COVENANT UNIVERSITY OMEGA SEMESTER TUTORIAL KIT (VOL. 2) 100 LEVEL

Transcription

1 COVENANT UNIVERSITY OMEGA SEMESTER TUTORIAL KIT (VOL. ) P R O G R A M M E : M AT H E M AT I CS 00 LEVEL

2 DISCLAIMER The contents of this document are intended for practice and learning purposes at the undergraduate level. The materials are from different sources including the internet and the contributors do not in any way claim authorship or ownership of them. The materials are also not to be used for any commercial purpose.

3 LIST OF COURSES MAT: Mathematics V (Calculus) MAT: Vector Algebra MAT: Statistics II *MAT5: Use of Statistical Package *MAT6: Use of Mathematical Package *Not included

4 COVENANT UNIVERSITY CANAANLAND, KM 0, IDIROKO ROAD P.M.B. 0, OTA, OGUN STATE, NIGERIA. COLLEGE: SCIENCE AND TECHNOLOGY SESSION: 05/06 {EXAMINATION} SEMESTER: OMEGA COURSE CODE: MAT CREDIT UNIT: COURSE TITLE: Mathematics V: CALCULUS TIME: 0 HOURS INSTRUCTION: *Answer all questions as you identify the choice that best answers each of the questions. Shade the correct answers appropriately on the provided OMR sheet. **The use of any electronic device is highly prohibited for this eamination. See working-sheet. Q: Which of the following mappings is not a function? (A) (B) (C) p, p (D) h, h Q: Find the domain of g. (A), (B) 4, (C), (D), Q: Find the domain of f : 4 Q4: Computing lim 5 lim. (A) 4, (B),4 (C) 4, (D),4 Q5: Which of the following is not in the range of gives: (A) (B) (C) (D) h (A) (B) :? 4 (C) (D) 4 Q6: Find the range of h cos. (A) y, (B), (C) y, (D), d Q7: Find g f m d (A) e (B) e (C) e (D) e if f ( ), m( ) & g( ) e. Q8: Compute h lim h h h0 : (A) (B) (C) (D) Q9: Find the range of dg d if g cos. (A), (B) 4, (C) 4, (D) 0, 4

5 Q0: One of the following functions is injective: (A) (B) g (D) g g g (C) Q: Compute t lim. t t (A) (B) (C) (D) Q: Which of the following is correct? (A) All mappings are functions. (B) All functions are mappings (C) Not all one-one functions have inverses. (D) All one-one functions are onto. Q: Find the inverse of f. ( A) (B) (C) (D) Q4: Compute 8 lim 5 4. (A) 5 (B) 5 (C) (D) Q5: Find gg if g Q6: Find y if. y. (A) (B) (C) (D) ( ) ( ) ( ) (A) (B) (C) (D) Q7: Compute 4 lim. (A) (B) (C) 0 (D) Q8: Find y if y cos sin. (C) cos sin (D) sin cos (A) 9 cos 4sin (B) cos sin Q9: Which of the following is a strictly monotone decreasing function? (A) f ( ) (B) f ( ) (C) f ( ) (D) f ( ) 0 Q0: Find the value of H if H lim. (A)4 (B) 7 (C) 4 (D) none Q: Let g f be a function of. Then the rate of change in g (A) composite function (B) differential product (C) derivative (D) rate of change Q: Find y at, given that y 4 w.r.t is known as. (A) (B) (C) (D) 5

6 d d Q: Find f 4g if f and (A) + (B) (C) (D) Q4: Find f if f g.. (A) (B) (C) (D) Q5: Obtain the differential coefficient of 0 Q6: Find y if y (A) (B) (C) (D) Q7: Find y y if y sin at 0 (A) (B) (C) 0 (D). (A) 4 (B) 4 (C) 4 (D) 4 Q8: The value of y at 0, given that 5 sin y is: (A) (B) (C) 0 (D) Q9: Evaluate lim 0 sin. (A) (B) 0 (C) (D) Q0: Obtain y if tan8 y. (A) cot 8 (B) sec (C) 6sec (D)6sec 8 Q: Find dy d if y 4 sin cot. (C) cos cos (D) cos cos ec ec (A) cos cos (B) cos sec ec Q: Find y if y cos sin sin cos cos e. (A) e (B) sin e (C) sin e (D) sin e Q: Calculate, given that y log cos y at 0. (A) 0 (B) (C) (D) Q4: Find 5 dy d given that y 5. (A) 5 ln5 (B) 5ln (C) 5 ln5 (D) 5 6

7 Q5: Find dy d y. if y y y y (A) (B) (C) (D) y y y y Q6: Find d y d given that t 4 and y t. Q7: Obtain the equation of tangent to the curve t (A) (B) (C) (D) t 8t 8t 6 y (A) y 4 4 (B) y 7 (C) y 7 (D) y 4 4 at Q8: Which of the following is true with regard to the stationary point(s) of the function? (A) ma :(,0) (B) min :(,0) (C) min :(,4) (D) ma(4, ) y. 5 Q9: Find the equation of Normal to the curve y at (A) 5y 6 (B) 5y 6 (C) 6y 4 (D) 5y 6. 4 Q40: A sphere is epanding in such a way that its volume v r cm s. Calculate the rate of increase of its radius, when the radius is 5cm. (A) cms (B) cms (C) cm s (D) 50cms I 5 d Q4: Evaluate (A) 5 k (B) k (C) k (D) 6 5 k. 8 6 Q4: Evaluate I tan d Q4: If is increasing at a uniform rate of (A) sec k (B) tan k (C) ln cos k (D) tan sec k. f d g k, then k is called a ---. (A) constant of variation (B) constant of a function (C) constant of integration (D) constant function. Q44: Integration is the reverse process of.. (A) a function (B) inverse function (C) anti-derivative (D) differentiation. Q45: If dy d a n a a a n n n I d gives: (A) (B) (C) (D). 6 0 b d f d (A) 4 (B) 8 (C) 0 (D) 4. a, then y (A) (B) (C) (D) Q46: Simplifying b Q47: If f 4, then 4 a n n n n c c c a n c. 7

8 Q48: Find dy d given that y y y (A) (B) (C) (D). y y y Q49: Evaluate dy d at, given that 4 4 y y 6 y 0. (A) (B) (C) (D). 4 4 Q50: Evaluate I d. 65 (A) ln 6 5 c (B) c (C) ln 6 5 c (D) ln c. I e sin d. Q5: Evaluate (A) cos (B) cos (C) sin (D) cos Q5: Find e c e c e c e c. dy d if y sin Q5: Find the area bounded by the curve (A) (B) (C) (D) y and the -ais with the ordinates (A) unit sq (B) unit sq (C) unit sq (D) 0 unit sq. 7 / Q54: Evaluate 4sin cos d. (A) 0 (B) (C) 45 (D). 0 /4 Q55: Evaluate 6 tan sec d. (A) (B) (C) (D) 4. 0 Q56: Find the area enclosed between the curves y 6 and y 5. (A) unit sq (B) unit sq (C) unit sq (D) unit sq Q57: Find the volume of the solid created when y is completely rotated about the. and. -ais with and. (A) unit sq (B) unit cube (C) unit sq (D) unit cube Q58: If y is completely rotated about the -ais with and, then find the volume generated. Q59: Evaluate y 9 9 (A) unit cube (B) unit sq (C) unit cube (D) unit sq I d, taking the constant of integration as zero. (A) (B) (C) (D). 8

9 Q60: Find I if I d. (A) (B) 4 (C) (D) none. Q6: Evaluate I d. (A) ln ln 5 c (B) ln 5 c 5 4 (C) ln 5 c (D) ln 5 c Q6: Evaluate I d. (A) ln c (B) ln k (C) ln c (D) ln k Q6: Find y if Q64: Find g f y if. f (A) (B) (C) (D). k k g. and Q65: Let the constant of integration be zero, then evaluate I e cos d. e e (A) cos sin (B) sin cos (C) e sin (D) e sin. Q66: Find the integral of ln with respect to. (A) (B) 6 9 (C) 6 (D). (A) ln c (B) ln c (C) ln c (D) ln c. Q67: Evaluate sin d. (A) sin c (B) sin c (C) cos sin c (D) cos sin c. Q68: Find y y if y cos. Q69: Evaluate (A) cos (B) sin (C) (D) 0. I d. (A) k (B) k (C) k (D) k. Q70: Evaluate I sin d. (A) cos (B) sin (C) sin (D) cos. c c c c 9

10 SOLUTIONS TO MAT OMEGA SEMESTER EXAMINATIONS_05/06 C A B 4C 5B 6A 7C 8C 9A 0B A B D 4C 5D 6A 7B 8A 9C 0B C D B 4A 5A 6D 7B 8C 9B 0D A D C 4C 5B 6A 7C 8B 9D 40B 4B 4C 4C 44D 45B 46A 47C 48D 49B 50A 5D 5A 5B 54D 55C 56D 57B 58D 59A 60B 6B 6C 6D 64A 65A 66D 67C 68D 69A 70C 0

11 COVENANT UNIVERSITY CANAANLAND, KM 0, IDIROKO ROAD P.M.B. 0, OTA, OGUN STATE, NIGERIA. TITLE OF EXAMINATION: 05/06 OMEGA SEMESTER EXAMINATION COLLEGE: SCIENCE AND TECHNOLOGY DEPARTMENT: MATHEMATICS SESSION: 05/06 SEMESTER: OMEGA COURSE CODE: MAT CREDIT UNIT: COURSE TITLE: VECTOR ALGEBRA INSTRUCTION: Attempt ALL questions. TIME: HOURS. Which epression is equivalent to the zero vector? A. QB YW BY B. CK KJ JC C. EU EP PU D. KJ KC JC. Which property of vectors is NOT correct? A. ab ba B. a ( b c) a b a c C. ( a b) c a ( b c) D. a b ab. If the two vectors u i p j k and v i 4jk are perpendicular, find the value of p. A. 4 5 B. 7 4 C. 7 4 D Which does NOT eist for a line in three-space? A. Vector equation B. Scalar equation C. Parametric equation D. Symmetric equation 5. Consider v, 4, and w,,. Determine a unit vector perpendicular to the plane containing v and w ij 5 A. i j B. i j C. i j D. 6. Let u,, and v,,. Determine the vector projection of u onto v A.,, B.,, C.,, D.,, If a,, and b 5,,, then the direction cosines of ( a b) are respectively: A.,, B.,, C.,, D.,, Find the angle between the vectors u,, and v,,. A. cos B. cos C. cos 9 D. cos 9. Tom has entered the wheelchair division of a marathon race. While training, he races his wheelchair up a 00-m hill with a constant force of 500 N applied at an angle of 0 to the surface of the hill. Find the work done by Tom.

12 A. 50kJ B J C J D. 4500kJ 0. Which plane does not contain the point P(0,,5)? A. 6y z 0 B. y z 0 C. y z 0 D. 4 5y z 0 0. Which one of the following is a scalar quantity? A. Force B. Position C. Mass D. Velocity. If the dot product of two non-zero vectors is 0, then the two vectors must be to each other. A. Parallel (pointing in the same direction) B. Parallel (pointing in the opposite direction) C. Perpendicular D. Cannot be determined.. If a 4,,9, b,8,5 and c 0, 7,, calculate a 5b c. A Given that B. u,, C.,, 4 D. 9, v,0, and w,,, find u( vw ). A. 8i 0j k B. 68 C. 8i 0j k D. 8, 0, 5. The value of sine of the angle between the vectors i j k and i j k A. B. C. D Find the area of the triangle whose vertices are having the position vectors i j4k, i 4jk, and 4i jk A. sq. units B. sq. units C. 8 sq. units D. sq. units 7. Find the area of the parallelogram that has u,, and v,, as adjacent sides. A. 5 B. 6 5 C. 5 6 D Calculate the volume of the parallelepiped whose three edges meeting at a point are i j4 k, i jk, i jk. A. 4 cubic units B. 7cubic units C. 7cubic units D. 7 cubic units 9. If the three vectors i jk, i jk and i j5k are coplanar, find the value of. A. 4 B. C. 0 D. 4 tan t 0. Find the limit (if it eists): lim t t t0 i j t k. A. k B. 0 C. k D. The limit does not eist. d. Given that r( t) t i t jt k and u( t) 4t i t jt k, find r( t) u ( t). dt A. t i (4 t t ) j ( t t )k B. 8 t i ( t 4 t ) j(t 4 t) k C. 4t i t jt k D. t (5t ). A vector field given by F (, y, z) is said to be rotational if A. F 0 B. F 0 C. F 0 D. F 0. Evaluate the definite integral: ( cos ) ( sin ) a t i a t j k dt. 0 A. ai a j B. ai a j ( ) k C. ai a j ( ) k D. ai a j ( ) k is

13 4. In the study of electricity and magnetism, a vector field that is divergenceless is said to be A. conservative B. irrotational C. solenoidal D. differential 5. Evaluate the definite integral: i t j t k. t 6 A. i j k B. j C. j 4 D. 0i 0 j 0k 6. Which of the following pairs are both vector quantities? A. force and acceleration B. velocity and speed C. acceleration and speed D. weight and mass 7. Find the parametric equations of the line passing through the points P(,,) and Q(,,4). A. t, y t, z t B. t, y t, z t C. t, y t, z t D. t, y t, z t t 8. Evaluate the definite integral: e i t j k dt. 0 A. e i 8 j k B. e i j C. 0.e i 8 j k D. ( e ) i 8 j k 9. For the vector-valued function, r( t) cos t i 4sin t j, find the angle between and as a function of t. r' () t 5sin tcost A. ( t) cos 6sin t 9 cos t 9sin t 6 cos t 7sin tcos t B. ( t) cos 6sin t 9cos t 9sin t 6cos t 7sin tcos t C. ( t) cos 6sin t 9cos t 9sin t 6cos t 9sin tcos t D. ( t) cos 9sin t 6 cos t 0. Find the unit tangent vector to the curve r( t) 4sin t i 4cos t j at t 4. A. B. C. D Find the domain of r( t) ln t i t jt k. i j i j i j i j A., B. 0, C. 0, D. 0, 4. Find the domain of the vector function, u A., 4 B., 4 C. 4, D., 7 ( t) sin t,ln t 4, t,. The position vector r ( t) cos t,sin t,t describes the path of an object moving in space. Find the velocity ( v () t ), speed ( v () t ) and acceleration ( a () t A. v( t) cos tsin t,sin tcos t,, v( t) sin tcost, a ( t) cos t(sin t cos t),sin t( cos t sin t), 0 ) of the object. r() t

14 B. v( t) cos tsin t,sin tcos t,, v( t) sin tcos t, a ( t) cos t(sin t cos t),sin t( cos t sin t), 0 C. v( t) costsin t,cos tsin t,, v( t) sin tcos t, a ( t) cos t(sin t cos t),sin t( cos t sin t), 0 D. v( t) cos tsin t, sin tcos t,, v( t) sin tcos t, a ( t) cos t(sin t cos t),sin t( cos t sin t), 0 t t t 4. The position vector of an object moving in space is given by r ( t) e cos t, e sin t, e. Find t t t t the speed of the object. A. e B. e C. e D. e 5. The vector product of two vectors is also known as A. Scalar product B. Cross product C. Dot product D. Inner product 6. Find a vector equation for the line passing through the point and is parallel to the vector. A. ( t) i ( t) j (5 t) k B. ( t) i ( t) j (5 t) k i j k C. ( t) i ( t) j (5 t) k D. ( t) i ( t) j (5 t) k 4 (,,5) 7. Find the equation of the plane containing the points (,,0), 0,, and,,. A. y 4z 6 B. y z C. y 4z 6 D. y 4z 6 8. Find f at 0,, given that the scalar field, A. 4, 0, B. 4, 0, C. 0,, D. 0,, 9. Calculate diva at,, if A. B. 0,, C. D. f y yz. A (, y, z) y, y, z. 40. Given that F(, y, z) i y jzk, is a constant. Then the divergence and curl of are respectively A. 0, B., 0 C., D., F 4. Which of the following is a vector equation of the line through the point, 5 with the direction vector,? A., y, 5 t, B., y, t 5, C., y, t, 5 D., y, 5 t, 4. Find the velocity of a particle whose displacement at time t, is r ( t) t, t, 8t (A) 8t 4 (B) 9t 4t 8 (C) t 4 (D) 6t 4t 8 i j i j k i j i j k d v( t) w ( t) at t if v( t) t,0, t dt and w( t) t,0,. A. 6 B. 4 C. D Evaluate 44. Evaluate ( ) ( ) integration C 0 : a t b t dt if a ( t ) t,0, and b ( t ) t,0, A. t B. 9 t C. t D. t, assuming the constant of

15 45. Calculate curl grad given that y. A. B. C. 0 D. y y 46. If A(, y, z) yi yz j yz k, determine A at the point (,,). A. 9 B. C. 5 D If F(, y, z) ( y z ) i ( y ) j yz k, determinefat the point (,, ). A. i 8j0 k B. i 8j0 k C. i 8j06 k D. i 8j06k d 48. Given that r( t) t i t j t k and u( t) 4t i t j t k, find ( t) ( t dt r g u ) A. 4t t t B. t t t C. 8t 9t 5 t D. t t 5 t A force i j k displaces a particle from the point to giving the work done 5. Find the value of λ. A. 5 B. C. D. (,,) (,,) 50. Find a set of symmetric equations of the line through the points P(,8,5) and Q 4,, A. y z B. y z C. y z D. y z

16 COVENANT UNIVERSITY CANAANLAND, KM. 0, IDIROKO ROAD P. M. B. 0, OTA, OGUN STATE. NIGERIA TITLE OF EXAMINATION: 05/06 OMEGA SEMESTER EXAMINATION COLLEGE: SCIENCE AND TECHNOLOGY DEPARTMENT: MATHEMATICS SESSION: 05/06 SEMESTER: OMEGA COURSE CODE: MAT CREDIT UNIT: COURSE TITLE: VECTOR ALGEBRA COURSE COORDINATOR: DR M.C. AGARANA COURSE LECTURERS: DR M.C. AGARANA, DR J.G. OGHONYON & MR O.O. AGBOOLA MARKING GUIDES / ANSWER KEY C D D 4B 5A 6A 7D 8B 9B 0B C C B 4C 5C 6B 7B 8C 9D 0C B C C 4C 5C 6A 7C 8D 9C 0D B C B 4D 5B 6B 7B 8D 9D 40B 4D 4D 4C 44D 45C 46B 47C 48C 49C 50B 6

17 COVENANT UNIVERSITY CANAANLAND, KM 0, IDIROKO ROAD P.M.B 0, OTA, OGUN STATE, NIGERIA TITLE OF EXAMINATION: B.Sc EXAMINATION COLLEGE: COLLEGE OF SCIENCE & TECHNOLOGY DEPARTMENT: MATHEMATICS SESSION: 05/06 SEMESTER: OMEGA COURSE CODE: MAT CREDIT UNIT: COURSE TITLE: STATISTICS II INSTRUCTION: ANSWER ANY THREE () QUESTIONS TIME: HOURS Question (a) Hospital records show that out of patients suffering from a certain disease, 75% die of it. What is the probability that out of 6 randomly selected patients, 4 will recover? (4 marks) (b) On the average, if three () applicants come to Covenant University for job interview monthly, find the probability that less than three () applicants attended job interview in Covenant University on a particular month. (6 marks) (c) The manufacturer of Hebron water claimed that the mean content of his product is 75cl per bottle with standard deviation of.5cl. A sample of 00 bottles of such product yielded a mean of 7cl. Comment on the validity of the manufacturer s claim. Use % level of significance. (6 marks) (d) Let X, X,, X n denote a random sample from a population having a mean μ and variance σ. Consider the following estimators of μ; θ = X + X + + X 7 7 θ = X X 6 + X 4 Show that θ and θ are both unbiased estimates. Question (a) Distinguish between Point Estimate and Interval Estimate. (b) State and discuss in details, the properties of a good estimator. (c) Define the term random variable. (d) Define the following terms; i. Type I Error ii. Type II Error (7 marks) ( marks) (4 marks) ( marks) 7

18 iii. P- Value iv. Power of a Test v. Level of Significance (5 marks) (e) Let X, X,... X n denote a random sample of size n. Show that X is an unbiased estimate of. ( marks) (f) If X is the sample mean of a random sample of size n from a normal population with known variance σ, derive the epression for a 00 % Confidence Interval on. (5 marks) (g) Define Statistics ( marks) Question (a) The customers accounts of a certain departmental store have an average balance of #0 for a period of 7 days and a standard deviation of #40. Find the probability that; i. the average account for a certain customer is over #50 (4 marks) ii. the average account for a certain customer is between #00 and #50 ( marks) (b) State the Central Limit theorem. ( marks) (c) In a sample of hip surgeries of a certain type, the average surgery time was 6.9 minutes with standard deviation of.6 minutes. Construct a 95% confidence interval for the mean surgery time for this procedure. (4 marks) (d) Define the following terms; i. Sample ii. Sampling iii. Probability Sampling iv. Non-probability Sampling (4 marks) 8

19 (e) State the properties of a probability density function (pdf). (f) State the properties of a probability mass function (pmf). (g) What are the disadvantages of Non-probability sampling? ( marks) ( marks) ( marks) Question 4 4. (a) Three judges scored 8 beauty contestants according to some observation attributes in them. The results are shown below. Contestants A B C D E F G H Judges Judges Judges i) Obtain Pearson Correlation Co-efficient for each pair of the judges? (6marks) ii) Obtain Co-efficient of determination for each pair of the judges? (mark) iii) Test for the significance of the correlation co-efficient at 5% level of significance. (4marks) iv) Obtain table of the rank data for each of the judges (marks) v) Obtain spearman rank correlation co-efficient for each pair of the judges? (6marks) vi) Which pair of the judges agrees most in their assessment of the contestant? (marks) Question 5 5. (a) In his eperiments with peas, Gregor Mendel observed that 5 were round and yellow, 08 were round and green, 0 were wrinkled and yellow, and were wrinkled and green. i) Draw the table of observed values from the information given. (mark) ii) Draw the table of epected values from the information given. (mark) iii) If according to his theory of heredity, the numbers should be in the proportion 9: : :, draw another table of epected value. (marks) iv) Based on table (ii), is there any evidence to doubt his theory at the 0.0 significance levels? (4marks) v) Based on table (iii), is there any evidence to doubt his theory at the 0.05 significance levels? (4marks) vi) Compare the result of question (iv) and (v) (mark) b) Given the information below, X Y i) Obtain the least-square regression line for the data? (4marks) ii) Obtain table of the residual for the data? (4marks) iii) Plot the graph for the residual information in question (ii) above? (marks)

20 COVENANT UNIVERSITY CANAANLAND, KM 0, IDIROKO ROAD P.M.B 0, OTA, OGUN STATE, NIGERIA. TITLE OF EXAMINATION: B.Sc EXAMINATION COLLEGE: Science & Technology SCHOOL: Natural & Applied Sciences DEPARTMENT: Mathematics SESSION: 04/05 SEMESTER: Omega COURSE CODE: MAT CREDIT UNIT: COURSE TITLE: Statistics II COURSE COORDINATOR: Mr. Odetunmibi O.A. COURSE LECTURERS: Mr. Odetunmibi O.A. & Mr. Oguntunde P.E. MARKING GUIDES QUESTION a) q p q n X 0.75, 0.5, 6, Pr 4? mark n n Pr X p q ; 0,,,..., n mark Pr X mark Pr X mark b) Pr X? X X X X Pr Pr 0 Pr Pr mark Recall that; ; 0,,,... mark! 0 e Pr X 0 0! mark e Pr X! Pr X e

21 0.494 mark e Pr X! mark Pr X Pr X 0.44 mark c) n 00, X 7,.5 Hypothesis: H : 75 0 Vs H : 75 mark Take 0.05 Test Statistic; X Z Z mark n 7 75 Z.96 mark Z.7 mark cal Z.96 ½ mark tab Decision Rule: Reject H 0 if Zcal Ztab ½ mark Decision: We do not reject 0 H since Z.7 Z.96 cal ½ mark Conclusion: The true mean content of the Hebron water per bottle is 75cl. ½ mark X X... X 7 d) 7 tab If is unbiased, then E ½ mark

22 E E X X X ½ mark 7 E X E X E X mark 7 7 mark is an unbiased estimate of For ; X X X 6 4 X X 6 X 4 E E mark EX EX 6 EX 4 mark mark mark QUESTION is also an unbiased estimate of. a) Distinction between Point Estimate & Interval Estimate A point estimate of a population parameter, say is a single numerical value of a statistic while an Interval estimate is an estimate of a population parameter given by two numbers between which the parameters may be considered to lie. marks b) Properties of a good estimator i. Unbiasedness ½ mark ii. Efficiency ½ mark

23 iii. Sufficiency ½ mark iv. Consistency ½ mark Discussion: i. Unbiasedness: An estimator is said to be an unbiased estimator of a population parameter if the mean or epected value of the statistic is equal to the parameter. i.e E(θ ) = θ. ½ mark ii. Efficiency: Let θ and θ be two unbiased estimators of θ. θ is said to be more efficient than θ if for a given sample size n, Var(θ ) < Var(θ ) ½ mark iii. iv. Sufficiency: An estimator is said to be sufficient if it make use of all the information contained in the population. Hence, a sufficient statistic is a statistic that summarizes all of the information in a sample about the desired parameter. ½ mark Consistency: Let θ be an estimator of θ, θ is said to be consistent if θ n θ as n. We can also say, as the sample size increases, θ n gets closer (or approaches) to θ. For any small ε > 0, lim n P( θ n θ ε) = 0 Equivalently; lim n P( θ n θ < ε) = ½ mark c) A random variable is a variable that can take on set of possible different values (similarly to other mathematical variables), but each with an associated probability, in contrast to other mathematical variables. marks d) i) Type I Error: This is the type of error committed when a null hypothesis that is true is being rejected. mark ii) Type II Error: This is the type of error committed when a false null hypothesis is being accepted. mark iii) P-value: This is the smallest level of significance that would lead to the rejection of the null hypothesis. mark iv) Power of a Test: This is the probability of rejecting the null hypothesis when the alternative hypothesis is true. mark 4

24 v) Level of Significance: This is the probability of committing Type I error. It is usually denoted by the Greek letter. mark e) If X.n n is unbiased of, then; EX mark X E X E i n X X... X n E n EX X... X n n E X E X... E X n n... n n mark Since EX, then, X f) Proof: Starting from; Z = X μ σ n is an unbiased estimate of. mark mark Pr { Zα X μ σ n Zα } = α mark Multiply the quantities inside the bracket by σ n Pr { Zα σ n X μ Zα Subtract X from each term; Pr { Zα σ n } = α σ X X σ μ X +Zα n X } = α n Multiply through by - Pr {X + Zα Hence; σ n μ X Zα σ n } = α marks 5

25 Pr {X Zα σ n μ X + Zα σ n } = α Therefore, 00( α)% C.I for μ is given by; X Zα QUESTION σ n μ X + Zα σ n mark a) n 7, 0, 40 ½ mark Pr X 50? Pr X, 600, 000 Pr X 50 ½ mark Recall that; X Pr Z z mark n 50 0 Pr X, 600, ½ mark Using the standard normal statistical table, Pr X ½ mark Pr X mark b) The central limit theorem (CLT) states that, let X, X,..., X n denote a sequence of random variables with a finite mean and a finite non-zero variance, the sampling distribution of the mean approaches a normal distribution with a mean of and a variance as n, the sample size increases. marks n c) n, X 6.9,.6 95% C.I on? 00 % C.I on is given by; X Z X Z mark n n Where; 00 % 90% 6

26 mark Z 95% C.I on is given by; mark Therefore; the 95% C.I on is; mark d) i) Sample: A sample is a subset of observations selected from a population. ii) Sampling: Sampling is the technique/method of selecting samples from a given population. iii) Probability Sampling: Probability sampling is a type of sampling in which every unit in the population has a chance (greater than zero) of being included in the selection. iv) Non-probability Sampling: Non-probability sampling is the sampling method where some elements of the population have no chance of being included in the selection. e) Properties of a Probability Density Function (pdf) i. f 0 ½ mark ½ mark ii. f d iii. Pr b a b f d ½ mark a iv. f is piecewise continuous ½ mark f) Properties of a Probability Mass Function (pmf) v. f 0 ½ mark ½ mark vi. f vii. Pr a b f b mark a g) Disadvantages of Non-probability Sampling It does not allow estimation of sampling errors. It gives rise to bias. mark mark 4A) TOPIC: CORRELATION ANALYSIS i) 7

27 Contestants y z A B C D E F G H y y 64 y 984 marks The correlation between Judge and Judge using the Pearson Correlation Coefficient formula is obtained as follows: r ny y [n( ) ( ) ][n( y ) ( y) ] /mark 8(6906) (468)(456) [8(768) (468) ][8(64) (456) ] mark The correlation between Judge and Judge using the Pearson Correlation Coefficient formula is obtained as follows: r ny y [n( ) ( ) ][n( y ) ( y) ] /mark 8

28 8(859) (468)(485) [8(768) (468) ][8(984) (485) ] mark The correlation between Judge and Judge using the Pearson Correlation Coefficient formula is obtained as follows: r ny y [n( ) ( ) ][n( y ) ( y) ] /mark 8(798) (456)(485) [8(64) (456) ][8(984) (485) ] mark ii) Co-efficient of determination between judge and judge. R r /mark Co-efficient of determination between judge and judge. R r /mark Co-efficient of determination between judge and judge. R r /mark (iii) HYPOTHESES BETWEEN JUDGE and : 9

29 H0: = 0 vs. HA: 0 /mark (no correlation) (correlation eists) The test statistic is t r r n mark =0.05, df = 8 - = 6 t/,df = t0.05, 6 =.447 (critical value) mark DECISION: Since our tcal (.45) < ttab (.447) we therefore do not reject H0 /mark CONCLUSION: There is no evidence of linear relationship between judges and at the 5% level of significance. mark HYPOTHESES FOR JUDGE AND : H0: = 0 (no correlation) vs. HA: 0 (correlation eists) The test statistic is 0

30 t r r n mark =0.05, df = 8 - = 6 t/,df = t0.05, 6 =.447 (critical value) mark DECISION: Since our tcal (.88) < ttab (.447) we therefore not reject H0 /mark CONCLUSION: There is no evidence of linear relationship at the 5% level of significance between judge and. mark HYPOTHESES FOR JUDGE AND : H0: = 0 (no correlation) vs. HA: 0 (correlation eists) The test statistic is t r r n mark =0.05, df = 8 - = 6

31 t/,df = t0.05, 0 =.447 (critical value) mark DECISION: Since our tcal (4.54) > ttab (.447) we therefore reject H0 /mark CONCLUSION: There is an evidence of linear relationship at the 5% level of significance between judge and. v) yzd zd yd dyz dz dy Rz Ry R Number 4 A B C D E F 0 0 G H 8 i i d 8 4 i i d 8 5 i i d marks v) 6 n( ) 8(5) 8(64 ) i i s d r n

32 marks Comment: There is relatively positive linear relationship or association between judge and but not strong direct relationship. mark r s 6 i d i n( n ) 8(4) 8(64) marks 0.49 Comment: There is relatively positive linear relationship or association between judge and but not strong direct relationship. mark r s 6 i d i n( n ) 8() 8(64)

33 marks Comment: There is relatively positive linear relationship or association between judge and. Also it has strong direct relationship. mark vi) Judge and agree most because they had highest spearman rank coefficient. mark 5A) TOPIC: CONTIGENCY TABLE i) Table of the observed values using the formula below: Round Wrinkle Total Yellow Green Total ii) Table of the epected values using this formula: marks e ij oi o n j Round Wrinkle Total Yellow Green Total marks iii) Table of the epected value according to the ratio 9::: using the formula below: 556 nij where is the ratio given in the question 6 4

34 HYPOTHESIS: Round Wrinkle Total Yellow Green Total Ho: The criterion used in purchasing a car is independent of gender. vs H: The criterion used in purchasing a car is dependent on gender mark marks The test statistic: /mark cal ( o e ) ij e i j ij ij ( ) ( ) (.49.5) cal cal 0.05 mark ᵡ tab = ᵡ (r-)(c-),(-α) = ᵡ (-)(-),(-0.0) = ᵡ,0.99 = 6.65 mark DECISION RULE: Reject H0 if ᵡ cal > ᵡtab otherwise accept H0 DECISION: Since our /mark (0.05) (6.65) we therefore do not reject H0 cal tab CONCLUSION: That is, at the percent level of significance, there is enough sample evidence to claim that the criterion used in purchasing a car is dependent on gender HYPOTHESIS: Ho: The criterion used in purchasing a car is independent of gender. vs 5

35 H: The criterion used in purchasing a car is dependent on gender mark The test statistic: /mark cal ( o e ) ij e i j ij ij (5.75) (0 04.5) ( 4.75) cal cal mark ᵡ tab = ᵡ (r-)(c-),(-α) = ᵡ (-)(-),(-0.0) = ᵡ,0.95 =.84 mark DECISION RULE: Reject H0 if ᵡ cal > ᵡtab otherwise accept H0 DECISION: Since our /mark (0.470) (.84) we therefore do not reject H0 cal tab CONCLUSION: That is, at the 5% percent level of significance, there is enough sample evidence to claim that the criterion used in purchasing a car is dependent on gender mark Number (X) (Y) y y

36 \ y y y 5407 marks The regression model of after (y) on before () administration of the contraceptive is given by: yi a b To get each parameter of the model we have: b n y y [n( ) ( ) ] /mark (5407) (800)(8) [(548) (800) ] /mark n /mark y 8 y n /mark a y b /mark (66.67) 5.8 / mark The linear regression model (regression equation) of (y) on () administration of the contraceptive is: ŷ () mark 7

37 8 RESIDUAL TABLE: Residual esty (Y) (X) SUM = 0

38 9