Math 3 Probablty ad Statstcal Methods Chapter 8 Fudametal Samplg Dstrbutos Samplg Dstrbutos I the process of makg a ferece from a sample to a populato we usually calculate oe or more statstcs, such as the Mea or Varace Sce, samples are radomly selected, the values that such statstcs assume may chage from sample to sample Thus sample statstcs are, themselves radom varables ad ther behavor ca be modeled by probablty dstrbutos The probablty dstrbuto of a sample statstc s called Samplg Dstrbuto Here, we wll study the ature ad propertes of samplg dstrbutos The sample copy of µ, amely, has a dstrbuto repeated samplg that ceters about µ We mght say, the that s a good estmator of µ The sample varace S ( ) s used to estmate the populato varace σ th Let deote the radom outcome of the sample observato to be selected (the sampled observatos are selected depedetly of oe aother ), the ad where ( ) E µ µ µ ( ) E ( ) E µ,,,, µ µ σ Var Var Var ( ) ( ) σ σ Samplg Dstrbutos of Meas Samplg Dstrbuto of the mea (σ kow) Cetral Lmt Theorem If s the mea of a radom sample of sze take from a populato wth mea µ ad fte varace σ, the the lmtg form of the dstrbuto of Z µ as, s the stadard ormal dstrbuto ( ) good f 30 σ / z ;0, The approxmato wll geerally be Souc Zorlu
Math 3 Probablty ad Statstcal Methods Example A electrcal frm maufactures lght bulbs that have a legth of lfe that s approxmately ormally dstrbuted wth mea equal to 800 hours ad a stadard devato of 40 hours Fd the probablty that a radom sample of 6 bulbs wll have a average lfe less tha 775 hours? Soluto Gve µ µ 800 hours, σ 40 hours, 6, σ σ / 40 / 6 0, µ 775 800 P ( < 775) P < P( Z < 5) P( Z < 5) 09938 0006 σ 0 Example If a certa mache makes electrcal resstors havg a mea resstace of 40 ohms ad a stadard devato of ohms, what s the probablty that a radom sample of 36 of these resstors wll have a combed resstace of more tha 458 ohms? Soluto Gve µ µ 40 ohms, σ ohms, 36, σ σ / / 36 / 3, 458 µ 405 40 P P P P < 458 > ( > 405) > 36 σ / 3 P Z > 5 < 5 0933 00068 ( ) P ( Z ) Samplg Dstrbuto of the mea (σ ukow) We are attemptg to estmate the mea of a populato whe the varace σ s ukow If we have a radom sample chose from a ormal populato, the the radom varable T µ S / has a studet s T-dstrbuto wth degrees of freedom υ Souc Zorlu
Math 3 Probablty ad Statstcal Methods Example Fd P ( t005 < T < t005 ) Soluto P ( t T t ) 005 < < 005 005 005 095 Example Fd k such that P ( k T ) µ ormal populato ad T S / < < 76 0045, for a sample of sze 5 selected from a Soluto From table A4 we ote that t 005 76 whe υ 4 Therefore t 005 76 ad let k t α The, we have 0045 005 α or α 0005 Hece from table A4 wth υ 4, k t0005 977 Example 3 (a) Fd P ( 356 < T < 79) (b) Fd P( T > 38) whe υ 4 (c) Fd P ( t T t ) < < wheυ 0 0005 00 Example 4 Gve a radom sample of sze 4 from a ormal dstrbuto, fd k such that P 069 < T < k 0965 (a) ( ) (b) P( k T ) < < 807 0095 (c) P ( k < T < k ) 090 Example 5 A radom sample of sze from a ormal populato has the mea x 78 ad s 34 Ca we say that the gve formato supports the clam that the mea of the populato f µ 85? Example 6 A maker of certa brad of low fat cereal bars clams that ther average saturated fat cotet s 05 gr I a radom sample of 8 cereal bars of ths brad the saturated fat cotet was 06,07,03,04,05,04 ad 0 Would you agree wth the clam? (If the computed t -value falls betwee t005 ad t005, teh maker s satsfed wth hs clam) Samplg Dstrbuto of S The beauty of the cetral lmt theorem les the fact that wll have approxmately a ormal samplg dstrbuto o matter what the shape of the probablstc model for the populato, so log as s large ad σ s fte For may other statstcs addtoal assumptos are eeded before useful samplg dstrbutos ca be derved Uder the ormalty assumpto for the ( ) S populato, a samplg dstrbuto ca be derved for S It turs out that has a σ samplg dstrbuto that s a specal case of Gamma desty fucto Souc Zorlu 3
Math 3 Probablty ad Statstcal Methods Note Exactly 95% of a Ch-squared dstrbuto les betwee 0975 ad fallg to the rght of Smlarly a 005 005 s ot lkely to occur uless our assumed value of -value fallg to the left of σ s too large I other words, t s possble to have a left of 005 whe σ s correct Example For a ch-squared dstrbuto fd α such that (a) P ( x α ) 099 Soluto P ( x α ) > whe υ 4 (b) P( x α ) 005 Soluto P ( x α ) > 099 097, υ 4 099 > whe υ 9 (c) P ( x α ) 005 A - value σ s too small 0975 s ulkely to occur uless our assumed value of > 005 3857, υ 9 3765 < < 0045 whe υ 5 Soluto 005 3765, υ 5 ; -value to the rght of 0975 or to the υ 005 0045 0005 4698, υ 5 Example Fd the probablty that a radom sample of 5 observatos, from a ormal populato wth varace σ 6, wll have a varace S of (a) greater tha 9 (b) betwee 346 ad 0745 Soluto (a) ( ) Souc Zorlu ( ) S 4(9) P S > 9 P > P ( > 364) 005, wth υ 4 σ 6 (b) 4(346) ( ) S 4(0745) P ( 346 < S < 0745) P < < 6 σ 6 ( ) P 3848 < < 498 table(3848, 4) table(498) 095 00 094 4
Math 3 Probablty ad Statstcal Methods Exercses Exercse (a) If s a ormal radom varable wth µ 0 ad σ 4 Suppose a sample of 40 s chose, fd P( > 5) (b) Fd P( T 3 < 650) whe ν 3 (c) If a radom samplg problem the sample sze s ad 08, fd ( ( ) > 336) P σ Exercse Fd the probablty that the average umber of cherres 36 cherry puffs wll be less tha 55 whe µ 6 ad σ Soluto Gve 36, µ 6, σ, µ 55 6 P ( < 55) P < P ( Z < 5 ) table( 5) 0006 σ / / 36 Exercse 3 The scores o a placemet test gve to college freshme for the past fve years are approxmately ormally dstrbuted wth a mea of 74 ad a varace of 8 If a radom sample of 0 studets takg ths course s selected, what s the probablty that varace of the scores of these 0 studets s more tha 383? Exercse 4 For a Ch-Squared dstrbuto fd 00 whe 6 Exercse 5 The amout of tme that a drve-through bak teller speds o a customer s a radom varable wth a mea µ 3m ad a stadard devato σ 6m, f a radom sample of 64 customers s observed fd the probablty that ther mea tme at the teller s couter s (a) at most 7 mutes (b) more tha 35 mutes Souc Zorlu 5