Download Tutorial Sheet 4

Department of Mathematics
MAL 522 (Statistical Inference)
Tutorial Sheet No. 4
1. Let s = 0.68 and x = 3.42 be the standard deviation and the mean respectively for a sample of size n = 16.
Assume that the population is a normal distribution with mean µ and variance σ 2 . Find 90% confidence
interval for µ.
2. In estimating the mean of a normal distribution N (µ, σ 2 ) having known standard deviation σ by using a
confident interval based on a sample of size n, atleast how large should n be in order for the probability to
be 0.99 that the confidence interval for µ be of length not greater than L?
3. Using asymptotic distribution of MLE, find a (1 − α)100% confidence interval for parameter θ of exponential
distribution.
4. Find a shortest length confidence interval for mean µ of N (µ, σ 2 ) population when σ 2 is unknown. Also, find
shortest length confidence interval for σ 2 .
5. Suppose that the variance of the speeds with which men and women can perform a certain task are respectively
σ1 = 12 seconds and σ2 = 15 seconds. If 20 men and 25 women required on the average 29.4 seconds and
32.5 seconds to perform the given task, obtain a 0.95 confidence interval for the difference between the true
average times it takes men and women to perform the task.
6. It is desired to test the hypothesis µ = 0 against the alternative µ > 0 on the basis of a random sample of size
9 from a normal population with variance σ 2 = 1. Show that X > 0.78 is the critical region of size α = 0.01.
7. A sample of size 1 is taken from an exponential pdf f (x, θ) = θ1 e− θ x , x > 0. To test H0 : θ = 1 against
H1 : θ > 1, the following test is used. Reject H0 if X > 2, otherwise accept H0 . Find the size, the level of
significance and the power function of the test.
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8. Let X1 , X2 be samples from U (θ, θ + 1) distribution. For testing H0 : θ = 0 against H1 : θ > 0 we have two
competing non
{ randomised tests.
{
1, X1 > 0.95
1, X1 + X2 > c
(I) ϕ1 (X1 ) =
(II) ϕ2 (X1 , X2 ) =
0, X1 ≤ 0.95
0, X1 + X2 ≤ c
(a) Choose c so that ϕ2 has the same size as ϕ1 .
(b) Find power function of each of the two tests.
9. A sample of size 1 is taken from Poisson distribution with mean λ. To test H0 : λ = 1 against H1 : λ = 2,
consider the nonrandomised test ϕ(X) = 1, if X > 3 and = 0 if X ≤ 3. Find probabilities of type I and type
II errors and the power of the test against λ = 2. If it is required to achieve the size of test equal to 0.05,
how should one modify the test ϕ to a randomised test.
10. An urn contains 10 marbles of which M are white and 10 − M are black. To test that M = 5 against
the alternating hypothesis that M = 6, one draws 3 marbles from the urn without replacement, the null
hypothesis is rejected if the sample contains 2 or 3 white marbles; otherwise it is accepted. Find the size and
the power function of this test.
11. Using a random sample X1 , X2 , . . . , Xn from Bernoulli distribution with parameter
it is desired to test
∑p,
n
H0 : p = 0.49 against H1 : p = 0.51. Consider a test that rejects H0 when
i=1 Xi is large. Find
approximately the sample size needed in order that the probabilities of errors of both types are approximately
0.01 (Use central limit theorem).
12. A sample of size 1 is taken from pdf f (x, θ) = θ22 (θ − x), 0 < x < θ. Find a MP test of size α for the
hypothesis H0 : θ = θ0 against H1 : θ = θ1 (θ1 < θ0 ).
13. Find Neyman-Pearson size α test of H0 : β = 1 against H1 : β = β1 (> 1) based on a sample of size 1 from
f (x, β) = βxβ−1 , 0 < x < 1.
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14. Let α be a real, 0 < α < 1 and ϕ∗ be a MP test of size α for testing H0 against H1 . Also let β = EH1 (ϕ∗ (X)) <
1. Show that 1 − ϕ∗ is an MP test for testing H1 against H0 at level 1 − β.
15. Let X1 , X2 , . . . , Xn be a random sample from B(1, p). Find a MP test of size α = 0.0547 of the hypothesis
H0 : p = 0.5 against H1 : p = 0.25. Find the power of the test so obtained. For what α level does there exist
a nonrandomised MP test. Is this test UMP for H0 : p ≤ 0.5 against H1 : p > 0.5?
16. Show that each of the following families has a MLR in a statistic T and find T . (a) N (θ, σ 2 ), σ 2 known.
Poisson distribution with parameter λ.
(c) B(n, θ), n known.
(b)
17. Find a UMP test for H0 : θ ≤ θ0 against H1 : θ > θ0 based on a sample of size n from pdf/pdf
−θ x
1
f (x, θ) = e x!θ , x = 0, 1, . . . , θ > 0 (b) f (x, θ) = θ1 x θ −1 , 0 < x < 1, θ > 0.
(a)
18. Show that f (x, θ) =
H1 : θ < θ0 .
θ
x2 , x
> θ, θ > 0 has MLR in X(1) and obtain a UMP test for H0 : θ ≥ θ0 against
19. For a sample of size 1 from double exponential distribution with pdf f (x, θ) = 12 exp− | x − θ |, −∞ < θ < ∞,
construct a UMP test of level α = 0.05 for H0 : θ ≤ 0 against H1 : θ > 0.
20. Find a UMP size α test for H0 : θ ≥ θ0 against H1 : θ < θ0 based on a sample of n observations from the
pdf (a) N (θ, 1) (b) Gamma(θ, 1). Verify that UMP test is unbiased.
21. Using a UMP test for H0 : θ = θ0 against H1 : θ < θ0 construct a (1 − α)100% confidence interval for mean
θ of exponential distribution.
χ29,0.05
Table Values
P( Z is a standard normal distribution ≥ Zα ) = α
P( χ2 r.v. with n degrees of freedom ≥ χ2n,α ) = α
P( t r.v. with n degrees of freedom ≥ tn,α ) = α
P( F r.v. with n1 and n2 degrees of freedom ≥ Fn1 ,n2 ,α ) = α
Z0.025 = 1.96; Z0.05 = 1.645; Z0.0764 = 1.43; Z0.01 = 2.33; Z0.035 = 1.81
= 16.917; χ26,0.05 = 12.6; χ25,0.05 = 11.1; χ24,0.05 = 9.48; χ23,0.05 = 7.81; χ22,0.05 = 5.99; χ29,0.95 = 3.334
t8,0.025 = 2.31; t9,0.025 = 2.26; t10,0.025 = 2.22
F9,11,0.025 = 0.1539; F10,15,0.025 = 3.5217; F15,10,0.025 = 3.0602
Note: If above table values are not matched, please leave the answer without numerical.
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