Bài tập xác suất thống kê bản đọc (sách thầy tô văn ban)

Nhu vfy, qua kh6u lya chgn biiSn ta dgOc (*) Y = -0,304+0,083x1 -0,031x3 +0,004x] +e Chuong 2: G BA lraP -1 ' -2 $6.1.8Ar TAP CHTI0NG r Hinh 5.9 PhAn du chuin ho6 theo quan sat c0a s6 tieu dQ tan Ki6m tra phan du cta m6 hinh ndy: Ching h4n, theo chi s5 i ta th6y c6 gi|triphAn du chu6n ho6 (tmg vdi quan s6t thri vi thrl l0) vuqt qu6 2; vi ph4m thir hai ld d; khA nhd t4i c6c quan s6t ll *24 Dir hai vi ph4m niy ! (*) ldm m6 hinh cu5i cirng c6 vi ph4m tl6ng kC Ta lga chqn cirng kh6ng a6n ndi nio Pfr6n du chuAn ho6 x6p theo x1, x2 hay ddu kh6ng # l.l The lifetime of a pC is defined the tirne it as takes for the pC to break down a) Define a random experiment that involves the lifetime ofthe pC b) What is the sample space? c) Define two events that are disjoint d) Define two events that have a nonempty intersection 1.1 Tu6i thg cria mQt chi6c pC tfnh tir hic n6 bit tlu-o.c ttAu ho4t riQng ttrin h6ng a) Xdc dinh ttri nghigm ngiu ntridn gin v6i tu6ithg cria pC b) Kh6ng gian miu d6y td gi? c) X6c ttlnh bi6n c6 xung khfc d) X6c tlinh birin c6 c6 giao khric trting D& b)S = IR* = (0; -); c) (0;1000) vi (>2000); d) (0;1000) and (900;2000) 1.2* Customers access an Automated 1.2* Circ khrich hing lui t6i mqt chiiic Teller Machine (ATM) They want m6y rrit tidn tu rlQng Hq muiin rrit to withdraw random amounts of mQt lugng ti6n ngSu nhi6n 50 ngdn money in multiples of 50 thousand tl6ng mQt Hiiy chi 16 kh6ng gian VND Specify the sample space Is m6u Edy phrii chnng ld kh6ng it the discrete one? Specift three gian mdu roi r4c? Chi bi6n ci5 events of interest quan tim D,S S = {50,1 00, ,1 04} ; Dfrng; (< t03), (t03; 5.103), 15.103+t04) 232 233 L3** xdt thi nghiQm ngiu nhi6n tung xirc xic don I litr vi di5m s6 diu ch6m hi€n tr6n mdt Gie sfr courrting the number of dots facing P({6})=0,3 vd t6t ci c6c m[t tup Assume that P({6})=0.3 and 1.3** Consider the random experiment of tossing a single die once and all other faces are equiprobable kh6c ,- ,.A A suat cua cac Dlen co A={2,4,6}, B={1,5}, = {1 3, 4} ancl D= Au(BnC) A= ?) 4,6\., a = {1, s} , g = {1, 2,3, 4}, D = A Let P(A; = 0.9; P(B) = 0.8 Show that P(A n e) > o.z 1.5* Given that P(A) = 0.9: a) b) 0,58; 0,28; 0,56vd0,72 P(A):0,9; P(B):0,8 Chitng t6r[ng e(anB)>0,7 1.5* Gia st n(nn P(A) = 0,9; P(B):0,8; B) = 0,75, tim DS; a) 0,95; b) 0,15; c) 0,05: 1.6 Prove the Boole's inequality \ i=t 1.7** Consider tlre switching network It is equally likely that a switch will or will not work Find the all alternatives are possible; ndng d6u c6 thii h) B-E (Bose-Einstein) the b) B-E (Bose-Einstein) particles carrnot be distinguished, all alternatives are possible; thr3 phAn biQt c) F-D (Ferrni-Dirac) - the particles cannot be distinguished, at most c) F-D (Fermi-Dirac) one particle is allowed in a box chrla nhi6u c6c prob that a closed path will exist between terminals A and B - kh6ng phdn biQt - kh6ng th6 c c6c hgt, mQt hQp ntr6t t tr4t duo n! n!(m-l)! a) "' : b) ,n,-, (m+n_l)1, n!(m n)! - 1.9* MOt thf nghiQm ngiu nhi€n c6 space S={a,b,c} Suppose that kh6ng gian miu S={a,b,c} Gi6 P{a,c} = 0.75 and P{b,c}=0.6 srlr P{a,c} =0,'15, P{b,c}=0,6 1.9* A random experiment has sample Find the probabilities of the elementary events 1.7** Xdt mOt mach tliQn nhu hinh v€ Cdc c6ng tic it6ng hoic md v6i Tim xic suSt cria c6c bitin cti so cap oS e(a) = O,+; P(b) =0,25; P(c) = 9,35' l.l0** lf m students born on independent days in adterrding a lecture D& 0,688 ' lg93 are Find the probabilitiy that at least two of them slrare a birthday arid show that p>l ,2 *n"nm=23 ft0** Gi6 sri c6 m sinh vi€n sinh nrm t993 ttang tham dU gio giAng Tim x6c su6t itnhit2 sinh vi6n cirng c6 ngiy sinh vd chftng t6 ring p >1 m=23 ESi 234 duo.'c; ttucr c cric het, t6t cd ndng ddu c6 thiS dugc; ]-l'm! '[!^,)=r(o,) nlng nhu Tim x6c su6t el6 c6 it mQt dudng d5n gita diu n5i A vi B tru6c (m5i hpt chi d c6c trudng hgp sau: a) M-B (Maxwell-Boltzmann) c6c hat coi ld kh6c nhau, t6t ci cdc (Maxwell-Boltzmann) g) 1.6 Chring minh b6t tling thric Boole i h4t a) M-B DS n Ue, l=I(n,) \i=r ) hQp chqn cl e(nne) c) P(AnB) ta d4t ng6u nhi6n n (ph6n t&) viro m > n hQp Tirn x6c su6t p cl€ c6c h4t du-o c tim th6y d n I h$p) X6t r(nuB); b) r(e - n); P(A-B); Pl each al P(AuB): (n in 1.8 Chtng box) Consider the following cases: the particles are distinct, u(Bn.C) 1.4 Cho 0.8: P(A n B) = 0.75, find P(B) = rS; preselected boxes (one ddng khn ndng Tim x6c Find the probability of the events 1.4 li 1.8 We place at random n particles in m > n boxes Find the probability p that the particles will be found in n l- 36s! (365 - m-)!365* 235 l.l1** A train and a bus arrive at the station at random between A.M and l0 A.M The train stops for l0 rninutes and the bus for a minutes ,1 _ 1.ll** Tiu ho6 vd xe bus tdi ga t4i m6t thoi di,3m ngSu nhi6n tir tl6n l0 gid Tdu dirng l0 phrit cdn xe bus dirng a phrit Tim a tl6 xric su6t xe kh6ch vi tdu ho6 g{p bing 0,5 l.ind a so that the probability that the bus and the train will meet \ 'r 1.17** Consider the experiment of 1.17** Xet thi nghiQm tung xirc throwing the two fair dice You are xic c6n aOi Si6t ring tting c6c n5t then informed that the sum is l6n hon greaterthan a) Tim x6c su6t biiin c6 mdt gi6ng kh6ng biiit th6ng a) Find the probability of the event equals 0.5 D.s 1.12 A fair die is rolled two tirnes Firrd oS: We have two coins; the first l is 1.13* C6 tt6ng tiBn, mQt cdn d6i, mQt c6 mf;t s6p Rtit ngiu nhi6n I d6ng tidn, tung n6 lin vd d6u hign m4t s6p Tim xdc su6t ddng it nvice and heads shows both times Find the probability vv€ toSS that the coin picked is fair tirSn rtt PS: l b) Find the probability of the same v6i th6ng tin dl cho duo manufacturing plants produce sirnilar parts Plant I produces 1000 parts, 100 of which are defective Plant produces 2000 parts, 150 of which are defective A part is selected at random and found to be defective c ld ddng ti6n cdn d6i plant that n(nln) defined by 1.I4 Chring t6 ring e(ele) theo Eq (1.2.1) satisfies the three (1.2.1) thod mdn ti€n d6 cta xdc lwo b) Tim x6c su6t cfra bii5n c6 tr€n ps, 1,1 6'33 \ xu6t I? D& 0,4 1.14 Show axioms of a probability, that is: dE n6u 1.18** Hai nhi m6y s6n xudt nhirng linh kiQn giting Nhd m6y I sin xu6t 1000 linh kiQn, d6 c6 100 li h6ng Nhd m6y san xu6t 2ooo linh ki€n, d6 c6 150 la h6ng Chgn ng5u nhi6n I ' linh kiQn vi th6y ring n6 bi h6ng Tim x6c suSt n6 nhi m6y I sin What is the prob that it came from 1.18*" two-headed We pick one of the coins at random, tin event with tlre information given xic cdn A5i Z tin Tim x6c sudt aii t6ng s5 n6t bing is fair and the second phirt 1.12 Tung xric the probability that the sum of dots l.l3* 60-J100 that two faces are the same without the information given .V 1.19** su6t, d6 ld: A lot of 100 semiconductor chips contains 20 that are defective Two 1.19** L6 hdng 100 chip bdn din c6 chira 20 clrip b! h6ng Chgn ngiu ay n(nle)>o; ay e(nln)>o; clrip.s are selected at random, uy e(sla)= uy n(sle)= r ; replacement, frorn the lot a) X5c su6t chitic thri nhSt b! h6ng a) phat is the probability that the li first one,selected is defective? b) X6c su5t chii5c thti b! h6ng li bao nhi6u, bii5t ring chii5c thfr nh6t bi h6ng? r ; c) n(n,t.lA2lB)= c; P(A,r-,rerln)= r(e, ls)+ r(e,In) if A'1n Az=A 1.15* Show that if P(AlB)>P(A) then P(BlA)>P(B) if P(A), P(B) rhen e(ele)> r(sle) 1.16 Show that e(n,la)+ n(nrle) ;r, niSu A1 b) What is the probability that the r\A2=A 1.15* Chirng minh ring second one selected niSu e(ele)>n(e) thi P(B|A)>P(B) ring n6u P(A) > P(B) 1.16 Chung minh thi P(AIB)> e(nla) nhi6n chi6c kh6ng without ' is defective bao nhi6u? giverr that the lth on" was defective? c) X6c suSt d6 c) What is the probability that both bao nhi6u? are defective? l{p lai ci chiiSc bi h6ng ld D,S: a) 0,2; b) 0,192; c) 0,0384 v I contains 1000 bulbs of 1.20** HQp I g6m 1000 b6ng ddn, d6 lO% bi hdng H$p g6m which l0 percent are defective 2000 b6ng, d6 5% bi h6ng Box contains 2000 bulbs of 1.20*,' Box which percent are defective Two bulbs are picked from a randomly ' du-o duo c rirt t& mQt h6p c chgn ng6u nhi6n selected box a) Tim x6c su6t a) Find the probability both bulbs h6ng are defective b) Gie st} ring cd b6ng ddu bi h6ng, tim *6c su6t d,6 chirng ttu-oc b) Assuming that both are defective, find the probability that they came fi'om box l: 1.21** SLrppose that laboratory test to detect a certain disease has the : , n(nla) B* B = o,ee; n(n[) {krit qui ki6m tra duorg tinh} %o ddn si5 bi bQnh niy bi6t rang kiSt is the probability that a bQnh, qui ki6m tra li duong 7, respectively Catch one mouse frorn the first cage and one mouqe from the second cage then 6rass them to the third cage Finally catch one mouse from the third cage Find the probability that this mouse is male 238 binary communication channel The channel input symbol X may :rssume the state or the state l Because of the channel noise, an 1.23* Xet k€nh th6ng tin nh! ph6n DAu viro X cfia k6nh trang th6i du-o c xem nhu ho4c l Do c6 nhi6u diu c6 thii ring k6nh truydn, and vice versa v6i dAu viro I vi ngucr.c l4i K€nh dugc tl{c truorg bdi x6c su6t truydn k€nh po,Qo,pr, Ql, x6c The channel is characterized by the tlinh theo channel transition probability Po,Qo,Pl, and ql, definedbY o, = R(vrl*o),p' =n(volxr), irrput may convert to an output I oo = oo e(lr l*o), p, = R(volxr), =e(volxs), andql =R(vrlxr), here xs and x1 denote the events =0) and (X = l), respectively, and y6 and y1 denote the events qo = P(vo lxo),q, = P(y, l*, ) d6 xs vi x1 kf hiQu biiSn ci5 (X = 0) vi (X = l), tuong ring; yo vi y1 kf hi€u bii5n c5 1v=0) vi (Y = l) tuong ring Chir f ring, p0+q0=l:pr+ql.DAt vi pt = 0,2 (Y = 0) and (Y = l), respectively P(xo) = 0,5, po = 0,1 Note that po + qo = I = Pl * q1 Let P(xo) 0.5, po 0.l, and pt = 0.2 a) a) Find P(ye) and P(y1) (de) la tr4ng thrii cfia dAu vdo? : : b) If a was observed at the Tim P(ys) P01) b) Ni5u thdy d diu ra, xac suit tI6 c) NiSu th5y d diu ra, x6c su6t d6 tinh output, what is the probability that I (de) Ia tr4ng th6i cta tliu D,S 0,165 a was the input state? d) Tfnh x6c su6t sai c) lf a I DS a) 0,55;0,45; b) 0,818; ry 1,22** We have two cages of 1.22t't' C6 l6ng chuQt thi nghiQm, experimental mice In the first I6ng thrl nh6t c6 l0 chuQt dgc there are l0 male and 15 female ones; in the second there are and 1.23* Consider the (X = o,oo5 Tinh x6c su6t m6t ngudi bi and 0.1 percent the population person has the disease given that the test restrlt is positive? : vit 0,1 = o.oo5 actually has the disease What I vdi A = {ngudi kiiim tra c6 bQnh}, {event that the test result is positive) It is known that r(ale) a) 0,0061 9; b) 0,80; c) 0,08 thu dugc kiSt qui sau ddy: iras the disease) = o.ee; D$ 1.21** Gid st ring, bing x€t nghiQm tt6 phet hiQn mQt lo4i bQnh nguoi ta leveni that the tested person e(ela) hai b6ng ttdu bi b6ng h6ng fol lowin g statisiicq Let ci rittirh$p 1; c) GiA sri ring ci b6ng ddu bi h6ng, tim x6c suSt d6 chiiSc b6ng tiiSp theo rfit ttr hQp tE chgn li c) Than find the probability that the next bulb picked from the selected box will be defective ' Hai b6ng vi 15 chuQt c6i; l6ng thti II c6 was observed at the output what is the probability that a I d) c) 0,889; d) lim vdo? P 0,1 was the input state? Calculate error P the probability of chuQt tllrc vd chuQt cdi Bit I tu l6ng I, mQt tir II rdi dua sang ldng III; sau d6 bit I tir l6ng III Tinh x6c l6ng su6t d.l niy ld chugt 0, I r- I ElXl =; vi V[X tL L,' Find Med(X) Which is equation DS c) n = I,2, ; 1, 2.16* Gqi X le BNN ph6n b6 mfr tham Verifr that, r-_l E[X]=: and V[X]= - V[X] b)xe [n;n + l) : F(x) = I -(1 -p)n, I DS:3-5: "12 = (t value E[X] the have to be replaced sai cria X X c) Tim k! vgng E[X] vd phuong sai V[X] DS 0,004 2.15 Let a RV X denote the outcome of 2.15 Gqi X le BNN chi kiSt qui rrlt mQt xirc xic cdn 3) DS 3; 0,5767 A RV X is the Pareto random variable with parameter a, b (a, b > 0) if its pdf is given by 2,22 fx(x) = (a/b) (b/x)a*l, x € [b; *) X ld BNN Pareto vdi c6c s6 a, b (a,b>O) n6u him 2.22 BNN tham mAt dg cria n6 cho bdi fx(x) =(a / b) (b/x)a+l, x e[b; o) 247 ' a) Show that E[Xn] exists if and ' onlyifn 0), k) vgng kh6ng tdn t4i x (x_a), N(pr, o2 ) , evatuate E[X3 ] 2.24 Giasti X - N(p,o2), tinh DS: 3o2p[l + p3] 2.25 Suppose.that Z - N(0,1) 2.25.Gie sir ring Z a) Evaluate ElZl a) - Ax r0 xelR +b J xfxlxlep;dx, ,-,, Ax P({xr I and then 1, < l 2,.26** O mQt vtng trdng cam, ngudita ln the 600 counted be parameter tr" Find ring 2.26)'*.ln the orange-region the number distribution X Ax ^i]o Chi ring l1 Etxl= k = 1.2, ,n ,r"]e ao fy(x I B) = Show that N(0,1) @ k = 1,2, ,n *n"r" ,11* I ,y = P({x -t6.65(8- l) ; thing D d t?-( fo=ffil8 =-0,6982 > -to.os (8 - l) ; tiling / \) 4.35** Let X denote the number of 4.35** Gqi X lA s6 v6t nut quan s6t flaws observed on a large coil of duqc t.en mQt cuQn l6n th6p ma galvanized steel Seventy-fi ve coils , -p2 product for one month results in an resulting weight loss data are reported below Use hypothesistesting procedures to answer the following questions -El =5,362>2,58=zop C6; * (0,19t 0,09) pound T{m ttdi tugng sri dpng s6n claim vd th6y c6 196 tt4t y6u cAu LiQu c6 DS: pounds Eight subjects use the product for one month, and the producer is considering changing its claim from "at least pounds" to "at least pounds" Test the new th6 kh6c dirng ch6t b6i tron thri hai confidence 4.34** A liquid dietary product implies its advertising that use of the 276 d6 vdi hdng thri hai, mdu ngiu nhi6n n2 =lf 16 cho r are inspected and the following data were observed for the values of X: Values (gia tri) I 11 I 13 11 12 10 I Obs Frequency (tAn s6) 75 cuQn tld dugc kh6o s6t vi thu ttugc sti liQu sau v6 c6c gi6 tri cria X: 277 a) Does the assumption Poisson of a) Ph6i chlng gii thuy6t ring the distribution ph6n br5 Poisson ld m6 hinh x6c appropriate as a probability model su6t cho s5 ligu for this data? Use a c6 : 0.05 b) Calculate the P-value for this test niy b) Tinh P - gi6 tri cho ki6m dlnh HD: a) 7'= 4,907,9h8p lai; (n,-nP,)' =f i]i nPi A B c D 41 20 11 16 31 12 I 15 15 11 15 11 xem nhu ld lj? Dirng cr:0,05 b)xz Machines (m6y) Shift (ca) Test the hypothesis (using o = 0.05) that breakdowns are independent of the shift Find the P - value for this =6,63 : * breakdowns are collected: 278 cHuoNG v IIdi quy tuy6n tfnh tlon 5.1* Roadway surface temperature (.r) 5.1* Nhiet tIQ mat tlulng (x) tlugc coi ln li6n quan d6n d0 bitin dang b6 is thought to be related to mat O).sii tlCu t6m tit Ii pavernent deflection (y) Summary Simple Linear Regression quantities were Iy, = tz,ts,lv? =8,86, Iy, = tz,ls,\v? =8,86, I*,= A78, lx! =143215.8 I*,=1a78, f xf =143215,8 I*,Y,=1083'67 'n:20' I*,Y, 'n=20' a) Calculate the least squares ' estimates of the slope and a) Tinh u6c lugng binh phuong cgc ti6u cria hQ siS g6c vi hQ sti intercept Craph the regression b) Dnng phuong trinh tlucrng h6i =1083'67 company operates four 4.37** MQt hdng sri dpng :nlrhy ca m5i ngiy Tir nh{t kf sin xu5t, dir From production records, the lipu sau vd s5 su c6 du-o c thu following data on the number of thfp: machines three shifts each day r4r line b) Use the equation of the fitted line to predict what pavement chfln Lflp ad th! Auone h6i quy , quy udc lugng a6 au uao d0 bi6n d4ng mflt tluong s6 quan s6t dugc n6u nhiQt tl$ b6 mat le 85oF 279 deflection would be observed when the surface temperature is 85oF c) What is the mean pavement deflection when the surface temperature is 90oF? b) Use the equation of the fiUed line to predict what percent yield would be observed wlren the c) DQ bii5n @ng b6 m{t trung binh nhiet dq bA m6t te 90oF? d) D0 bitin dang bd m{t trung binh biiin dOi cO bao nhi€u nhiQt d0 bC mat UitSn a6i tof'z temperature is 1600 c) What is the mean percent yield d) What change in mean pavement deflection would be expected for a loF change in surface temperature? Find c) HiQu su6t trung binh bing bao nhi6u nhi€t d0 ld l70o? d) Find the estimate of o2 d) Tim u6c lugng cria s2 D& a) Y =-4,473+0,496x+e; linear regression 5.2 Tim MHHQ tuytin tinh don cho s6 model to the oxygen purity data in Iieu d0 s4ch oxy d B6ng 5.1; ki6m Table 5.1; test for significance of tllrrh 1i nghla cfra vi_6c dirng b)74,95%; c)79,9\Yo; d)62=11,98 5.4* The MHHQ ,,S; Y = 4,284 + 4,947 x + e; Consider the following pairs (x;,y1 ), i = l0 data 1, , 10, relating y, the percent yield of a laboratory experiment, to x, the temperature at which the experirnent was run fiber is stored in a location without (xi,yi), i = I, , content (y) of a sample of the raw l0 li6n h€ bi6n y, hiQu su6t (%) cta mQt thi nghiQm v6i x, nhiQt tlQ t4i material were taken over 15 days with the following data (in tl6 thi nghi€m thgc hiQn percentages) resulting 5.3 Xet I0 c{p st5 tieu X1 Yi 100 45 110 52 160 120 54 I 170 76 130 63 o 180 92 a) ua Calculate 62 the least squares estimates of the slope and intercept Graph the regression line 10 X1 Vi 150 68 190 a) Tim udc lugng binh phuong cuc vi hC s6 nhqn Lap d6 thi cfra tlud'ng hOi quy tii3u cria h€ sti g6c kho l5 ngdy th6 hipn d bdng sau X 46 53 29 61 36 39 47 49 v 12 15 17 10 11 11 12 x 52 38 55 32 57 54 44 v 14 o 16 I 18 14 12 75 88 noi kh6ng c6 ki6m so6t d9 Am D0 6m tuong a5l 1xy d ncri luu kho vi d0 6m (y) cia m6u v{t tiQu th6 luu a humidity control Measurements of the relative humidity (x) in the storage location and the noisture (l 9); c6 i 280 16,625 raw material used in the 5.4* Vat liQu th6 dtrng tlti sin xuAt m6t of a certain synthetic lo4i sgi hfru co duo.rc c5t vio kho production ltrl=ffi=n,78e > 2,093 = sdt dugc sE bing bao nhi6u nhi6t ttQ la 1600 when the ternperature is l70o? a simple regression using the model b) Dirng phuong trinh h6i quy thuc nghi€m a6 ag Uao hiQu sudt quan Find the mean square estimate I I I rl ti, u6c luo ng binh phuong cgc tieu ,Sry =134,517 +147,600x+e; rrlt.- ,l:::1 ," 5.5** An article in the rournar sound and vibration described a I I rroise exposure and I vi tr6n rap chi Am Dao dQng m6 ta mQt study investigating the relationship nghi€n criu vA m6i quan between luc ti6ns 6n vi hQ gifra 6p su gia eia tlng tlns huy6t huv6t 281 hypertension The following data I are representative ofthose reported I in the article, where y means blood I pressure rise in rnillimeters of I mercury and x mealls sound pressure level in decibels 'ro so lieu sau d6y r6y tir bii bao, d6 y la d0 gia ting 6p su6t m6u theo mitirndt thu! rrgan, x tlo mirc dn theo decibel Obser Strength y Number (tcrc y) (TD r (TT) Age x Obser (psi) (weeks) y y) (psi) Strength (tuc Number (tu6i) Age x (tu6i) (weeks) I 2158,7 15,5 11 2165,2 13 1678,15 23,75 12 2399,55 3,75 3 2316 I 13 1779,8 25 2061,3 17 14 2336,75 9,75 I I v 1 x 60 63 65 70 70 70 80 90 80 80 v I 2207,5 15 1765,3 22 x 85 89 90 g4 100 100 100 1709,3 19 16 2053,5 18 1784,7 24 17 2414,4 2575 2,s 18 2200,5 12,5 I 2357,9 7,5 19 2654,2 10 2277,7 11 20 1753,7 21,5 90 90 90 a) Draw a scatter diagram Does a sirnple linear regression model a) Vc tld thi rdi tti6m MQt rludng hdi quy tuy6n tinh don dudng nlrtr in this situation? Find the least squares estimates of the slope and intercept in the c6 lli cho tinh hu6ng ndy? Tim u6c lugng binh phuong cyc ti6u hQ si5 simple linear regression model tuy6n tinh tlon b) Find an estimate of o2 b) Tim u6c lugng cho o2 seem reasonable c) Find the predicted mean rise in blood pressrrre level associated vrith a sound pressure level of 85 decibels g6c vi h$ s6 ch{n cria MHHQ 5.6** A rocket nrotor is manufactured by bonding together two types of propellants, an igniter and a sustainer The shear strength ofthe bond y is thought to be a linear function of the age of the propellant.r when the motor is cast 'l wenty observations are shown in the folowing table a) Draw a scatter diagram of data Does the c) Tim dg b6o cho mtic gia tdng 6p a) Ve d6 thi rni di6m cria s6 ti6u Phii chlng md hinh hdi quy dirng the straight-line regression model seem to tluong thing Id c6 lj? plausible? 6n 85 decibel b) Lqc MHHQ b) Fit the simple linear regression tuyiSn D.S model usilrg least dirng phuong ph6p binh phuong b) Calculate R2 forthe model sudt meu trung binh ring r5.6** bo =-9,3131,61 o.2 = 1,981 ; 282 vdi mtic =-0,17148 c) 4,76301 c sin su6t bing c6ch ngp tt6ng thoi logi ch6t n6: kich n6 vi tri Lgc ldm phd hu! li6n kiit cia ch6t nO (y) xem nhtr ld him tuyiin tfnh cfra DQng co t€n lfta tuOi 1x) cia duo chAt nr5 cho d6n"khi tlQng co tlugc ph6ng 20 quan s6t chi d bing du6i tl6y be squares cgc tidu Tinh hQ s6 tinh don x6c clinh R2 c) Estimate the mean shear strength of a motor made from propellant c) U'6c lugng tpc pna nr5r li6n ktit trung binh cria tlgng co ch6t n6 c6 20 tuAn tur5i; phin du bing bao that is 20 weeks old What is the nhi€u? residual? d) Tim gi6 tr! dg b6o ring v6i m5i giri tri quan s6t y; Lgp d6 thi yi d) Obtain the fitred vatues that correspond to each observed value y; Plot f, verus yi , and comment on what this plot would look like tlre linear theo y; vi xdt xem tt6 thi 6y tr6ng giting nhu thii nAo n6u quan hQ giira lsc ph6 hu! Ii€n k6t vi tu6i duo c x6c ttinh tiit (kh6ng c6 sai sO; efrai chlng tt6 th! ndy chi rdng tu6i li mgt lga chgn kha dT if relatiorrship between shear strength and age were perfectly deterministic (no error) Does this plot indicate that age is a , cria biiin h6i quy m6 hinh? I 283 reasonable clroice of D,S a)Y = 2625,38 -36,96x regressor variable in this model? data Does a straight-line relationship seem plausible? b)R2 = 0,890; c)1830,7; d) 5.7 Suppose the appropriate model is -77;dtng 5.7 Ci6 srt m6 hinh y=bx+e b) Fit a sirnple linear plir hqP le the point (0, 0)) Assirme that we have n pairs of data (x1,y,), ,(x,.,,yn) Find the c) Test for least squares estimate of b cria b estimate on the slope +e (the true regression line passes tlrrough to 5.8** The time of failure an electronic cOmponent is related to the temperature in the application environment in which A component was used sarnple of n:25 x, as: d) Find a u=(Ixivi)/(I*?) e) a) Cho r = 0,85, ki'3m tra gii p=0,vdi cr=0,05 Tinh - b) Tim khoing tin b) Find a 95% confidence interval c) Ki6m dinh gie thuYtit Ho:P=0,8/ Hl:P+0,8 v6i p 0.05) H6 :P = Q.3 I cr cfy drain current y (in rnilliarnperes) as ground-to-source function 95Yo cia Ta = 1,544 / 0,003 = 502,26 voltage x (in volts) The data are as follows: lidu nhu sau: }/ 284 0,734 0,886 1,1 1,2 1,04 1,3 1,19 1,4 1,35 '1,5 1,50 1,6 1,66 1,7 1,81 1,8 1,97 1,9 2,12 2,0 1,50 1,1 2) ; (1,537; 1,55 l) trdm cria chSt xo nguydn liQu bQt gi6y (x) Trong diAu kiQn chuin, nhd mriy sin xu6t tht 16 miu tir nhirng md vdi nhtng bQt gi6y kh6c St5 ligu chi d a pilot plant manufactures l6 samples, each from a different batch ofpulp, and measures the tensile strength The data are shown in the table that follows: bing sau v 101,4 117,4 117,1 106,2 131,9 146,9 146,8 133,9 x 1,0 1,5 1,5 1,5 2,0 2,0 2,2 2,4 v 111,0 123,0 125,1 145,2 134,3 144,5 143,7 146,9 x 2,5 2,5 2,8 2,8 3,0 3,0 3,2 3,3 a) Fit a simple linear x - of paper used in the 5.10 D9 dai cria gi6ry tt6 sdn xu6t hQp carton (y) li6n quan d6n t! l€ phAn conditions, = 0,05 Tim P gi6 tri cho ki6rn rrit k6t lufn gi? Y = -0,967 +1,544x +e; hardwood concentration in the origirral pulp (x) Under controlled P bi6u di6n ddng tliQn dd (Y) (theo milliampe) li him c0a dien thli ngu6n to,o5(l tr! cfra ki6m tlinh niY * 0.8 P - d) Tim khoing tin c4y 95% cho h0 random hypothesis a = 0.05 What is the P-value for H, :p nhi6n gdrn 25 quan s6t the r = 0.85, test the that P = 0, using (cr = c) Ki6m dinh j nghTa cfra m6 hinh dirng o = 0,05 Tinh P - gi6 tr! What is tlre P-value for this test? 5.8** Thdi gian sdng cria mgt linh ki€n i i T6ng Einh Qu!, Gido trinh Xdc suiit thiingfA, NXB Ci6o duc, 1999 Nguy6n Duy Tiiin Vfi Vigt Li thuyet xdcsarit, NXB Gi6o dpc, 2001 t8j ISt D G Childer, Probability.and random processes,IRwlN, 1997 D.C Montgomery engineers,3th - G.C Runger, Applied statistics and probability John Wiley - tl0] A Papoulis, Probability, " , YQn, Hwei P Hsu, Schaunt's outline of theory and problem of probabitity' random variables, and random processes, Mc Graw - Hill, 1993 Mc Graw Illl - r c rcmdom variables: and stochastic processes,3th - Hill, 1991 ien t is ts, 3th - for engineers and Elsevier, 2004 ll2l A.N Shiryaev, Probability,2th - Springer,1996 tl3] Y Viniotis, Probability and random processits for - lor & Sons,2003 Ross S.M., Introduction to probability and statistics Mc Graw -Il - t71 a)2 12 1lL l - Kh6ng tdn tai x2 e z l(22f(nl2);x 12) t5l t6l l.e-tr; l.,x > fi{r.*)'-'"-*; T6 Vdn Ban, Md hinh h6i Quy, HSc viQn K! thuft'Qu6n sg, 2006 K!'vqng (ilJzno21exp{-(r - tll t3l t4l p2 n.-xn Khi binh 342 E(r) c(a,0) Rayleigh : i b-a rHAM KHAo NXB Dai hgc Qudc gia HdNQi, 2003 1-p 1lp Mat d0 hiQu Cauchy /t *P'*=0,1,,n Bing 82 Luf,t ph6n Kf -p) np(1 l" 0,1, p(1-p)k-l, k=1,2, H(N,n,p) T6n Phuong sai l$u Hill, electrical engineers, 1998 o1rl"_l"_axi ;o,i,",x > #"'"-'{-%#1 I 3"-tr-")2 6" l(2a2), x> a ,r- rr(t * tlr) "rpt* * $) a+JiiT o (,-;)o 303