Interval Estimation. Consider a random variable X with a mean of X. Let X be distributed as X X

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1 ECON 37: Ecoomercs Hypohess Tesg Iervl Esmo Wh we hve doe so fr s o udersd how we c ob esmors of ecoomcs reloshp we wsh o sudy. The queso s how comforble re we wh our esmors? We frs exme how o produce cofdece ervl, d he udersd how o use for ferece usg some of he sscl dsrbuos d coceps we hve lered before. Cosder rdom vrble wh me of. Le be dsrbued s ~ N µ, The we kow by sdrdzg, we ge sdrd orml vrble wh me of d vrce of, h s µ Z ~ N(,) Le Φ be he dsrbuo of he sdrd orml vrble Z. Suppose we w o mmze he probbly our esmor flls o he rgh sde of he dsrbuo by, h s, Z z ( ) ( Z z ) µ z µ + z d we would kow our esmor s o ccure whe s greer h µ + z. We fc wre hs oe sded ervl s, µ + z. We c smlrly do wo sded boud, or cofdece ervl so h does o fll off of boh sdes (we c do hs sce he sdrd orml dsrbuo s symmerc, d s esselly jus µ z, µ + z where he probbly of he esmor fllg ousde he bouds s wh probbly, d he probbly hs o fllg beyod eher of he bouds s jus hlf of h, /.

2 ECON 37: Ecoomercs Hypohess Tesg Hypohess Tesg The bove provdes h for esg dffere hypohess for he esmors we re cocered wh. Before we c do so, we hve o defe he hypohess. Whe esg hypoheses, we hve o defe frs wh we re esg, d wh he lerves would be. Tkg he exmple we hd before, suppose we hve md h µ should ke o he vlue of. Ths would sd s our ull hypohess, d we wre s, H : µ Depedg o he vlues we suspec he lerve could ke, we c srucure he lerve hypohess s eher oe sded or wo sded hypohess. If we oly wsh o provde for he lerve h µ c oly be lrger, h he lerve hypohess s; H : µ > wh he src equly swchg whe we wsh o llow for µ o ke vlues smller h. However f we w o llow for boh sgs, we h wre he lerve hypohess s H : µ We ypclly wre he hypohess s H : µ H : µ Type I d Type II Errors As reserchers, we re lble o mke wo ypes of error;. Type I error refers o he mske of rejecg he ull hypohess whe he ull s rue.. Type II error refers o he mske of o rejecg he ull whe s cully flse. As you should see, here s rde off o boh, s you reduce he lkelhood of oe, you crese he lkelhood of he oher. Wh s ypclly doe s h we focus o ype I error d decde how much ype I error we re wllg o ccep, d we selec es ssc h mmzes he lkelhood of ype II error gve h gve o choce o ype II error.

3 ECON 37: Ecoomercs Hypohess Tesg Rewrg he hypohess; Sscl Decso Rule (Two Sded Alerve) H : µ H : µ Suppose we w rejeco re of % of he me, he we buld he cofdece ervl s follows; z + z, or equvlely ± z. Ths he mes h f our smple me flls ousde hs bouds, we would rejec he ull hypohess. We c lso lervely express our decso rule s z P-Vlues The de wh he pproch o esg s h we w o rejec he ull hypohess f he smple evdece s very ulkely were he ull hypohess rue. Typclly, we se 5%, hough we would lso lke o exme somemes % or % eve. Aoher more drec pproch, we could clcule he probbly h gve he ull hypohess, wh s he probbly we cully ge he esme we go! Ths s kow s he p-vlue, d you could wre s such z 3

4 ECON 37: Ecoomercs Hypohess Tesg Hypohess esg of Ordry Les Squres Esmors We hve exmed he smples esmo echque for smples of seups, h of hvg oe depede, d oe depede vrble. Gve he esmors, d her esmed vrces, wh he? We sll do o kow wheher our esmors re good, rher how cofde should we be? We wll dscuss hs here ow. Cofdece Iervls Gve h we kow he dsrbuo of our prmeer esmes, we c cree cofdece ervls o ell us how sgfc our esmors re, or how cofde we should be off our esmors. To do so we eed o sdrdze our esmors (do you remember hs from our ssc revso). Frs oe h, we c defe boud ~ N (,) z o be he vlue of Z, he sdrdzed vlue of our esmor so h Φ z, where Φ s he dsrbuo fuco for he sdrd orml dsrbuo. Wh hs sys s h he probbly of Z beg greer h z s. I follows he h, z z z Z z z + z Whch s he ( )% cofdece ervl for. However, s oed before, we ypclly do o kow he populo prmeer ε, whch mes we hve o replce wh s. However, hs would me h he sdrdzo we perform before does o pply excly sce he ssc s o loger dsrbued s sdrd orml dsrbuo. e Frs oe h e s. Furher, recll we ssume he dosycrc error erms re ormlly dsrbued. Nex exme he followg, 4

5 ECON 37: Ecoomercs Hypohess Tesg 5 ( ) ( ) ( ) ( ) ~ s x x e However, hs does o chge he mer we do hypohess esg, d he cofdece ervl s jus, +,, s s