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Exam Details

2019 Statistics Paper - II

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Questions 80
Duration 180 mins
Package National Defence Academy and Naval Academy Examination (I) & (II) - Previous Year Papers

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Difficulty distribution

Hard 27 33.8%
Easy 27 33.8%
Medium 26 32.5%

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Multiple Choices 80 100%

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1
2019 · Unclassified
2019 Statistics Paper - II
If y1, y2, y3 and y4 are independent with E(y1) = E(y2) = (θ1 + θ2), E(y3) = E(y4) = (θ1 + θ3), and V(yi) = σ2, i = 1, 2, 3 and 4. For estimability of parametric function l'θ = l1θ1 + l2θ2 + l3θ3, consider the following conditions : 1. l1 = l2 + l3 2. l2 = l1 + l3 3. l3 = l1 + l2 Which of the above condition(s) is/are correct ?
A
1 only
B
2 only
C
3 only
D
1, 2 and 3
2
2019 · Unclassified
2019 Statistics Paper - II
Let an n × 1 vector y ~ N(Xβ, σ2I) where X is an n × p matrix. Let H = (hij) = X(X' X)−1 X' be an n × n matrix. Let e = y - ŷ; ŷ = Xβ̂ be the n × 1 residual vector where β̂ is the least squares estimate of β. Then which one of the following is not correct ?
A
Rank(H) = p
B
0 < hii < 1; 1 ≤ i ≤ n
C
V(ŷ) = σ2 H
D
Cov(ei, ej) = σ2 hij
3
2019 · Unclassified
2019 Statistics Paper - II
For n observations (x1, y1), (x2, y2), ...., (xn, yn) two models were fitted. 1. yi = α + βxi + error; i = 1, 2, ..., n 2. yi = α + βxi + γxi2 + error; i = 1, 2, ..., n Let ŷi and ŷi* be the estimated values of yi from the two models and if E = ∑i=1n (yi - ŷi)2 and E* = ∑i=1n (yi - ŷi*)2, then which one of the following is correct ?
A
E ≥ E*
B
E ≤ E*
C
E = E*
D
It is possible that E = 0, E* > 0
4
2019 · Unclassified
2019 Statistics Paper - II
Let Y1, Y2, Y3 and Y4 be uncorrelated observations. It is given that 1. E(Y1) = β1 + β2 + β3 = E(Y2) E(Y3) = β1 - β2 = E(Y4) 2. V(Yi) = σ2; i = 1, 2, 3, 4 Define e1 = 1/√2 (Y1 - Y2) and e2 = 1/√2 (Y3 - Y4). Then an unbiased estimator of σ2 is given by
A
1⁄2 (e12 - e22)
B
1⁄2 (e12 + e22)
C
e12 + e22
D
1⁄4 (e12 + e22)
5
2019 · Unclassified
2019 Statistics Paper - II
Consider the following ANOVA table :
Source of variationDegrees of freedomSum of squaresMean SSTest statistic
Treatmentwxy5·22
Error12z25
Total14561
The values of w, x, y and z that complete the above table are respectively
A
2, 261, 130·5, 300
B
2, 300, 150, 261
C
2, 250, 125, 311
D
2, 311, 155·5, 250
6
2019 · Unclassified
2019 Statistics Paper - II
For estimating the mode of a normal distribution, which one of the following is the most efficient estimator ?
A
Sample mean
B
Sample median
C
Sample mode
D
Largest observation