MATH1725 Introduction to Statistics: Worked examples

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{ 1 From the figure above, pr{0 < X + Y < z} = 2 z2 if 0 < z < 1, 1 − 1 2 (2 − z)2 if 1 ≤ z < 2. An alternative derivation uses integration. For example, in the case z < 1, ∫ ∫ ∫ z ∫ z−y ∫ z pr{0 < X + Y < z} = f(x,y)dxdy = dxdy = (z − y)dy = 1 2 z2 . 0
Answer: E[T] = E[a 1 X 1 + a 2 X 2 ] = a 1 E[X 1 ] + a 2 E[X 2 ] = a 1 µ + a 2 µ = (a 1 + a 2 )µ. If we require E[T] = µ, then a 1 + a 2 = 1, so that a 2 = 1 − a 1 . Since E[T] = µ, then T is said to be an unbiased estimator of the mean µ. Var[T] = Var[a 1 X 1 + a 2 X 2 ] = a 2 1 Var[X 1] + a 2 2 Var[X 2] = a 2 1 σ2 + a 2 2 σ2 = (a 2 1 + a2 2 )σ2 . Since a 2 = 1 − a 1 , Var[T] = {a 2 1 + (1 − a 1) 2 }σ 2 = (2a 2 1 − 2a 1 + 1)σ 2 . Differentiate this with respect to a 1 to find the minimum. d da 1 Var[T] = (4a 1 − 2)σ 2 , which is zero when a 1 = 1 2 . Hence Var[T] is a minimum when a 1 = a 2 = 1 2 so T = 1 2 (X 1 + X 2 ). Alternative derivation: write a 1 = 1 2 + ε, a 2 = 1 2 − ε. Then and is a minimum if ε = 0. Var[T] = (a 2 1 + a2 2 )σ2 = {( 1 2 + ε)2 + ( 1 2 − ε)2 }σ 2 = ( 1 2 + 2ε2 )σ 2 , What does this question show In part (a) you chose a 2 to restrict attention to linear combinations of the X i which were unbiased estimators of the mean µ, so E[T] = µ. In part (b) you then showed that of all such unbiased estimators, the sample mean ¯X is the one with smallest variance, so giving values closest to the true mean µ. Worked Example: Lecture 15. The following data give the noise level (in decibels) generated by fourteen different chain saws powered in one of two different ways. Petrol-powered chain saws 103 103 105 106 108 105 106 Electric-powered chain saws 97 95 94 93 91 95 94 At the 5% level of significance, test whether the average noise level of petrol-powered chain saws is higher than for electric-powered chain saws. Answer: Testing H 0 : µ 1 = µ 2 vs. H 1 : µ 1 > µ 2 , i.e. H 0 : µ 1 − µ 2 = 0 vs. H 1 : µ 1 − µ 2 > 0. Have two independent samples with unknown variance. Need to assume variances are equal. Worked Example: Lecture 15. The following data give the length (in mm.) of cuckoo (cuculus canorus) eggs found in nests belonging to wrens (A) and reed warblers (B). A: 19.8 22.1 21.5 20.9 22.0 21.0 22.3 21.0 20.3 20.9 B: 23.2 22.0 22.2 21.2 21.6 21.6 21.9 22.0 22.9 22.8 Assuming the variances for each group are the same, is there any evidence at the 5% level to suggest that the egg size differs between the two host species 9
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MATH1725 Introduction to Statistics: Worked examples

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