Sample Mean - Mathematics and Statistics - Old Exam Paper, Exams of Mathematical Statistics

Download Sample Mean - Mathematics and Statistics - Old Exam Paper and more Exams Mathematical Statistics in PDF only on Docsity! Cork Institute of Technology Bachelor of Engineering (Honours) in Structural Engineering- Stage 3 (NFQ Level 8) Summer 2007 Mathematics and Statistics (Time: 3 Hours) Instructions Answer FIVE questions, including at least one question from Section A.. Use separate answer books for each section. All questions carry equal marks. Statistical tables are available. Examiners: Mr. P. Anthony Prof. P. O’Donoghue Mr. D. O Hare Mr. T O Leary Section A 1. (a) The statement X ~       σ µ n N is fundamental to certain statistical inference procedures. (i) Put this statement into words, taking care to say what each of the symbols involved represents. (ii) A random sample of 16 observations is taken from a normal population with mean 220 and standard deviation 18. What is the probability that the sample mean is greater than 230? (7 marks) (b) In an air-pollution study, ozone measurements were taken in a city at 5.00 p.m. The following 12 readings (in parts per million) were obtained: 7.5, 11.1, 6.5, 8.4, 9.2, 12.5, 13.2, 8.9, 9.8, 10.8, 9.5, and 11.6. Establish 95% confidence intervals for the corresponding population mean and population variance. (8 marks) (c) (i) Distinguish between type 1 error and type 2 error as these terms apply in the context of hypothesis testing. (ii) In a z-test of H0: µ =54 versus H1: µ >54, the value of the test statistic is 2.37. If the standard error of the sample mean is 8.63, find the value of x and the p-value of the test statistic. What conclusion do you draw from the test? (5 marks) 2 2. (a) A study of the load-bearing properties of two materials gave the following data on breaking strength: Material 1 1 20n = 1 380x = 2 1 28.7s = Material 2 2 15n = 2 370x = 2 2 16.2s = Is there a significant difference in mean breaking strength between the two materials? Justify your answer by carrying out appropriate test(s) of hypothesis. (10 marks) (b) Corrosion of steel reinforcing bars is an important durability problem for reinforced concrete structures. Carbonation of concrete results from a chemical reaction that lowers the pH value by enough to initiate corrosion of the bar. Data on x = carbonation depth (mm) and y = strength (MPa) for a sample of core specimens taken from a particular building are shown below. x 8.0 38.0 16.5 45.0 50.0 27.5 30.0 59.0 40.0 y 22.8 19.5 23.7 13.2 10.3 18.6 16.1 12.0 12.4 The following quantities are calculated from the observed data: 792.089.3995.4621 0.26456.148130420.314 2 22 === ==== ∑ ∑∑∑∑ rSSExy yyxx (i) Plot the data on a scatter diagram. (ii) Deduce the value of the correlation coefficient and test the significance of the value you obtain. (iii) Determine the least squares regression line of y on x. (10 marks) 5 6. Select any three parts of the following parts (a) to (d). The maximum mark awarded for correctly answering this question is 20 marks. (a) A light beam is of span 4m and the deflection y at any point on a beam is found by solving the differential equation =2 2 dx ydEI -24(x-1)2U(x-1)+RU(x-2). By using Laplace Transforms solve this differential equation where the end x=0 is fixed. At the point x=4 the defection is zero. (5 marks) (b) (i) Find the Inverse Z-transform of 2 3 4z 8 (z-2) z+ (ii) By using z-transforms solve the difference equation yn+2-5yn+1+6yn=6(4n) y0=y1=0 (8 marks) (c) Using the Method of Frobenius find three terms in two series solutions of the differential Equation 0)y 4 1(xyxyx 22 =−+′+′′ . Write down the Maclaurin Series for cosx and sinx. Compare these series with those obtained in the solution to express the solution in terms of sinx and cosx. (8 marks) (d) Evaluate the surface integral ∫∫ S dS.nv ) where v is the vector v=3xzi+3yzj+z2k andn) is a unit vector (outward) to the surface S of the volume described by x2+y2 ≤ 4 0 ≤ z ≤ 3 (6 marks) 6 7. (a) Gauss’ Divergence Theorem states: If S is the surface of a closed bounded volume V then for a vector function a(x,y,z) the surface integral about S and the volume integral about V are related by the formula dSˆ S n.a∫∫ = ∫∫∇ V dV .a where n̂ is a unit vector normal (outward) to the surface S. (i) Verify that this theorem holds for the vector a=6xzi+6yxj+6z2k and where V is the prism with vertices (0,0,0), (1,0,0), (0,1,0), (0,0,4), (1,0,4) and (0,1,4). (ii) If the volume is of unit density find the moment of inertia of the volume about the z-axis. (15 marks) (b) Evaluate the triple integral ∫∫∫ + V )dV6(6x where V is the spherical region x2+y2+z2 ≤ 9. Note: In using spherical coordinates (r,θ,φ) the Jacobian is given by J=rsinφ. (5 marks) f(x) (x)f ′ a=constant sinx cosx cosx -sinx uv dx duv dx dvu + f(x) ∫ f(x)dx a=constant sinx -cosx cosx sinx 2sinAcosB=sin(A+B)+sin(A-B) 2cosAcosB=cos(A+B)+cos(A-B) 2sinAsinB=cos(A-B)-cos(A+B) sin(-A)=-sinA cos(-A)=cosA sin2A= 1 2 (1-cos2A) cos2A= 1 2 (1-cos2A) 7 Z-TRANSFORMS For a sequence f(n) the Z-Transform is defined by ∑ ∞ = −= 0n nf(n)zF(z) f(t) F(z) U(n)=1 1z z − Na az z − n 21)(z z − 2n 31)(z 1)z(z − + f(n)a n       a zF nf(n) -zF(z) f(n+1) zF(z)-zf(0) f(n+2) zf(1)f(0)zF(z)z 22 −−