Department of Statistics and Operations Research College of Science, King Saud University. Summer Semester

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1 Departmet of Statistics ad Operatios Research College of Sciece, Kig Saud Uiversity Summer Semester STAT 34 Probability ad Statistics for Egieers EXERCISES A Collectio of Questios Selected from Midterm ad Fial Examiatios' Papers for the Years from 4 to 47 PREPARED BY: Dr. Abdullah Al-Shiha

2 . COMBINATIONS =The umber of combiatios of distict objects take r at a time (r objects i each combiatio) r = The umber of differet selectios of r objects from distict objects. = The umber of differet ways to select r objects from distict objects. = The umber of differet ways to divide a set of distict objects ito subsets; oe subset cotais r objects ad the other subset cotais the rest.! r = r! ( r)!! = ( ) ( ) L 0! = Q. Compute: 6 6 (a) (b). 4 Q. Show that =. x x Q3. Compute: (a), (b), (c) 0 Q4. A ma wats to pait his house i 3 colors. If he ca choose 3 colors out of 6 colors, how may differet color settigs ca he make? (A) 6 (B) 0 (C) 8 (D) 0 Q5. The umber of ways i which we ca select two studets amog a group of 5 studets is (A) 0 (B) 0 (C) 60 (D) 0 (E) 0 Q6. The umber of ways i which we ca select a presidet ad a secretary amog a group of 5 studets is (A) 0 (B) 0 (C) 60 (D) 0 - -

3 . PROBABILITY, CONDITIONAL PROBABILITY, AND INDEPENDENCE Q. Let A, B, ad C be three evets such that: P(A)=0.5, P(B)=0.4, P(C A c )=0.6, P(C A)=0., ad P(A B)=0.9. The (a) P(C) = (A) 0. (B) 0.6 (C) 0.8 (D) 0. (E) 0.5 (b) P(B A) = (A) 0.0 (B) 0.9 (C) 0. (D).0 (E) 0.3 (c) P (C A) = (A) 0.4 (B) 0.8 (C) 0. (D).0 (E) 0.7 (d) P (B c A c ) = (A) 0.3 (B) 0. (C) 0. (D). (E) 0.8 Q. Cosider the experimet of flippig a balaced coi three times idepedetly. (a) The umber of poits i the sample space is (A) (B) 6 (C) 8 (D) 3 (E) 9 (b) The probability of gettig exactly two heads is (A) 0.5 (B) (C) (D) (E) 0.45 (c) The evets exactly two heads ad exactly three heads are (A) Idepedet (B) disjoit (C) equally likely (D) idetical (E) Noe (d) The evets the first coi is head ad the secod ad the third cois are tails are (A) Idepedet (B) disjoit (C) equally likely (D) idetical (E) Noe Q3. Suppose that a fair die is throw twice idepedetly, the. the probability that the sum of umbers of the two dice is less tha or equal to ٤ is; (A) (B) (C) (D) the probability that at least oe of the die shows ٤ is; (A) (B) (C) (D) the probability that oe die shows oe ad the sum of the two dice is four is; (A) (B) (C) (D) the evet A={the sum of two dice is 4} ad the evet B={exactly oe die shows two} are, (A) Idepedet (B) Depedet (C) Joit (D) Noe of these. Q4. Assume that P( A) = 0.3, P( B) = 0. 4, P ( A B C) = 0. 03, ad P ( A B) = 0. 88, the. the evets A ad B are, (A) Idepedet (B) Depedet (C) Disjoit (D) Noe of these.. P( C A B) is equal to, (A) 0.65 (B) 0.5 (C) (D) 0.4 Q5. If the probability that it will rai tomorrow is 0.3, the the probability that it will ot rai tomorrow is: (A) 0.3 (B) 0.77 (C) 0.77 (D)

4 Q6. The probability that a factory will ope a brach i Riyadh is 0.7, the probability that it will ope a brach i Jeddah is 0.4, ad the probability that it will ope a brach i either Riyadh or Jeddah or both is 0.8. The, the probability that it will ope a brach: ) i both cities is: (A) 0. (B) 0.9 (C) 0.3 (D) 0.8 ) i either city is: (A) 0.4 (B) 0.7 (C) 0.3 (D) 0. Q7. The probability that a lab specime is cotamiated is 0.0. Three idepedet specime are checked. ) the probability that oe is cotamiated is: (A) (B) 0.00 (C) 0.79 (D) 0. 3 ) the probability that exactly oe sample is cotamiated is: (A) 0.٢٤٣ (B) 0.0٨١ (C) 0.7٥٧ (D) 0. 3 Q8. 00 adults are classified accordig to sex ad their level of educatio i the followig table: Sex Educatio Male (M) Female (F) Elemetary (E) 8 50 Secodary (S) College (C) 7 If a perso is selected at radom from this group, the: ) the probability that he is a male is: (A) 0.38 (B) 0.44 (C) 0.8 (D) 78 ) The probability that the perso is male give that the perso has a secodary educatio is: (A) (B) (C) 0.9 (D) ) The probability that the perso does ot have a college degree give that the perso is a female is: (A) (B) 0.58 (C) (D) ) Are the evets M ad E idepedet? Why? [P(M)=0.44 P(M E)=0.359 depedet] Q idividuals are classified below by sex ad smokig habit. SEX Male (M) Female (F) Daily (D) SMOKING Occasioally (O) HABIT Not at all (N) A perso is selected radomly from this group.. Fid the probability that the perso is female. [P(F)=0.4]. Fid the probability that the perso is female ad smokes daily. [P(F D)=0.05] 3. Fid the probability that the perso is female, give that the perso smokes daily. [P(F D)=0.49] 4. Are the evets F ad D idepedet? Why? [P(F)=0.4 P(F D)=0.49 depedet] Q0. Two egies operate idepedetly, if the probability that a egie will start is 0.4, ad the probability that the other egie will start is 0.6, the the probability that both will start is: (A) (B) 0.4 (C) 0. (D)

5 Q. If P( B) = 0.3 ad P ( A B) = 0. 4, the P( A B) equals to; (A) 0.67 (B) 0. (C) 0.75 (D) 0.3 Q. The probability that a computer system has a electrical failure is 0.5, ad the probability that it has a virus is 0.5, ad the probability that it has both problems is 0.0, the the probability that the computer system has the electrical failure or the virus is: (A).5 (B) 0. (C) 0.5 (D) 0.30 Q3. From a box cotaiig 4 black balls ad gree balls, 3 balls are draw idepedetly i successio, each ball beig replaced i the box before the ext draw is made. The probability of drawig gree balls ad black ball is: (A) 6/7 (B) /7 (C) /7 (D) 4/7 Q4. 80 studets are erolled i STAT-34 class. 60 studets are from egieerig college ad the rest are from computer sciece college. 0% of the egieerig college studets have take this course before, ad 5% of the computer sciece college studets have take this course before. If oe studet from this class is radomly selected, the: ) the probability that he has take this course before is: (A) 0.5 (B) (C) 0.80٢١ (D) ) If the selected studet has take this course before the the probability that he is from the computer sciece college is: (A) 0.4٢٩ (B) (C) 0.80 (D) 0. 5 Q5. Two machies A ad B make 80% ad 0%, respectively, of the products i a certai factory. It is kow that 5% ad 0% of the products made by each machie, respectively, are defective. A fiished product is radomly selected.. Fid the probability that the product is defective. [P(D)=0.06]. If the product were foud to be defective, what is the probability that it was made by machie B. [P(B D)=0.3333] Q6. If P(A )=0.4, P(A A )=0., ad P(A 3 A A )=0.75, the () P(A A ) equals to (A) 0.00 (B) 0.0 (C) 0.08 (D) 0.50 () P(A A A 3 ) equals to (A) 0.06 (B) 0.35 (C) 0.5 (D) 0.08 Q7. If P(A)=0.9, P(B)=0.6, ad P(A B)=0.5, the: () P(A B C ) equals to (A) 0.4 (B) 0. (C) 0.5 (D) 0.3 () P(A C B C ) equals to (A) 0. (B) 0.6 (C) 0.0 (D) 0.5 (3) P(B A) equals to (A) (B) (C) (D) 0.0 (4) The evets A ad B are (A) idepedet (B) disjoit (C) joit (D) oe (5) The evets A ad B are (A) disjoit (B) depedet (C) idepedet (D) oe - 4 -

6 Q8. Suppose that the experimet is to radomly select with replacemet childre ad register their geder (B=boy, G=girl) from a family havig boys ad 6 girls. () The umber of outcomes (elemets of the sample space) of this experimet equals to (A) 4 (B) 6 (C) 5 (D) 5 () The evet that represets registerig at most oe boy is C (A) {GG, GB, BG} (B) {GB, BG} (C) {GB} (D) {GB, BG, BB} (3) The probability of registerig o girls equals to (A) (B) (C) 0.49 (D) (4) The probability of registerig exactly oe boy equals to (A) (B) (C) 0.04 (D) (5) The probability of registerig at most oe boy equals to (A) (B) (C) 0.49 (D)

7 3. BAYES RULE: Q. 80 studets are erolled i STAT-34 class. 60 studets are from egieerig college ad the rest are from computer sciece college. 0% of the egieerig college studets have take this course before, ad 5% of the computer sciece college studets have take this course before. If oe studet from this class is radomly selected, the: ) the probability that he has take this course before is: (A) 0.5 (B) (C) 0.80٢١ (D) ) If the selected studet has take this course before the the probability that he is from the computer sciece college is: (A) 0.4٢٩ (B) (C) 0.80 (D) 0. 5 Q. Two machies A ad B make 80% ad 0%, respectively, of the products i a certai factory. It is kow that 5% ad 0% of the products made by each machie, respectively, are defective. A fiished product is radomly selected. (a) Fid the probability that the product is defective. [P(D)=0.06] (b) If the product were foud to be defective, what is the probability that it was made by machie B. [P(B D)=0.3333] Q3. Dates' factory has three assembly lies, A, B, ad C. Suppose that the assembly lies A, B, ad C accout for 50%, 30%, ad 0% of the total product of the factory. Quality cotrol records show that 4% of the dates packed by lie A, 6% of the dates packed by lie B, ad % of the dates packed by lie C are improperly sealed. If a pack is radomly selected, the: (a) the probability that the pack is from lie B ad it is improperly sealed is (A) 0.08 (B) 0.30 (C) 0.06 (D) 0.36 (E) 0.53 (b) the probability that the pack is improperly sealed is (A) 0.6 (B) 0.0 (C) 0.06 (D) 0. (E) 0.5 (c) if it is foud that the pack is improperly sealed, what is the probability that it is from lie B? (A) (B) 0.03 (C) (D) 0.03 (E) Q4. Two brothers, Mohammad ad Ahmad ow ad operate a small restaurat. Mohammad washes 50% of the dishes ad Ahmad washes 50% of the dishes. Whe Mohammad washes a dish, he might break it with probability O the other had, whe Ahmad washes a dish, he might break it with probability 0.0. The, (a) the probability that a dish will be broke durig washig is: (A) (B) 0.5 (C) 0.8 (D) 0.5 (b) If a broke dish was foud i the washig machie, the probability that it was washed by Mohammad is: (A) (B) 0.5 (C) 0.8 (D) 0.5 Q5. A vocatioal istitute offers two traiig programs (A) ad (B). I the last semester, 00 ad 300 traiees were erolled for programs (A) ad (B), respectively. From the past experiece it is kow that the passig probabilities are 0.9 for program (A) ad 0.7 for program (B). Suppose that at the ed of the last semester, we selected a traiee at radom from this istitute. () The probability that the selected traiee passed the program equals to (A) 0.80 (B) 0.75 (C) 0.85 (D) 0.79 () If it is kow that the selected traiee passed the program, the the probability that he has bee erolled i program (A) equals to (A) 0.8 (B) 0.9 (C) 0.3 (D)

8 4. RANDOM VARIABLES, DISTRIBUTIONS, EXPECTATIONS AND CHEBYSHEV's THEOREM: 4.. DISCRETE DISTRIBUTIONS: Q. Cosider the experimet of flippig a balaced coi three times idepedetly. Let X= Number of heads Number of tails. (a) List the elemets of the sample space S. (b) Assig a value x of X to each sample poit. (c) Fid the probability distributio fuctio of X. (d) Fid P( X ) (e) Fid P( X < ) (f) Fid µ=e(x) (g) Fid σ =Var(X) Q. It is kow that 0% of the people i a certai huma populatio are female. The experimet is to select a committee cosistig of two idividuals at radom. Let X be a radom variable givig the umber of females i the committee.. List the elemets of the sample space S.. Assig a value x of X to each sample poit. 3. Fid the probability distributio fuctio of X. 4. Fid the probability that there will be at least oe female i the committee. 5. Fid the probability that there will be at most oe female i the committee. 6. Fid µ=e(x) 7. Fid σ =Var(X) Q3. A box cotais 00 cards; 40 of which are labeled with the umber 5 ad the other cards are labeled with the umber 0. Two cards were selected radomly with replacemet ad the umber appeared o each card was observed. Let X be a radom variable givig the total sum of the two umbers. (i) List the elemets of the sample space S. (ii) To each elemet of S assig a value x of X. (iii) Fid the probability mass fuctio (probability distributio fuctio) of X. (iv) Fid P(X=0). (v) Fid P(X>0). (vi) Fid µ=e(x) (vii) Fid σ =Var(X) Q4. Let X be a radom variable with the followig probability distributio: x f(x) ) Fid the mea (expected value) of X, µ=e(x). ) Fid E(X ). 3) Fid the variace of X, Var (X) = σ X. 4) Fid the mea of X+, E ( X + ) = µ X

9 5) Fid the variace of X+, Var(X+)= σ X +. Q5. Which of the followig is a probability distributio fuctio: (A) ( ) = x + f x ; x=0,,,3,4 (B) f ( x) = x ; x=0,,,3, x (C) f ( x) = ; x=0,,,3,4 (D) f ( x) = ; x=0,,,3 5 6 Q6. Let the radom variable X have a discrete uiform with parameter k=3 ad with values 0,, ad. The probability distributio fuctio is: f(x)=p(x=x)=/3 ; x=0,,. () The mea of X is (A).0 (B).0 (C).5 (D) 0.0 () The variace of X is (A) 0.0 (B).0 (C) 0.67 (D).33 Q7. Let X be a discrete radom variable with the probability distributio fuctio: f(x) = kx for x=,, ad 3. (i) Fid the value of k. (ii) Fid the cumulative distributio fuctio (CDF), F(x). (iii) Usig the CDF, F(x), fid P (0.5 < X.5). Q8. Let X be a radom variable with cumulative distributio fuctio (CDF) give by: 0, x < 0 0.5, 0 x < F ( x) = 0.6, x <, x (a) Fid the probability distributio fuctio of X, f(x). (b) Fid P( X<). (usig both f(x) ad F(x)) (c) Fid P( X>). (usig both f(x) ad F(x)) Q9. Cosider the radom variable X with the followig probability distributio fuctio: X 0 3 f(x) 0.4 c The value of c is (A) 0.5 (B) 0. (C) 0. (D) 0.5 (E) 0. Q0. Cosider the radom variable X with the followig probability distributio fuctio: X 0 f(x) c Fid the followig: (a) The value of c. (b) P( 0 < X ) (c) µ = E(X) (d) E(X ) - 8 -

10 (e) σ = Var(X) Q. Fid the value of k that makes the fuctio 3 f ( x) = k for x=0,, x 3 x serve as a probability distributio fuctio of the discrete radom variable X. Q. The cumulative distributio fuctio (CDF) of a discrete radom variable, X, is give below: 0 for x < 0 /6 for 0 x < 5/6 for x < F ( x ) = /6 for x < 3 5/6 for 3 x < 4 for x 4. (a) the P( X = ) is equal to: (A) 3/8 (B) /6 (C) 0/6 (D) 5/6 (b) the P( X < 4) is equal to: (A) 0/6 (B) /6 (C) 0/6 (D) 5/6 Q3. If a radom variable X has a mea of 0 ad a variace of 4, the, the radom variable Y = X, (a) has a mea of: (A) 0 (B) 8 (C) 0 (D) (b) ad a stadard deviatio of: (A) 6 (B) (C) 4 (D) 6 Q4. Let X be the umber of typig errors per page committed by a particular typist. The probability distributio fuctio of X is give by: x f ( x) 3k 3k k k k () Fid the umerical value of k. () Fid the average (mea) umber of errors for this typist. (3) Fid the variace of the umber of errors for this typist. (4) Fid the cumulative distributio fuctio (CDF) of X. (5) Fid the probability that this typist will commit at least errors per page. Q5. Suppose that the discrete radom variable X has the followig probability fuctio: f( )=0.05, f(0)=0.5, f()=0.5, f()=0.45, the: () P(X<) equals to (A) 0.30 (B) 0.05 (C) 0.55 (D) 0.50 () P(X ) equals to (A) 0.05 (B) 0.55 (C) 0.30 (D) 0.45 (3) The mea µ=e(x) equals to (A). (B) 0.0 (C). (D) 0.5 (4) E(X ) equals to (A).00 (B).0 (C).50 (D)

11 (5) The variace σ =Var(X) equals to (A).00 (B) 3.3 (C) 0.89 (D).0 (6) If F(x) is the cumulative distributio fuctio (CDF) of X, the F() equals to (A) 0.50 (B) 0.5 (C) 0.45 (D) CONTINUOUS DISTRIBUTIONS: Q. If the cotiuous radom variable X has mea µ=6 ad variace σ =5, the P ( X = 6) is (A) (B) 0.5 (C) 0.0 (D) Noe of these. Q. Cosider the probability desity fuctio: k x, 0 < x < f ( x) = 0, elsewhere. ) The value of k is: (A) (B) 0.5 (C).5 (D) ) The probability P( 0.3 < X 0.6) is, (A) (B) (C) (D) ) The expected value of X, E(X ) is, (A) 0.6 (B).5 (C) (D) / x [Hit: x dx = + c ] (3/ ) Q3. Let X be a cotiuous radom variable with the probability desity fuctio f(x)=k(x+) for 0<x<. (i) Fid the value of k. (ii) Fid P(0 < X ). (iii) Fid the cumulative distributio fuctio (CDF) of X, F(x). (iv) Usig F(x), fid P(0 < X ). Q4. Let X be a cotiuous radom variable with the probability desity fuctio f(x)=3x / for <x<.. P(0<X<) =... E(X) = 3. Var(X) =. 4. E(X+3)= 5. Var(X+3)= Q5. Suppose that the radom variable X has the probability desity fuctio: k x; 0< x< f ( x) = 0; elsewhere.. Evaluate k.. Fid the cumulative distributio fuctio (CDF) of X, F(x). 3. Fid P(0 < X < ). 4. Fid P(X = ) ad P( < X < 3). Q6. Let X be a radom variable with the probability desity fuctio: - 0 -

12 . Fid µ = E(X).. Fid σ = Var(X). 3. Fid E (4X+5). 4. Fid Var(4X+5). 6 x( x); 0< x< f ( x) = 0; elsewhere. Q7. If the radom variable X has a uiform distributio o the iterval (0,0) with the probability desity fuctio give by: ; 0< x< 0 f ( x) = 0, 0; elsewhere.. Fid P(X<6).. Fid the mea of X. 3. Fid E(X ). 4. Fid the variace of X. 5. Fid the cumulative distributio fuctio (CDF) of X, F(x). 6. Use the cumulative distributio fuctio, F(x), to fid P(<X 5). Q8. Suppose that the failure time (i moths) of a specific type of electrical device is distributed with the probability desity fuctio: x,0 < x < 0 f ( x) = 50 0, otherwise (a) the average failure time of such device is: (A) (B).00 (C).00 (D) 5.00 (b) the variace of the failure time of such device is: (A) 0 (B) 50 (C) 5.55 (D) 0 (c) P(X>5) = (A) 0.5 (B) 0.55 (C) 0.65 (D) 0.75 Q9. If the cumulative distributio fuctio of the radom variable X havig the form: 0 ; x < 0 P( X x) = F( x) = x /( x + ) ; x 0 The () P( 0 < X < ) equals to (a) (b) (c) (d) oe of these. () If P( X k) = 0.5, the k equals to (a) 5 (b) 0.5 (c) (d).5 Q0. If the diameter of a certai electrical cable is a cotiuous radom variable X ( i cm ) havig the probability desity fuctio : 3 0x ( x) 0< x f ( x) = 0 otherwise - -

13 () P( X > 0.5) is: a) 0.85 b) c) d) () P( 0.5 < X <.75) is: a) 0.85 b) c) d) (3) µ=e(x) is: a) b) c) d) oe of these. (4) σ =Var(X) is: a)0.375 b) 3.75 c) d).375 (5) For this radom variable, P ( µ σ X µ + σ ) will have a exact value equals to: a) b) c) d) 0.50 (6) For this radom variable, P ( µ σ X µ + σ ) will have a lower boud value accordig to chebyshev s theory equals to: a) b) c) d) 0.50 (7) If Y= 3X.5, the E(Y) is: a) 0.5 b) c) 0.5 d) oe of these. (8) If Y= 3X.5, the Var(Y) is: a).8575 b) 0.95 c) d) CHEBYSHEV S THEOREM : Q. Accordig to Chebyshev s theorem, for ay radom variable X with mea µ ad variace σ, a lower boud for P ( µ σ < X < µ + σ ) is, (A) (B) (C) (D) 0.50 Note: P ( µ σ < X < µ + σ ) = P( X µ < σ ) Q. Suppose that X is a radom variable with mea µ=, variace σ =9, ad ukow probability distributio. Usig Chebyshev s theorem, P ( 3 < X < ) is at least equal to, (A) 8/9 (B) 3/4 (C) /4 (D) /6 Q3. Suppose that E(X)=5 ad Var(X)=4. Usig Chebyshev's Theorem, (i) fid a approximated value of P(<X<9). (ii) fid some costats a ad b (a<b) such that P(a<X<b) 5/6. Q4. Suppose that the radom variable X is distributed accordig to the probability desity fuctio give by: ; 0< x < 0 f ( x) = 0, 0; elsewhere. Assumig µ=5 ad σ =.89,. Fid the exact value of P(µ.5σ < X <µ+.5σ).. Usig Chebyshev's Theorem, fid a approximate value of P(µ.5σ < X <µ+.5σ). 3. Compare the values i () ad (). Q5. Suppose that X ad Y are two idepedet radom variables with E(X)=30, Var(X)=4, E(Y)=0, ad Var(Y)=. The: () E(X 3Y 0) equals to (A) 40 (B) 0 (C) 30 (D)

14 () Var(X 3Y 0) equals to (A) 34 (B) 4 (C).0 (D) 4 (3) Usig Chebyshev's theorem, a lower boud of P(4<X<36) equals to (A) (B) (C) (D)

15 DISCRETE UNIFORM DISTRIBUTION: Q. Let the radom variable X have a discrete uiform with parameter k=3 ad with values 0,, ad. The: (a) P(X=) is (A).0 (B) /3 (C) 0.3 (D) 0. (E) Noe (b) The mea of X is: (A).0 (B).0 (C).5 (D) 0.0 (E) Noe (c) The variace of X is: (A) 0/3=0.0 (B) 3/3=.0 (C) /3=0.67 (D) 4/3=.33 (E) Noe 5. BINOMIAL DISTRIBUTION: Q. Suppose that 4 out of buildigs i a certai city violate the buildig code. A buildig egieer radomly ispects a sample of 3 ew buildigs i the city. (a) Fid the probability distributio fuctio of the radom variable X represetig the umber of buildigs that violate the buildig code i the sample. (b) Fid the probability that: (i) oe of the buildigs i the sample violatig the buildig code. (ii) oe buildig i the sample violatig the buildig code. (iii) at lease oe buildig i the sample violatig the buildig code. (c) Fid the expected umber of buildigs i the sample that violate the buildig code (E(X)). (d) Fid σ =Var(X). Q. A missile detectio system has a probability of 0.90 of detectig a missile attack. If 4 detectio systems are istalled i the same area ad operate idepedetly, the (a) The probability that at least two systems detect a attack is (A) (B) (C) (D) (E) (b) The average (mea) umber of systems detect a attack is (A) 3.6 (B).0 (C) 0.36 (D).5 (E) 4.0 Q3. Suppose that the probability that a perso dies whe he or she cotracts a certai disease is 0.4. A sample of 0 persos who cotracted this disease is radomly chose. () What is the expected umber of persos who will die i this sample? () What is the variace of the umber of persos who will die i this sample? (3) What is the probability that exactly 4 persos will die amog this sample? (4) What is the probability that less tha 3 persos will die amog this sample? (5) What is the probability that more tha 8 persos will die amog this sample? Q4. Suppose that the percetage of females i a certai populatio is 50%. A sample of 3 people is selected radomly from this populatio. (a) The probability that o females are selected is (A) (B) (C) (D) 0.5 (b) The probability that at most two females are selected is (A) (B) (C) (D) 0.5 (c) The expected umber of females i the sample is (A) 3.0 (B).5 (C) 0.0 (D)

16 (d) The variace of the umber of females i the sample is (A) 3.75 (B).75 (C).75 (D) 0.75 Q5. 0% of the traiees i a certai program fail to complete the program. If 5 traiees of this program are selected radomly, (i) Fid the probability distributio fuctio of the radom variable X, where: X = umber of the traiees who fail to complete the program. (ii) Fid the probability that all traiees fail to complete the program. (iii) Fid the probability that at least oe traiee will fail to complete the program. (iv) How may traiees are expected to fail completig the program? (v) Fid the variace of the umber of traiees who fail completig the program. Q6. I a certai idustrial factory, there are 7 workers workig idepedetly. The probability of accruig accidets for ay worker o a give day is 0., ad accidets are idepedet from worker to worker. (a) The probability that at most two workers will have accidets durig the day is (A) (B) (C) (D) (b) The probability that at least three workers will have accidets durig the day is: (A) (B) 0.35 (C) (D) (c) The expected umber workers who will have accidets durig the day is (A).4 (B) 0.35 (C).57 (D) Q7. From a box cotaiig 4 black balls ad gree balls, 3 balls are draw idepedetly i successio, each ball beig replaced i the box before the ext draw is made. The probability of drawig gree balls ad black ball is: (A) 6/7 (B) /7 (C) /7 (D) 4/7 Q8. The probability that a lab specime is cotamiated is 0.0. Three idepedet samples are checked. ) the probability that oe is cotamiated is: (A) (B) 0.00 (C) 0.79 (D) 0. 3 ) the probability that exactly oe sample is cotamiated is: (A) 0.٢٤٣ (B) 0.0٨١ (C) 0.7٥٧ (D) 0. 3 Q9. If X~Biomial(,p), E(X)=, ad Var(X)=0.75, fid P(X=). Q0. Suppose that X~Biomial(3,0.). Fid the cumulative distributio fuctio (CDF) of X. Q. A traffic cotrol egieer reports that 75% of the cars passig through a checkpoit are from Riyadh city. If at this checkpoit, five cars are selected at radom. () The probability that oe of them is from Riyadh city equals to: (A) (B) (C) (D) () The probability that four of them are from Riyadh city equals to: (A) (B) (C) 0 (D) 0.49 (3) The probability that at least four of them are from Riyadh city equals to: (A) (B) (C) (D) (4) The expected umber of cars that are from Riyadh city equals to: (A) (B) 3.75 (C) 3 (D) 0-5 -

17 6. HYPERGEOMETRIC DISTRIBUTION: Q. A shipmet of 7 televisio sets cotais defective sets. A hotel makes a radom purchase of 3 of the sets. (i) Fid the probability distributio fuctio of the radom variable X represetig the umber of defective sets purchased by the hotel. (ii) Fid the probability that the hotel purchased o defective televisio sets. (iii) What is the expected umber of defective televisio sets purchased by the hotel? (iv) Fid the variace of X. Q. Suppose that a family has 5 childre, 3 of them are girls ad the rest are boys. A sample of childre is selected radomly ad without replacemet. (a) The probability that o girls are selected is (A) 0.0 (B) 0.3 (C) 0.6 (D) 0. (b) The probability that at most oe girls are selected is (A) 0.7 (B) 0.3 (C) 0.6 (D) 0. (c) The expected umber of girls i the sample is (A). (B). (C) 0. (D) 3. (d) The variace of the umber of girls i the sample is (A) 36.0 (B) 3.6 (C) 0.36 (D) 0.63 Q3. A radom committee of size 4 is selected from chemical egieers ad 8 idustrial egieers. (i) Write a formula for the probability distributio fuctio of the radom variable X represetig the umber of chemical egieers i the committee. (ii) Fid the probability that there will be o chemical egieers i the committee. (iii) Fid the probability that there will be at least oe chemical egieer i the committee. (iv) What is the expected umber of chemical egieers i the committee? (v) What is the variace of the umber of chemical egieers i the committee? Q4. A box cotais red balls ad 4 black balls. Suppose that a sample of 3 balls were selected radomly ad without replacemet. Fid,. The probability that there will be red balls i the sample.. The probability that there will be 3 red balls i the sample. 3. The expected umber of the red balls i the sample. Q5. From a lot of 8 missiles, 3 are selected at radom ad fired. The lot cotais defective missiles that will ot fire. Let X be a radom variable givig the umber of defective missiles selected.. Fid the probability distributio fuctio of X.. What is the probability that at most oe missile will ot fire? 3. Fid µ = E(X) ad σ = Var(X). Q6. A particular idustrial product is shipped i lots of 0 items. Testig to determie whether a item is defective is costly; hece, the maufacturer samples productio rather tha usig 00% ispectio pla. A samplig pla costructed to miimize the umber of defectives shipped to cosumers calls for samplig 5 items from each lot ad rejectig the lot if more tha oe defective is observed. (If the lot is rejected, each item i the lot is the tested.) If a lot cotais 4 defectives, what is the probability that it will be accepted

18 Q7. Suppose that X~h(x;00,,60); i.e., X has a hypergeometric distributio with parameters N=00, =, ad K=60. Calculate the probabilities P(X=0), P(X=), ad P(X=) as follows: (a) exact probabilities usig hypergeometric distributio. (b) approximated probabilities usig biomial distributio. Q8. A particular idustrial product is shipped i lots of 000 items. Testig to determie whether a item is defective is costly; hece, the maufacturer samples productio rather tha usig 00% ispectio pla. A samplig pla costructed to miimize the umber of defectives shipped to cosumers calls for samplig 5 items from each lot ad rejectig the lot if more tha oe defective is observed. (If the lot is rejected, each item i the lot is the tested.) If a lot cotais 00 defectives, calculate the probability that the lot will be accepted usig: (a) hypergeometric distributio (exact probability.) (b) biomial distributio (approximated probability.) Q9. A shipmet of 0 digital voice recorders cotais 5 that are defective. If 0 of them are radomly chose (without replacemet) for ispectio, the: () The probability that will be defective is: (A) 0.40 (B) (C) (D) () The probability that at most will be defective is: (A) (B) 0.64 (C) (D) 0.57 (3) The expected umber of defective recorders i the sample is: (A) (B) (C) 3.5 (D).5 (4) The variace of the umber of defective recorders i the sample is: (A) (B).5 (C) (D).875 Q0. A box cotais 4 red balls ad 6 gree balls. The experimet is to select 3 balls at radom. Fid the probability that all balls are red for the followig cases: () If selectio is without replacemet (A) 0.6 (B) (C) (D) () If selectio is with replacemet (A) (B) (C) (D)

19 7. POISSON DISTRIBUTION: Q. O average, a certai itersectio results i 3 traffic accidets per day. Assumig Poisso distributio, (i) what is the probability that at this itersectio: () o accidets will occur i a give day? () More tha 3 accidets will occur i a give day? (3) Exactly 5 accidets will occur i a period of two days? (ii) what is the average umber of traffic accidets i a period of 4 days? Q. At a checkout couter, customers arrive at a average of.5 per miute. Assumig Poisso distributio, the () The probability of o arrival i two miutes is (A) 0.0 (B) 0.3 (C) (D) (E) () The variace of the umber of arrivals i two miutes is (A).5 (B).5 (C) 3.0 (D) 9.0 (E) 4.5 Q3. Suppose that the umber of telephoe calls received per day has a Poisso distributio with mea of 4 calls per day. (a) The probability that calls will be received i a give day is (A) (B) (C) (D) (b) The expected umber of telephoe calls received i a give week is (A) 4 (B) 7 (C) 8 (D) 4 (c) The probability that at least calls will be received i a period of hours is (A) (B) (C) (D) Q4. The average umber of car accidets at a specific traffic sigal is per a week. Assumig Poisso distributio, fid the probability that: (i) there will be o accidet i a give week. (ii) there will be at least two accidets i a period of two weeks. Q5. The average umber of airplae accidets at a airport is two per a year. Assumig Poisso distributio, fid. the probability that there will be o accidet i a year.. the average umber of airplae accidets at this airport i a period of two years. 3. the probability that there will be at least two accidets i a period of 8 moths. Q6. Suppose that X~Biomial(000,0.00). By usig Poisso approximatio, P(X=3) is approximately equal to (choose the earest umber to your aswer): (A) 0.65 (B) 0.75 (C) 0.85 (D) 0.95 (E) Q7. The probability that a perso dies whe he or she cotracts a certai disease is A sample of 000 persos who cotracted this disease is radomly chose. () What is the expected umber of persos who will die i this sample? () What is the probability that exactly 4 persos will die amog this sample? Q8. The umber of faults i a fiber optic cable follows a Poisso distributio with a average of 0.6 per 00 feet

20 () The probability of faults per 00 feet of such cable is: (A) (B) 0.90 (C) 0.30 (D) 0.5 () The probability of less tha faults per 00 feet of such cable is: (A) 0.35 (B) (C) (D) (3) The probability of 4 faults per 00 feet of such cable is: (A) (B) (C) (D)

21 CONTINUOUS UNIFORM DISTRIBUTION: Q. If the radom variable X has a uiform distributio o the iterval (0,0), the. P(X<6) equals to (A) 0.4 (B) 0.6 (C) 0.8 (D) 0. (E) 0.. The mea of X is (A) 5 (B) 0 (C) (D) 8 (E) 6. The variace X is (A) (B) 8.33 (C) 8.33 (D) 5 (E) Noe Q. Suppose that the radom variable X has the followig uiform distributio: 3, < x < f ( x) = 3 0, otherwise P( 0.33 () X < 0.5 ) = (A) 0.49 (B) 0.5 (C) 0 (D) 3 P( X () >.5 ) = (A) 0 (B) (C) 0.5 (D) 0.33 (3) The variace of X is (A) (B) (C) 9 (D) Q3. Suppose that the cotiuous radom variable X has the followig probability desity fuctio (pdf): f(x)=0. for 0<x<5. The () P(X>) equals to (A) 0.4 (B) 0. (C) 0. (D) 0.8 () P(X ) equals to (A) 0.05 (B) 0.8 (C) 0.5 (D) 0.4 (3) The mea µ=e(x) equals to (A).0 (B).5 (C) 3.0 (D) 3.5 (4) E(X ) equals to (A) (B) (C) (D) (5) Var(X) equals to (A) (B) (C) (D).0833 (6) If F(x) is the cumulative distributio fuctio (CDF) of X, the F() equals to (A) 0.75 (B) 0.5 (C) 0.8 (D)

22 8. NORMAL DISTRIBUTION: Q. (A) Suppose that Z is distributed accordig to the stadard ormal distributio. ) the area uder the curve to the left of z =. 43 is: (A) (B) (C) 0 (D) ) the area uder the curve to the left of z =.39 is: (A) (B) (C).73 (D) ) the area uder the curve to the right of z = is: (A) (B) (C) (D) ) the area uder the curve betwee z =. 6 ad z = is: (A) (B) (C) (D) ) the value of k such that P ( 0.93< Z < k) = is: (A) (B). (C). (D).00 (B) Suppose that Z is distributed accordig to the stadard ormal distributio. Fid: ) P(Z< 3.9) ) P(Z> 4.5) ) P(Z< 3.7) ) P(Z> 4.) Q. The fiished iside diameter of a pisto rig is ormally distributed with a mea of cetimeters ad a stadard deviatio of 0.03 cetimeter. The, ) the proportio of rigs that will have iside diameter less tha.05 cetimeters is: (A) (B) (C) (D) ) the proportio of rigs that will have iside diameter exceedig.97 cetimeters is: (A) (B) (C) (D) ) the probability that a pisto rig will have a iside diameter betwee.95 ad.05 cetimeters is: (A) (B) (C) (D) Q3. The average life of a certai type of small motor is 0 years with a stadard deviatio of years. Assume the live of the motor is ormally distributed. The maufacturer replaces free all motors that fail while uder guaratee. If he is willig to replace oly.5% of the motors that fail, the he should give a guaratee of : (A) 0.03 years (B) 8 years (C) 5.66 years (D) 3 years Q4. A machie makes bolts (that are used i the costructio of a electric trasformer). It produces bolts with diameters (X) followig a ormal distributio with a mea of iches ad a stadard deviatio of 0.00 iches. Ay bolt with diameter less tha iches or greater tha 0.06 iches must be scrapped. The () The proportio of bolts that must be scrapped is equal to (A) (B) 0.08 (C) (D) (E) () If P(X>a)= 0.949, the a equals to: (A) (B) (C) (D) (E)

23 Q5. The diameters of ball bearigs maufactured by a idustrial process are ormally distributed with a mea µ = 3.0 cm ad a stadard deviatio σ = cm. All ball bearigs with diameters ot withi the specificatios µ ± d cm (d > 0) will be scrapped. () Determie the value of d such that 90% of ball bearigs maufactured by this process will ot be scrapped. () If d = 0.005, what is the percetage of maufactured ball bearigs that will be scraped? Q6. The weight of a large umber of fat persos is icely modeled with a ormal distributio with mea of 8 kg ad a stadard deviatio of 9 kg. () The percetage of fat persos with weights at most 0 kg is (A) 0.09 % (B) 90.3 % (C) 99.8 % (D).8 % () The percetage of fat persos with weights more tha 49 kg is (A) 0.09 % (B) 0.99 % (C) 9.7 % (D) 99.8 % (3) The weight x above which 86% of those persos will be (A) 8.8 (B) 8.8 (C) 54.8 (D) 8.8 (4) The weight x below which 50% of those persos will be (A) 0.8 (B) 8 (C) 54.8 (D) 8 Q7. The radom variable X, represetig the lifespa of a certai electroic device, is ormally distributed with a mea of 40 moths ad a stadard deviatio of moths. Fid. P(X<38). (0.587). P(38<X<40). (0.343) 3. P(X=38). (0.0000) 4. The value of x such that P(X<x)= (4.4) Q8. If the radom variable X has a ormal distributio with the mea µ ad the variace σ, the P(X<µ+σ) equals to (A) (B) (C) (D) (E) Q9. If the radom variable X has a ormal distributio with the mea µ ad the variace, ad if P(X<3)=0.877, the µ equals to (A) 3.84 (B).84 (C).84 (D) 4.84 (E) 8.84 Q0. Suppose that the marks of the studets i a certai course are distributed accordig to a ormal distributio with the mea 70 ad the variace 5. If it is kow that 33% of the studet failed the exam, the the passig mark x is (A) 67.8 (B) 60.8 (C) 57.8 (D) 50.8 (E) 70.8 Q. If the radom variable X has a ormal distributio with the mea 0 ad the variace 36, the. The value of X above which a area of 0.96 lie is (A) 4.44 (B) 6.44 (C) 0.44 (D) 8.44 (E).44. The probability that the value of X is greater tha 6 is (A) (B) (C) (D) (E) Q. Suppose that the marks of the studets i a certai course are distributed accordig to a ormal distributio with the mea 65 ad the variace 6. A studet fails the exam if he obtais a mark less tha 60. The the percetage of studets who fail the exam is (A) 0.56% (B) 90.56% (C) 50.56% (D) 0.56% (E)40.56% - -

24 Q3. The average raifall i a certai city for the moth of March is 9. cetimeters. Assumig a ormal distributio with a stadard deviatio of.83 cetimeters, the the probability that ext March, this city will receive: () less tha.84 cetimeters of rai is: (A) (B) 0.76 (C) 0.5 (D) 0.08 () more tha 5 cetimeters but less tha 7 cetimeters of rai is: (A) (B) (C) (D) 0.34 (3) more tha 3.8 cetimeters of rai is: (A) (B) (C) 0.30 (D)

25 9. EXPONENTIAL DISTRIBUTION Q. If the radom variable X has a expoetial distributio with the mea 4, the:. P(X<8) equals to (A) (B) (C) (D) (E) The variace of X is (A) 4 (B) 6 (C) (D) /4 (E) / Q. Suppose that the failure time (i hours) of a certai electrical device is distributed with a probability desity fuctio give by: x / 70 f ( x) = e, x > 0, 70 ) the probability that a radomly selected device will fail withi the first 50 hours is: (A) (B) (C) (D) ) the probability that a radomly selected device will last more tha 50 hours is: (A) (B) (C) 0.73 (D) ) the average failure time of the electrical device is: (A) /70 (B) 70 (C) 40 (D) 35 4) the variace of the failure time of the electrical device is: (A) 4900 (B) /49000 (C) 70 (D) 5 ax ax [Hit: e dx = e + c ] a Q3. The lifetime of a specific battery is a radom variable X with probability desity fuctio give by: x / 00 f ( x) = e, x > 0 00 () The mea life time of the battery equals to (A) 00 (B) /00 (C) 00 (D) /00 (E) No of these () P(X>00) = (A) 0.5 (B) (C) (D) (E) 0.63 (3) P(X=00) = (A) 0.5 (B) 0.0 (C) (D) (E).0 Q4. Suppose that the lifetime of a certai electrical device is give by T. The radom variable T is modeled icely by a expoetial distributio with mea of 6 years. A radom sample of four of these devices are istalled i differet systems. Assumig that these devices work idepedetly, the: () the variace of the radom variable T is (A) 36 (B) (36) (C) 6 (D) 36 () the probability that at most oe of the devices i the sample will be fuctioig more tha 6 years is (A) (B) 0.63 (C) 0.53 (D) (3) the probability that at least two of the devices i the sample will be fuctioig more tha 6 years is (A) (B) 0.63 (C) 0.53 (D) (4) the expected umber of devices i the sample which will be fuctioig more tha 6 years is approximately equal to (A) 3.47 (B).47 (C) 4.47 (D)

26 Q5. Assume the legth (i miutes) of a particular type of a telephoe coversatio is a radom variable with a probability desity fuctio of the form:. P( 3 < x < 0) is: f ( x) 0. = 0 0.x e ; x 0 ; elsewhere (a) (b) (c) 0.43 (d) For this radom variable, P ( µ σ X µ + σ ) will have a exact value equals: (a) 0.50 (b) (c) (d) For this radom variable, P ( µ σ X µ + σ ) will have a lower boud valued accordig to chebyshev s theory equals: (a) (b) 0.50 (c) (d) Q6. The legth of time for oe customer to be served at a bak is a radom variable X that follows the expoetial distributio with a mea of 4 miutes. () The probability that a customer will be served i less tha miutes is: (A) (B) 0.3 (C) (D) () The probability that a customer will be served i more tha 4 miutes is: (A) 0.63 (B) (C) (D) 0.00 (3) The probability that a customer will be served i more tha but less tha 5 miutes is: (A) (B) 0.3 (C) (D) (4) The variace of service time at this bak is (A) (B) 4 (C) 8 (D) 6-5 -

27 0. SAMPLING DISTRIBUTIONS 0.. Sigle Mea: Q. A machie is producig metal pieces that are cylidrical i shape. A radom sample of size 5 is take ad the diameters are.70,.,.0,.3 ad.8 cetimeters. The, ) The sample mea is: (A). (B).3 (C).90 (D).0 (E). ) The sample variace is: (A) (B) (C) (D) (E) Q. The average life of a certai battery is 5 years, with a stadard deviatio of year. Assume that the live of the battery approximately follows a ormal distributio. ) The sample mea X of a radom sample of 5 batteries selected from this product has a mea E(X ) = µ x equal to: (A) 0. (B) 5 (C) 3 (D) Noe of these ) The variace Var(X ) = σ x of the sample mea X of a radom sample of 5 batteries selected from this product is equal to: (A) 0. (B) 5 (C) 3 (D) Noe of these 3) The probability that the average life of a radom sample of size 6 of such batteries will be betwee 4.5 ad 5.4 years is: (A) (B) 0.35 (C) (D) ) The probability that the average life of a radom sample of size 6 of such batteries will be less tha 5.5 years is: (A) (B) 0.08 (C) 0.93 (D) Noe of these 5) The probability that the average life of a radom sample of size 6 of such batteries will be more tha 4.75 years is: (A) (B) (C) (D) Noe of these 6) If P ( X > a ) = where X represets the sample mea for a radom sample of size 9 of such batteries, the the umerical value of a is: (A) (B) 6.5 (C) (D) Noe of these Q3. The radom variable X, represetig the lifespa of a certai light bulb, is distributed ormally with a mea of 400 hours ad a stadard deviatio of 0 hours.. What is the probability that a particular light bulb will last for more tha 380 hours?. Light bulbs with lifespa less tha 380 hours are rejected. Fid the percetage of light bulbs that will be rejected. 3. If 9 light bulbs are selected radomly, fid the probability that their average lifespa will be less tha 405. Q4. Suppose that you take a radom sample of size =64 from a distributio with mea µ=55 ad stadard deviatio σ=0. Let X = X i i= be the sample mea. (a) What is the approximated samplig distributio of X? (b) What is the mea of X? (c) What is the stadard error (stadard deviatio) of X? - 6 -

28 (d) Fid the probability that the sample mea X exceeds 5. Q5. The amout of time that customers usig ATM (Automatic Teller Machie) is a radom variable with the mea 3.0 miutes ad the stadard deviatio of.4 miutes. If a radom sample of 49 customers is observed, the () the probability that their mea time will be at least.8 miutes is (A).0 (B) (C) (D) () the probability that their mea time will be betwee.7 ad 3. miutes is (A) (B) (C) (D) Q6. The average life of a idustrial machie is 6 years, with a stadard deviatio of year. Assume the life of such machies follows approximately a ormal distributio. A radom sample of 4 of such machies is selected. The sample mea life of the machies i the sample is X. () The sample mea has a mea µ x = E(X ) equals to: (A) 5 (B) 6 (C) 7 (D) 8 () The sample mea has a variace σ x = Var( X ) equals to: (A) (B) 0.5 (C) 0.5 (D) 0.75 (3) P (X < 5.5) = (A) (B) (C) (D) (4) If P ( X > a ) = 0. 49, the the umerical value of a is: (A) (B).04 (C) 6.5 (D) Two Meas: Q. A radom sample of size = 36 is take from a ormal populatio with a mea µ = 70 ad a stadard deviatio σ = 4. A secod idepedet radom sample of size = 49 is take from a ormal populatio with a mea µ = 85 ad a stadard deviatio σ = 5. Let X ad X be the averages of the first ad secod samples, respectively. a) Fid E( X ) ad Var( X). b) Fid E( X X ) ad Var( X X ). c) Fid P( 70 < X < 7). d) Fid P( X X > 6). Q. A radom sample of size 5 is take from a ormal populatio (first populatio) havig a mea of 00 ad a stadard deviatio of 6. A secod radom sample of size 36 is take from a differet ormal populatio (secod populatio) havig a mea of 97 ad a stadard deviatio of 5. Assume that these two samples are idepedet. () the probability that the sample mea of the first sample will exceed the sample mea of the secod sample by at least 6 is (A) (B) (C) 0.00 (D) () the probability that the differece betwee the two sample meas will be less tha is (A) (B) (C) (D) Sigle Proportio: - 7 -

29 Q. Suppose that 0% of the studets i a certai uiversity smoke cigarettes. A radom sample of 5 studets is take from this uiversity. Let p be the proportio of smokers i the sample. () Fid E( p) = µ p, the mea p. () Fid Var( p) = σ, the variace of p. p (3) Fid a approximate distributio of p. (4) Fid P( p >0.5). Q: Suppose that you take a radom sample of size =00 from a biomial populatio with parameter p=0.5 (proportio of successes). Let p = X / be the sample proportio of successes, where X is the umber of successes i the sample. (a) What is the approximated samplig distributio of p? (b) What is the mea of p? (c) What is the stadard error (stadard deviatio) of p? (d) Fid the probability that the sample proportio p is less tha Two Proportios: Q. Suppose that 5% of the male studets ad 0% of the female studets i a certai uiversity smoke cigarettes. A radom sample of 5 male studets is take. Aother radom sample of 0 female studets is idepedetly take from this uiversity. Let p ad p be the proportios of smokers i the two samples, respectively. () Fid E( p p ) = µ p p, the mea of p p. () Fid Var( p p ) = σ, the variace of p p p p. (3) Fid a approximate distributio of p p. (4) Fid P(0.0< p p <0.0). 0.5 t-distributio: Q. Usig t-table with degrees of freedom df=4, fid t0.0, t Q. From the table of t-distributio with degrees of freedom ν = 5, the value of t 0.05 equals to (A).3 (B).753 (C) 3.68 (D)

30 . ESTIMATION AND CONFIDENCE INTERVALS:.. Sigle Mea: Q. A electrical firm maufacturig light bulbs that have a legth of life that is ormally distributed with a stadard deviatio of 30 hours. A sample of 50 bulbs were selected radomly ad foud to have a average of 750 hours. Let µ be the populatio mea of life legths of all bulbs maufactured by this firm. () Fid a poit estimate for µ. () Costruct a 94% cofidece iterval for µ. Q. Suppose that we are iterested i makig some statistical ifereces about the mea, µ, of a ormal populatio with stadard deviatio σ=.0. Suppose that a radom sample of size =49 from this populatio gave a sample mea X =4.5. () The distributio of X is (A) N(0,) (B) t(48) (C) N(µ, 0.857) (D) N(µ,.0) (E) N(µ, ) () A good poit estimate of µ is (A) 4.50 (B).00 (C).50 (D) 7.00 (E).5 (3) The stadard error of X is (A) (B).0 (C) (D) (E) (4) A 95% cofidece iterval for µ is (A) (3.44,5.56) (B) (3.34,5.66) (C) (3.54,5.46) (D) (3.94,5.06) (E) (3.04,5.96) (5) If the upper cofidece limit of a cofidece iterval is 5., the the lower cofidece limit is (A) 3.6 (B) 3.8 (C) 4.0 (D) 3.5 (E) 4. (6) The cofidece level of the cofidece iterval (3.88, 5.) is (A) 90.74% (B) 95.74% (C) 97.74% (D) 94.74% (E) 9.74% (7) If we use X to estimate µ, the we are 95% cofidet that our estimatio error will ot exceed (A) e=0.50 (B) E=0.59 (C) e=0.58 (D) e=0.56 (E) e=0.5 (8) If we wat to be 95% cofidet that the estimatio error will ot exceed e=0. whe we use X to estimate µ, the the sample size must be equal to (A) 59 (B) 53 (C) 537 (D) 534 (E) 530 Q3. The followig measuremets were recorded for lifetime, i years, of certai type of machie: 3.4, 4.8, 3.6, 3.3, 5.6, 3.7, 4.4, 5., ad 4.8. Assumig that the measuremets represet a radom sample from a ormal populatio, the a 99% cofidece iterval for the mea life time of the machie is (A) 5.37 µ 3. 5 (B) 4.7 µ 9. (C) 4.0 µ (D) 3.37 µ 5. 5 Q4. A researcher wats to estimate the mea lifespa of a certai light bulbs. Suppose that the distributio is ormal with stadard deviatio of 5 hours.. Determie the sample size eeded o order that the researcher will be 90% cofidet that the error will ot exceed hours whe he uses the sample mea as a poit estimate for the true mea.. Suppose that the researcher selected a radom sample of 49 bulbs ad foud that the sample mea is 390 hours. (i) Fid a good poit estimate for the true mea µ. (ii) Fid a 95% cofidece iterval for the true mea µ

31 Q5. The amout of time that customers usig ATM (Automatic Teller Machie) is a radom variable with a stadard deviatio of.4 miutes. If we wish to estimate the populatio mea µ by the sample mea X, ad if we wat to be 96% cofidet that the sample mea will be withi 0.3 miutes of the populatio mea, the the sample size eeded is (A) 98 (B) 00 (C) 9 (D) 85 Q6: A radom sample of size =36 from a ormal quatitative populatio produced a mea X = 5. ad a variace S = 9. (a) Give a poit estimate for the populatio mea µ. (b) Fid a 95% cofidece iterval for the populatio mea µ. Q7. A group of 0 college studets were asked to report the umber of hours that they spet doig their homework durig the previous weeked ad the followig results were obtaied: 7.5, 8.5, 5.0, 6.75, 8.0, 5.5, 0.5, 8.5, 6.75, 9.5 X = , X = } It is assumed that this sample is a radom sample from a ormal distributio with ukow variace σ. Let µ be the mea of the umber of hours that the college studet sped doig his/her homework durig the weeked. (a) Fid the sample mea ad the sample variace. (b) Fid a poit estimate for µ. (c) Costruct a 80% cofidece iterval for µ. Q8. A electroics compay wated to estimate its mothly operatig expeses i thousads riyals (µ). It is assumed that the mothly operatig expeses (i thousads riyals) are distributed accordig to a ormal distributio with variace σ = (I) Suppose that a radom sample of 49 moths produced a sample mea X = (a) Fid a poit estimate for µ. (b) Fid the stadard error of X. (c) Fid a 90% cofidece iterval for µ. (II) Suppose that they wat to estimate µ by X. Fid the sample size () required if they wat their estimate to be withi 0.5 of the actual mea with probability equals to Q9. The tesile stregth of a certai type of thread is approximately ormally distributed with stadard deviatio of 6.8 kilograms. A sample of 0 pieces of the thread has a average tesile stregth of 7.8 kilograms. The, (a) A poit estimate of the populatio mea of the tesile stregth (µ) is: (A) 7.8 (B) 0 (C) 6.8 (D) 46.4 (E) Noe of these (b) Suppose that we wat to estimate the populatio mea (µ) by the sample mea (X ). To be 95% cofidet that the error of our estimate of the mea of tesile stregth will be less tha 3.4 kilograms, the miimum sample size should be at least: (A) 4 (B) 6 (C) 0 (D) 8 (E) Noe of these (c) For a 98% cofidece iterval for the mea of tesile stregth, we have the lower boud equal to: (A) (B) 69.6 (C) 7.44 (D) (E) Noe of these (d) For a 98% cofidece iterval for the mea of tesile stregth, we have the upper boud equal to: