M1 M1 A1 M1 A1 M1 A1 A1 A1 11 A1 2 B1 B1. B1 M1 Relative efficiency (y) = M1 A1 BEWARE PRINTED ANSWER. 5

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1 Q L e σ π ( W μ e σ π ( W μ M M A Product form. Two Normal terms. Fully correct. (ii ln L const ( W ( W d ln L ( W + ( W dμ 0 σ W σ μ W σ W W ˆ μ σ Chec this is a maximum. d ln L E.g. < 0 dμ σ σ σ μ σ μ μ + E( ˆ μ σ unbiased. σ μ 0 M A M A A A M Differentiate w.r.t. μ. BEWARE PRINTED ANSWER. A M A (iii Var( ˆ μ σ σ σ σ ( σ σ σ B B First factor. Second factor. Simplification not required at this point. (iv T W + W ( ( σ Var(T B Var( ˆ μ M Relative efficiency (y Var( T M σ σ σ σ ( σ σ ( σ σ + σ A Any attempt to compare variances. If correct. A BEWARE PRINTED ANSWER. 5 (v E.g. consider σ σ ( σ σ 0 M Denominator numerator, fraction E [Both μˆ and T are unbiased,] μˆ has smaller variance than T and is therefore better. E E

2 Q f ( x + x e! x, [ x > 0 -u Given: u m e du m! 0 ( > 0, integer 0] M X ( θ E[e 0 θx ]! + x!( θ e + ( θ x θ dx Put ( θx u u e u du M M M A A A A For obtaining this expression after substitution. Tae out constants. (Dep on subst. Apply given : integral! (Dep on subst. BEWARE PRINTED ANSWER. 7 (ii (iii Y X + X + + X n By convolution theorem:- mgf of Y is {M X (θ} n n +n B i.e. θ μ M (0 n + n n n M ( θ ( n n( θ ( M A n + n μ A σ M (0 μ n + n n n M ( θ ( n + n ( n n ( θ ( M M (0 ( n + n( n + n + / A n n n σ n + n n n ( + ( + + ( + n [Note that M Y (t is of the same functional form as M X (t with + replaced by n + n, i.e. replaced by n + n. This must also be true of the pdf.] M A 8 Pdf of Y is [for y > 0] n + n n + n y y ( n + n! e B B B One mar for each factor of the expression. Mar for third factor shown here depends on at least one of the other two earned. 3 (iv,, n 5, Exact P(Y > Use of N(5, 5 M M Mean. ft (ii. Variance. ft (ii.

3 0 5 P(this > 0 P N(0, > Reasonably good agreement CLT woring for only small n. A c.a.o. A c.a.o. E (E, E [Or other sensible comments.] 6

4 Q3 x 36.8 y 5.5 s s.89 s s B If all correct. [No mars for use of s n which are and.83 respectively.] Assumptions: Normality of both populations equal variances B B H 0 : μ A μ B H : μ A μ B B Do NOT accept X Y or similar. Where μ A, μ B are the population means. B Pooled + s Test statistic is B (.7 M A Refer to t 0. M No ft from here if wrong. Double tailed 5% point is 086. A No ft from here if wrong. Not significant. A ft only c s test statistic. No evidence that population mean times differ. A ft only c s test statistic. (ii Assumption: Normality of underlying B population of differences. H 0 : μ D 0 H : μ D > 0 B Do NOT accept D 0 or similar. Where μ D is the population mean of before after differences. B The direction of D must be CLEAR. Allow μ A μ B etc. Differences are 6.,., 3.9, -.0, 5.6, 8.8, -.8, M. ( x.8 s.6393 [A can be awarded here if NOT awarded in part ]. Use of s n (.3396 is NOT acceptable, Test statistic is M.9(6 A even in a denominator of Refer to t 7. M No ft from here if wrong. Single tailed 5% point is.895. A No ft from here if wrong. Significant. A ft only c s test statistic. Seems mean is lowered. A ft only c s test statistic. 0 s n n (iii The paired comparison in part (ii eliminates the variability between worers. E (E, E

5 Q Latin square. Layout such as: B Locations 3 5 I A B C D E Surf II B C D E A -aces III C D E A B IV D E A B C V E A B C D B B (letters paints Correct rows and columns. A correct arrangement of letters. SC. For a description instead of an example allow max out of. 3 (ii X ij μ + α i + e ij B μ population grand mean for whole experiment. α i population mean amount by which the i th treatment differs from μ. e ij are experimental errors ~ ind N(0, σ. (iii Totals are: 3, 35, 307, 355, 9 (each from sample of size 5 Grand total: Correction factor CF Total SS CF Between paints SS CF CF 6.96 Residual SS (by subtraction Source of SS df MS variation Between paints Residual Total MS ratio B B B B B B B B M M A B B M M A Allow uncorrelated. Mean. Variance. 9 For correct methods for any two SS. If each calculated SS is correct. Degrees of freedom between paints. Degrees of freedom residual. MS column. Independent of previous M. Dep only on this M.

6 Refer to F, 0 M No ft if wrong. But allow ft of wrong d.o.f. above. Upper 5% point is.87 A No ft if wrong. Significant. A ft only c s test statistic and d.o.f. s. Seems performances of paints are not all the same. A ft only c s test statistic and d.o.f. s.