Kig Saud Uiversity College of Sciece Statistics ad Operatios Research Departmet Stat 39 Theory of Statistics () Exercises Refereces:. Itroductio to Mathematical Statistics, Sixth Editio, by R. Hogg, J. McKea, ad A. Craig, Pretice Hall.. Itroductio to Theory of Statistics, A. Mood, F. Graybill ad B. Boes, McGrow-Hill. 3. Itroductio to Mathematical Statistics, Fourth Editio, by Hogg ad Craig, Macmilla Publishig Co., Ic. 4. Statistical Iferece, secod Editio, by G. Casella ad R. Berger, Duxbury. By Dr. Samah Alghamdi
Stat 39 Exercises: Theory of Statistics Dr. Samah Alghamdi Cofidece Itervals. Let X be the mea of a radom sample from the expoetial distributio, Exp(θ). (a) Show that X is a ubiased poit estimator of θ. (b) Usig the MGF techique determie the distributio of X. (c) Use (b) to show that Y = X θ has a χ distributio with degrees of freedom. (d) Based o Part (c), fid a 95% cofidece iterval for θ if = 0. Hit: Fid c ad d such that P (c < X θ < d) = 0.95 ad solve the iequalities for θ.. Let X,, X be a radom sample from the Г(, θ) distributio, where θ is ukow. Let Y = i= X i. (a) Fid the distributio of Y ad determie c so that cy is a ubiased estimator of θ. (b) If = 5, show that P (9.59 < Y < 34.) = 0.95. θ (c) Usig Part (b), show that if y is the value of Y oce the sample is draw, the the iterval ( y, y ) is a 95% cofidece iterval for θ. 34. 9.59 (d) Suppose the sample results i the values, 44.8079.55.99.5734 43.305 Based o these data, obtai the poit estimate of θ as determied i Part (a) ad the computed 95% cofidece iterval i Part (c). What does the cofidece iterval mea? 3. Let the observed value of the mea X of a radom sample of size 0 from a distributio that N(μ, 80) be 8.. Fid a 95% cofidece iterval for μ. 4. Let X be the mea of a radom sample of size from a distributio that is N(μ, 9). Fid such that P(X < μ < X + ) = 0.90, approximately. 5. Let a radom sample of size 7 from the ormal distributio N(μ, σ ) yield x = 4.7 ad s = 5.76. Determie a 90% cofidece iterval for μ. 6. Let X deote the mea of a radom sample of size from a distributio that has mea μ ad variace σ = 0. Fid so that the probability is approximately 0.954 that the radom iterval (X < μ < X + ) icludes μ. 7. Let Y be Bi(300, p). If the observed value of Y is y = 75, fid a approximate 90% cofidece iterval for p. 8. Let x be the observed mea of a radom sample of size from a distributio havig mea μ ad kow variace σ. Fid so that x σ to x + σ is a approximate 95% cofidece 4 4 iterval for μ. 9. Let X,, X be a radom sample from N(μ, σ ), where both parameters μ ad σ are ukow. A cofidece iterval for σ ca be foud as follows: We kow that ( )S σ is a radom variable with a χ ( ) distributio. Thus, we ca fid costats a ad b so that P(( )S σ < b) = 0.975 ad P(a < ( )S σ < b) = 0.95. (a) Show that this secod probability statemet ca be writte as P(( )S b < σ < ( )S a) = 0.95.
Stat 39 Exercises: Theory of Statistics Dr. Samah Alghamdi (b) If = 9 ad s = 7.93, fid a 95% cofidece iterval for σ. 0. Let X,, X be a radom sample from gamma distributios with kow parameter α = 3 ad ukow β > 0. Discuss the costructio of a cofidece iterval for β. X i β Hit: What is the distributio of i=? Follow the procedure outlied i Exercise 9.. Whe 00 tacks were throw o a table, 60 of them laded poit up. Obtai a 95% cofidece iterval for the probability that a tack of this type will lad poit up. Assume idepedece.. Let two idepedet radom samples, each of size 0, from two ormal distributios N(μ, σ ) ad N(μ, σ ) yield x = 4.8, s = 8.64, y = 5.6, s = 7.88. Fid a 95% cofidece iterval for μ μ. 3. Let two idepedet radom variables, Y ad Y, with biomial distributio that have parameters = = 00, p ad p, respectively, be observed to be equal to y = 50 ad y = 40. Determie a approximate 90% cofidece iterval for p p. 4. Let X ad Y be the meas of two idepedet radom samples, each of size, from the respective distributios N(μ, σ ) ad N(μ, σ ), where the commo variace is kow. Fid such that P (X Y σ 5 < μ μ < X Y + σ 5 ) = 0.90. 5. If 8.6, 7.9, 8.3, 6.4, 8.4, 9.8, 7., 7.8, 7.5 are the observed values of a radom sample of size 9 from a distributio that is N(8, σ ), costruct a 90% cofidece iterval for σ. 6. Let X,, X ad Y,, Y m be two idepedet radom samples from the respective ormal distributios N(μ, σ ) ad N(μ, σ ), where the four parameters are ukow. To costruct a cofidece iterval for the ratio, σ σ, of the variaces, from the quotiet of the two idepedet chi-square variables, each divided by its degrees of freedom, amely F = S σ S σ, where S ad S are the respective sample variaces. (a) What kid of distributio does F have? (b) From the appropriate table, a ad b ca be foud so that P(F < b) = 0.975 ad P(a < F < b) = 0.95. (c) Rewrite the secod probability statemet as P (a S S < σ σ < b S S ) = 0.95. The observed values, s ad s, ca be iserted i these iequalities to provide a 95% cofidece iterval for σ σ. 7. Let two idepedet radom samples of sizes = 6, m = 0, take from two idepedet ormal distributios N(μ, σ ) ad N(μ, σ ), respectively, yield x = 3.6, s = 4.4, y = 3.6, s = 7.6. Fi a 90% cofidece iterval for σ σ whe μ ad μ are ukow. 8. Fid a 90 percet cofidece iterval for the mea of a ormal distributio with σ = 3 give the sample (3.3, -0.3, -0.6, -0.9). What would be the cofidece iterval if σ were ukow? 9. The breakig stregths i pouds of five specimes of maila rope of diameter 3 6 ich were foud to be 660, 460, 540, 580 ad 550. 3
Stat 39 Exercises: Theory of Statistics Dr. Samah Alghamdi (a) Estimate the mea breakig stregth by a 95% cofidece iterval assumig ormality. (b) Estimate σ by a 90% cofidece iterval; also σ. 0. A sample was draw from each of five populatios assumed to be ormal with the same variace. The values of ( )S = (X i X ) ad, the sample size, were S : 40 30 0 4 50 : 6 4 3 7 8 Fid 98% cofidece iterval for the commo variace.. To test two promisig ew lies of hybrid cor uder ormal farmig coditios, a seed compay selected eight farms at radom i Iowa ad plated both lies i experimetal plots o each farm. The yields (coverted to bushels per acre) for the eight locatios were Lie A: 86 87 56 93 84 93 75 79 Lie B: 80 79 58 9 77 8 74 66 Assumig that the two yields are joitly ormally distributed, estimate the differece betwee the mea yields by a 95% cofidece iterval.. Let X deote the mea of a radom sample of size 5 from a gamma distributio with α = 4 ad β > 0. Use the Cetral Limit Theorem to fid a approximate 0.954 cofidece iterval for μ, the mea of the gamma distributio. Hit: Use the radom variable (X 4β) 4β 5 = 5X β 0. 3. Let S ad S deote, respectively, the variaces of radom samples, of sizes ad m, from two idepedet distributios that are N(μ, σ ) ad N(μ, σ ). (a) Derive ( α)00% cofidece iterval for the commo ukow variace σ. (b) Fid 99% cofidece iterval for σ if =, s = 4.74, m = 5, s = 5.66. 4
Stat 39 Exercises: Theory of Statistics Dr. Samah Alghamdi Type I Error, Type II Error ad Power Fuctio. Let X have a biomial distributio with the umber of trials = 0 ad with p either /4 or /. The simple hypothesis H 0 : p = is rejected, ad the alterative simple hypothesis H : p = is accepted, if the observed value of X 4, a radom sample of size, is less tha or equal to 3. Fid the sigificace level ad the power of the test.. Let X, X., X 5 be a radom sample from N(μ, 6). If the test: Reject H 0 : μ = ad accept H : μ = 3 whe x. Fid the Type I error ad the power fuctio of the test. 3. Let X, X be a radom sample of size = from the distributio havig pdf formf(x; θ) = e x θ, x > 0, θ > 0. We reject H θ 0 : θ = vs H : θ = if the observed values of X, X, say x, x are such that f(x, )f(x, ) f(x, )f(x, ) Here Ω = {θ: θ =,}. Fid the sigificace level of the test. 4. Let X have a Poisso distributio with mea θ. Cosider the simple hypothesis H 0 : θ = ad the alterative composite hypothesis H : θ <. Thus Ω = {θ: 0 < θ }. Let X,., X deote a radom sample of size from this distributio. We reject H 0 if ad oly if observed value of Y = X +. +X. If γ(θ) is the power fuctio of the test, fid the power γ ( ), γ (), γ 3 (), γ 4 () ad γ ( ). What is the sigificace level of the 6 test? 5. Let X, X., X 8 be a radom sample of size = 8 from a Poisso distributio with mea μ. Reject the simple ull hypothesis H 0 : μ = 0.5 ad accept H : μ > 0.5 if the observed 8 sum i= x i 8. (a) Compute the sigificace level α of the test. (b) Fid the power fuctio γ(μ) of the test as a sum of Poisso probabilities. (c) Usig the Poisso Table, determie γ(0.75), γ() ad γ(.5). 6. Let X be a Beroulli radom variable with probability of success p. suppose we wat to 0 test at size α, H 0 : p = 0.7 vs H : p = 0.5. Reject H 0 if i= x i c. Fid c if the size of test is equal to 0.048. The, fid the power fuctio of the test. 7. Let X,, X is ormally distributed with mea μ ad variace 00. Reject H 0 : μ = 75 vs H : μ = 77 if i= x i > c. Determie ad c so that the power fuctio γ(μ) of the test has the values γ(75) = 0.59 ad γ(77) = 0.84. 8. Let X have a pdf of the form f(x, θ) = θx θ, 0 < x <, zero elsewhere, where θ {θ: θ =, }. To test the simple hypothesis H 0 : θ = agaist the alterative simple hypothesis H : θ =, use a radom sample X, X of size = ad defie the critical regio to be C = {(x, x ): 3 4 x x }. Fid the power fuctio of the test. 5
Stat 39 Exercises: Theory of Statistics Dr. Samah Alghamdi 9. Let Y < Y < Y 3 < Y 4 be the order statistics of a radom sample of size = 4 from a distributio with pdf f(x; θ) = θ, 0 < x < θ, zero elsewhere, where 0 < θ. The hypothesis H 0 : θ = is rejected ad H : θ > is accepted if the observed Y 4 c. (a) Fid the costat c so that the sigificace level is α = 0.05. (b) Determie the power fuctio of the test. 0. Let X be a sigle observatio from the desity f(x, θ) = (θx + θ), 0 < x <, where θ. To test H 0 : θ = 0 vs H : θ > 0, the followig procedure was used: Reject H 0 if X exceeds. Fid the power ad the size of the test. Let X,., X deote a radom sample from f(x; θ) = ( θ ), 0 < x < θ, ad let Y,., Y be the correspodig ordered sample. To test H 0 : θ = θ 0 versus H : θ θ 0, the followig test was used: Accept H 0 if θ 0 ( α) Y θ 0 ; otherwise reject. Fid the power fuctio of this test.. Let X,., X be a radom sample of size from f(x; θ) = θ xe θx x > 0. I testig H 0 : θ = versus H : θ > for = (a sample of size ) the followig test was used: Reject H 0 if ad oly if X. Fid the power fuctio ad size of the test. 6
Stat 39 Exercises: Theory of Statistics Dr. Samah Alghamdi Testig Hypotheses Case : Simple Hypotheses (Neyma-Pearso Lemma). Let the radom variable X have the pdf f(x; θ) = ( θ )e x θ, 0 < x <, zero elsewhere. Cosider the simple hypothesis H 0 : θ = ad the alterative hypothesis H : θ = 4. Let X, X deote a radom sample of size from this distributio. Show that the best of H 0 agaist H may be carried out by use of the statistic X + X.. Let X, X,.. X 0 be a radom sample of size 0 from a ormal distributio N(0, σ ). Fid a best critical regio of size α = 0.05 for testig H 0 : σ = agaist H : σ =. 3. If X, X,.. X is a radom sample from a distributio havig pdf of the formf(x; θ) = θx θ, 0 < x <, zero elsewhere, show that a best critical regio for testig H 0 : θ = agaist H : θ = is C = {(x, x,.. x ): c i= x i }. 4. Let X, X,.. X 0 be a radom sample from a distributio that is N(θ, θ ). Fid a best test of the simple hypothesis H 0 : θ = 0, θ = agaist the alterative simple hypothesis H : θ =, θ = 4. 5. Let X, X,.. X deote a radom sample from a ormal distributio N(θ, 00). Show that C = {(x, x,.. x ): c x } is a best critical regio for testig H 0 : θ = 75 agaist H : θ = 78. Fid ad c so that P[(X, X,., X ) C; H 0 ] = P(X c; H 0 ) = 0.05 ad P[(X, X,., X ) C; H ] = P(X c; H ) = 0.90, approximately. 6. If X, X,.. X is a radom sample from a beta distributio with parameters α = β = θ > 0, fid a best critical regio for testig H 0 : θ = agaist H : θ =. 7. Let X, X,.. X be iid with pmf f(x; p) = p x ( p) x, x = 0,, zero elsewhere. Show that C = {(x, x,.. x ): x i c} is a best critical regio for testig H 0 : p = agaist H : p =. Use the Cetral Limit Theorem to fid ad c so that approximately 3 P( X i c; H 0 ) = 0.0 ad P( X i c; H ) = 0.80. 8. Let X, X,.. X 0 deote a radom sample of size 0 from a Poisso distributio with mea θ. Show that the critical regio C defied by 0 x i c is a best critical regio for testig H 0 : θ = 0. agaist H : θ = 0.5. Determie, for this test, c ad the power at θ = 0.5 whe the sigificace level α = 0.08. 9. Let X be a radom variable has the pdf f(x) = β, x, β > 0 xβ+ Fid the best critical regio of size α for testig H 0 : β = β 0 versus H : β = β, β > β 0. 7
Stat 39 Exercises: Theory of Statistics Dr. Samah Alghamdi 0. Let X, X,.. X be idepedet radom variables have N(μ, σ ). Fid the best critical regio of size α for testig: (i) H 0 : μ = μ 0 agaist H : μ = μ whe μ 0 > μ ad σ is kow. (ii) H 0 : σ = σ 0 agaist H : σ = σ whe σ 0 < σ ad μ is kow. 8
Stat 39 Exercises: Theory of Statistics Dr. Samah Alghamdi Case : Simple Hypothesis agaist Oe-Sided Hypothesis. Cosider a ormal distributio of the form N(θ, 4). Test the simple hypothesis H 0 : θ = 0 agaist the alterative composite hypothesis H : θ > 0 ad fid the best rejectio regio.. Let X, X,., X be a radom sample from a distributio with pdf f(x; θ) = θx θ, 0 < x <, zero elsewhere, where θ > 0. Calculate the best critical regio to reject H 0 : θ = 6 versus H : θ < 6. 3. Let X have the pdf f(x; θ) = θ x ( θ) x, x = 0,, zero elsewhere. We test H 0 : θ = agaist H : θ < by takig a radom sample X, X,., X 5 of size = 5. Fid the best critical regio of size α. 4. Suppose that X, X,., X is a radom sample of size from expoetial populatio with parameter θ. Determie the best rejectio regio of size α for H 0: θ = θ 0 agaist H : θ > θ 0. 5. If X~Gamma (, θ ). Fid the best critical regio for H 0: θ = agaist H : θ <. 6. Let X be a radom sample whose probability mass fuctio is biomial distributio with parameters = 0 ad p. Test the hypotheses H 0 : p = 0.5 agaist H : p > 0.5 ad fid the best critical regio of size α of this test. 9