Summary of the lecture Bostatstcs Probablty Desty Fucto For a cotuos radom varable, a probablty desty fucto s a fucto such that: 0 dx a b) b a dx A probablty desty fucto provdes a smple descrpto of the probabltes assocated wth a radom varable. Example The tme utl a chemcal reacto s complete s defed by ts F(x) as the followg. 0 x < 0 F( x) 0. 0x e x 0 a) Fd ts f(x). b) What proporto of reacto s complete wth 00 ms? a) df( x) 0 x < 0 0. 0x dx 0.0e x 0 b) p ( x < 00) F(00) e 0. 8647 Hstogram A hstogram s a approxmato to a probablty desty fucto. For each terval of the hstogram, the area of the bar equals the relatve frequecy (proporto) of the measuremets the terval. The relatve frequecy s a estmate of the probablty that a measuremet falls the terval. Cumulatve Dstrbuto Fucto The cumulatve dstrbuto fucto of a cotuous radom varable s defed as x F ( x) x) f ( u) du for ay x. Mea or Expected Value If s a cotuous radom varable wth probablty desty fucto f(x), the mea or expected value of s defed as µ E ( ) x dx
However, f we have just some observatos of, deoted as x, x,, x, the sample mea s: x µ. x. If we have the whole populato, the the populato mea s Varace & Stadard Devato If s a cotuous radom varable wth probablty desty fucto f(x), the varace of s defed as V ( ) ( x µ ) dx x dx µ, where s the stadard devato of. ( x x) I case of havg observatos of, the sample varace s defed as s, where s s the sample stadard devato. Smlarly the populato varace s defed as ( x µ) s. (Do you remember the reasog behd havg the - dvsor sample varace?) ormal Dstrbuto A radom varable wth probablty desty fucto ( x ( x) e π µ ) f for has a ormal dstrbuto wth parameters µ, where µ, ad 0. Some useful results cocerg a ormal dstrbuto for ay radom varable are: µ < < µ + 68% µ < < µ + ) 95% µ 3 < < µ + 3 ) 99% ) x, µ The radom varable defed as s a ormal radom varable wth E()0 ad V(). That s, s a stadard ormal radom varable. Ceteral Lmt Theorem (oe of the most useful theorems Statstcs) If s a radom varable wth sample sze, take from a populato wth mea µ ad fte varace, ad f s the sample mea, the the lmtg form of the dstrbuto of µ as, s the stadard ormal dstrbuto.
Correlato Coeffcet The correlato betwee radom varables ad, deoted as ρ ( ( ) )( ) ( ) cov(, ) ρ s defed as E [( µ )( µ )] E[ ( ] Covarace (ad hece, correlato) s a measure of lear assocato betwee two radom varables. If the relatoshp s o-lear, covarace s ot sestve to the relatoshp. If two radom varables are depedet, the ther covarace (ad therefore ther correlato) would be zero, whle the opposte s ot ecessarly true. Wde Sese Statoary Sgals Whe both the mea ad varace of a radom process are tme varat ad also ts autocorrelato fucto s ot a fucto of tme but oly a fucto of the tme dfferece betwee the two pots, the the process s called statoary the wde sese. Ergodcty Whe the statstcal averagg over the esamble equals the tme averagg over the tme axs of ay sample fucto, the process s called ergodc. It s obvous that a ergodc process has to be a statoary process as well. Hypothess, p-value, Type I & II Errors, Power A statstcal hypothess (H 0 ) s a statemet about the parameters of oe or more populato. The p-value s the smallest level of sgfcace that would lead to rejecto of the ull hypothess wth the gve data. Rejectg the ull hypothess whe t s true, s defed as a type I error ad alpha s the probablty of makg such a error. That s, α type I error) reject H 0 whe H 0 s true). Alpha s fact same as p-value. Falg to reject the ull hypothess, H 0, whe t s false s defed as a type II error ad beta s the probablty of makg such a error. That s, β type II error) fal to reject H 0 whe H 0 s false). The power of a statstcal test s the probablty of rejectg the ull hypothess whe the alteratve hypothess s true. The power s computed as -β. Power s a very descrptve ad cocse measure of the sestvty of a statstcal test, where by sestvty we mea the ablty of the test to detect dffereces. Example To llustrate the above deftos by a example, cosder the ull hypothess that states the heght of a average Caada fat s 50 cm wth a stadard devato of.5 cm. Let's assume that we would lke to test ths hypothess by a sample of 0 fats. Therefore our H 0 ad H are: H 0 : µ 50cm H : µ 50cm µ µ 3
The sample mea s a estmate of the true populato mea µ. A value of the sample mea x that falls close to the hypotheszed value of µ 50 cm, s evdece that the true mea µ s really 50 cm; that s, such evdece supports the ull hypothess H 0. O the other had, a sample mea that s cosderably dfferet from 50 cm s evdece support of the alterate hypothess H. Thus, the sample mea s the test statstc ths case. ow, let's assume that f the sample mea falls betwee 48.5 to 5.5, we wll ot reject the ull hypothess. Therefore, these values costtute the crtcal rego for the test ad essece determes the p-value or the cofdece of ot makg type I error. Let's see how t defes the p-value. α x < 48.5 whe µ 50) + x >5.5 whe µ 50), ow to fd ths probablty we have to use Cetral Lmt Theorem ( µ ) to fd the z-values so that the we ca use the ormal dstrbuto table for gettg the probabltes. The z-values that correspods to the crtcal values of 48.5 ad 5.5 are 48.5 50.90 ad.5 0 5.5 50.90. Therefore,.5 0 α z <.9) + z >.9) 0.0877 +.0877 0.057434 5% Ths mples that about 5% of all radom samples would lead to rejecto of the ull hypothess (µ 50 cm) whe the true mea s really 50 cm. We ca decrease ths rate (or o the other word crease the cofdece terval) by wdeg the acceptace rego. For example, f we make the crtcal values 48 ad 5 stead of 48.5 ad 5.5, the 48 50 5 50 α < ) + > ) 0.04 %..5 0.5 0 We could also reduce α by creasg the sample sze wthout wdeg the acceptace rego. For example f we choose 6 stead of a sample sze of 0, the we have 48.5 50 5.5 50 α < ) + > ) 0.064.6%..5 6.5 6 ow, let's see what s the probablty of type II error. To calculate β we must have a specfc alterate hypothess; that s, we must have a partcular value of µ. For example, suppose that t s mportat to reject the ull hypothess wheever the mea heght s greater tha 5 cm or less tha 48 cm. Because of symmetry of the ormal dstrbuto, t s eough to calculate oly oe of the cases - say fd the probablty of acceptg the ull hypothess (µ 50 cm) whe the true mea s 5 cm. Therefore, β 48.5 x 5.5 whe µ 5). The the correspodg z - values are : 48.5 5 4.43 ad.5 0 5.5 5 0.63. Therefore,.5 0 β 4.43 z 0.63) 0.63) 4.43) 0.643 + 0.0000 0.643 6.4%. Thus, f we are testg H : µ 50cm agast H : µ 50cm wth 0, ad the true 0 4
value of the mea s µ 5 cm, the probablty that we wll fal to reject the false ull hypothess s about 6.4%. By symmetry, f the true value of the mea s 48, the value of β wll also be 6.4%. The probablty of makg type II error creases rapdly as the true value of µ approaches the hypotheszed value. The type II error probablty also depeds o the sample sze ; creasg the sample sze results a decrease the probablty of type II error. The above results ca be summarzed as the followg.. The sze of the crtcal rego, ad cosequetly the probablty of type I error, α, ca always be reduced by approprate selecto of the crtcal values.. Type I ad II errors are related. A decrease the probablty of oe type of error always results a crease the probablty of the other, provded that the sample sze does't chage. 3. A crease sample sze wll geerally reduce the both α ad β, provded that the crtcal values are held costat. 4. Whe the ull hypothess s false, β creases as the true value of the parameter approaches the value hypotheszed the ull hypothess. The value of β decreases as the dfferece betwee the true mea ad the hypotheszed value creases. Geerally, the aalyst cotrols the type I error probablty α whe s/he selects the crtcal values. Thus, t s usually easy for the aalyst to set the type I error probablty at (or ear) ay desred value. Sce the aalyst ca drectly cotrol the probablty of wrogly rejectg H 0 as a strog cocluso. Aother mportat cocept s the power of the statstcal test, whch s fact -β. The power ca be terpreted as the probablty of correctly rejectg a false ull hypothess. Statstcal tests are ofte beg compared wth ther power propertes. For example, cosder the case that true mea heght s 5 cm ad that case, we foud β 0.643. So, the power of ths test s -β 0.7357. Power s a very descrptve ad cocse measure of the sestvty of a statstcal test, where by sestvty we mea the ablty of the test to detect dffereces. For example, the latest case dscussed above, a power of 0.7357 whle the true mea s 5 cm, meas that the test wll correctly reject the ull hypothess H 0 50 cm, 73.57% of the tme. If ths value of power s judged to be too low, the aalyst ca crease ether α or the sample sze. 5