of times they ca be heated o Buse burer before they cracked gave x =1,230, s=270. Costruct 99% cofidece iterval for μ. df=-1=15-1=14 for 99% cofidece t=2.977 s x ± ( t critical value) = 1,230 ± = (1,020,1,440)

8 8 LARGE SAMPLE CONFIDENCE INTERVAL FOR A POPULATION PROPORTION I may populatios each item belogs to oe of two categories (S & F). We are iterested i estimatig the proportio (or percetage) of the populatio who belog to each category. e.g. proportio of adults who smoke cigarettes, proportio who vote for FF, proportio who drik Budweiser etc.

9 9 Example Each year 1 st year sciece studets may (S) or may ot (F) choose to study Statistics. To estimate the fractio who do study Statistics a sample of 1000 studets was chose from the past 10 years ad 637 had chose Statistics as a 1 st year subject. If the fractio of studets who choose Statistics is p the radomly selectig somebody gives a probability p of S ad 1-p of F. The obvious solutio is to use the sample proportio ( $p ) to estimate the populatio proportio (p) where umber of S's i sample 637 p $ = = =637. Sample size 1000

10 10 The samplig distributio of $p is approx ormal with μ $p =p ad σ $p =p(1-p)/ 2 Cosequetly the cofidece iterval for p is p( 1 p) p$ ± ( zcritical value) Approximate p with $p to get p$( 1 p$) p$ ± ( zcritical value) for the approximate cofidece iterval. For our example the 95% cofidece iterval for p is p$( 1 p$) p$ ± ( zcritical value) = (. 607,. 667)

11 11 Example The Graduate Schools of Busiess ad Law ad Huma Scieces take a radom sample of recet PhD graduates (=135) ad fids that 12 of these are uemployed. Compute a 90% cofidece iterval for the proportio of all PhD graduates who fail to fid a job.

12 12 $p =12/135=.089 Check for large sample p$ ± 3σ p$ ± 3 p$ p$( 1 p$) = (. 015,. 163) 90% cofidece iterval for p p( 1 p) p$( 1 p$) p$ ± z p$ ± z = (. 049,. 129 ) Suppose we kow that the uemploymet rate i the coutry is 3%. Ca we coclude that PhD Graduates are more likely to be uemployed tha the populatio i geeral? Sice.03 lies outside the cofidece iterval it is ot cosistet with our data ad we would coclude that the uemploymet rate amog PhD graduates is above that of the populatio as a whole.

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