Stat 319 Theory of Statistics (2) Exercises

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1 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

2 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.) = 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 (d) Suppose the sample results i the values, 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 = 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 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) = ad P(a < ( )S σ < b) = (a) Show that this secod probability statemet ca be writte as P(( )S b < σ < ( )S a) = 0.95.

3 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 = 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 ) = 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) = ad P(a < F < b) = (c) Rewrite the secod probability statemet as P (a S S < σ σ < b S S ) = 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

4 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 : : 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: Lie B: 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 cofidece iterval for μ, the mea of the gamma distributio. Hit: Use the radom variable (X 4β) 4β 5 = 5X β 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 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 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) = 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

6 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 = (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

7 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 = 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 ) = 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 = 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

8 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

9 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 < 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 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