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1 Samplg Dstrbutos Learg Outcomes fal 1. Deftos you should kow 1.1. Parameter A umber that descrbes the populato. (hs s the defto, what follows are other statemets that descrbe parameters.) Fxed (ot radom), but ukow. (hs s a assumpto uderlyg classcal statstcs, whch s what we study ths class. I Bayesa statstcs, the uderlyg assumpto s that all ukow quattes are radom varables.) 1.. Statstc A umber that descrbes the sample. (hs s the defto, what follows are other statemets that descrbe statstcs.) 1... A fucto of radom varables, so t s also a radom varable Not lmted to lear combatos of radom varables, though lear combatos are easy to work wth (or, more formal laguage, lear combatos are qute tractable) Some of the most mportat statstcs we ll deal wth ths class: he sample mea, whch we usually use to estmate the populato mea he sample total he Z statstc: he sample varace Z. s or stadard devato s he statstc (we dd t talk about ths oe yet, but we wll): s If we re terested a populato proporto, ad we have the bomal settg, the sample proporto, where deotes the umber of successes. We use the sample proporto to estmate the populato proporto, also kow as the probablty of success. 1

2 1.3. Radom sample he radom varables,, 1 are cosdered to be a radom sample from the populato of terest f hey are depedet hey are detcally dstrbuted Samplg dstrbuto he probablty dstrbuto of a statstc What creatures have samplg dstrbutos? 1.5. Lear combato of radom varables (are examples of?) a. Radom samples.1. Defe a radom sample d.. Recogze the otato 1, Dst( parm1, parm) as defg a radom sample..3. Recogze whe the problem stuato has a radom sample t Because you were explctly told t does..3.. Because, as the bomal stuato, you kow that t does or that t does approxmately based o characterstcs of the samplg gve the problem..4. Recogze whe you eed to assume that the problem stuato volves a radom sample, eve though you were ot gve ths formato..5. Wheever you assume the problem stuato volves a radom sample, provde justfcato for dog so You may have had to make ths assumpto to use the methods you leared ths course, but s t realstc to make that assumpto?.5.. If ot, ad f the problem s very mportat, there are methods that you could use that allow for volato of the detcally dstrbuted part, ad/or for the volato of the depedece part, ad you may eed to cosult a statstcal cosultat, read some books, or take aother stat class or two.

3 3. Samplg Dstrbutos 3.1. Defe samplg dstrbuto (of a statstc) 3.. Kow what creatures have samplg dstrbutos (see above) 3.3. Kow ad use the followg methods for determg the samplg dstrbuto of a statstc For a dscrete radom varable wth ot very may possble values, the samplg dstrbutos of some statstcs ca be derved by wrtg out every possblty possbly a clever way as the followg examples / problems we dd Example 5.0 from Samplg Dstrbuto Notes Part 1 (sample meas) Samplg Dstrbuto Assgmet (sample total ad sample maxmum) Work out the theory ad/or use theorems that I taught you (see below uder the headg heorems) Use a smulato study We mght do ths for the dstrbuto of the sample proporto for the bomal dstrbuto usg the bead example. 4. Use your calculator to do the followg: 4.1. Calculate ormal probabltes (less thas, greater thas, betwees) gve mea, varace or stadard devato, ad the value of a radom varable. (ormcdf problems) 4.. Gve a mea, varace or stadard devato, ad the ormal probablty that s bouded by oe or two values of the ormal radom varable, be able to fd the value(s) of the ormal radom varable. (vorm problems) 5. heorems about the samplg dstrbutos of statstcs based o certa populatos what you should be able to do wth the theorems Be able to state the theorems eve though I allow a formula sheet, t s your best terest to kow these theorems ad ot be totally relat o your formula sheet. 5.. Recogze whe oe of the theorems below s ad s ot applcable to a problem stuato If a theorem s ot applcable to a problem stuato, be able to tell what codto of the theorem s ot met Determe whch of the theorems below most completely matches the codtos gve the problem stuato, ad use that theorem to do the problem as quckly as possble, wthout havg to do further dervatos or proofs. 3

4 6. heorems about the samplg dstrbutos of statstcs based o certa populatos statemets of the theorems Mea ad varace of a lear combato of radom varables: Let radom varables,, 1,, have meas 1, ad varaces,, 1, respectvely. he Whether or ot the are depedet, E( a a ) a E( ) a E( ) a a If the are depedet, Var( a a ) a Var( ) a Var( ) a a For ay,, 1, Var( a a ) a a Cov(, ) 1 1 j j 1 j1 6.. If,, 1 are depedet, ormally dstrbuted radom varables ot ecessarly detcally dstrbuted the ay lear combato of the also has a ormal dstrbuto From the prevous two theorems, we have that f,, 1 are depedetly dstrbuted ormal radom varables wth meas,, 1 ad varaces,, 1, respectvely or dep, 1,,, wrtte aother way, f, for N the a Na, a Note that a s oe ew radom varable. 1 4

5 6.4. Let,, 1 be a radom sample from ay dstrbuto wth mea value ad varace. Wrtte aother way, for 0. he, 1 d 1,,?,,. Deote wth 0 the sample total E we actually oly eed that the RV s have the same mea for ths to be true. We do t eed them to have the same varace or be depedet. V depedet for ths to be true. We do t eed them to have the same mea E we eed that the RV s have the same varace ad are we actually oly eed that the RV s have the same mea for ths to be true. We do t eed them to have the same varace or be depedet. V we eed that the RV s have the same varace ad are depedet for ths to be true. We do t eed them to have the same mea Ea a we actually oly eed that the RV s have the same mea for 1 1 ths to be true. We do t eed them to have the same varace or be depedet V a a V 0 0 we eed that the RV s have the same 1 1 varace ad are depedet for ths to be true. We do t eed them to have the same mea Whe all of the codtos 5.4 are met, the all of the above statemets are true. Whe some of the codtos 5.4 met, the some of the statemets are true based o prevous theorems. 5

6 6.5. Let,, 1 be a radom sample from a ormal dstrbuto wth mea value ad d varace. Wrtte aother way, for 1,,, N,. Deote wth 0 the sample total 0. he , N, ad N, , a Na, a Cetral Lmt heorem (CL): Let,, 1 be a radom sample from ay dstrbuto wth mea value d ad varace. Wrtte aother way, for 1,,?,,. Deote wth 0 the sample total 0. he, f s suffcetly large, approx, N, approx a Na, a, provded that there are ot oe or two of the a that are very large compared to the others. Whe doubt, do a smulato study. 6