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N(0,02). We wish to use a 1. [18 marks] Suppose X hypothesis single value X = x to test the null Ho : 0 = 1 against the...
18 marks] Suppose X~N(0,0). We wish to use a single value X hypothesis to test the null against the alternative hypothe...
18 marks] Suppose X~N(0,0). We wish to use a single value X hypothesis to test the null against the alternative hypothesis Denote by C aa) the critical region of a test at the significance level of -0.05 (a) 2 marks] What is the sample space S, the parameter space 9 space Θο of the test? and the null parameter (b) 12 marks) Computea (c) 12 marks Compute the power of the test (i.e., at 2) (d) [2 marks] Compute the...
Exercise 4.8: Suppose that X1, X2,..., Xn is a random sample of observations on a r.v. X, which takes values only in the range (0, 1). Under the null hypothesis Ho, the distribution of X is uniform o...
Exercise 4.8: Suppose that X1, X2,..., Xn is a random sample of observations on a r.v. X, which takes values only in the range (0, 1). Under the null hypothesis Ho, the distribution of X is uniform on (0, 1), whereas under an alternative hypothesis, њ, the distribution is the truncated exponential with p.d.f. 0e8 where 6 is unknown. Show that there is a UMP test of Ho vs Hi and find, roximately, the critical region for such a test...
5. Let X ~ N( ,o2) and assume ơ--3. We are to test the null hypothesis Ho: 10 against e) Let a -0...
5. Let X ~ N( ,o2) and assume ơ--3. We are to test the null hypothesis Ho: 10 against e) Let a -0.05. Design a test to accept or reject He (b) Determine and the power of your test. (c) Plot the probability density functions under the two hypotheses in the same coordinate systern and locate the critical value, acceptance and critical regions, and designate the quantities α and β graphically. revised hypothesis we have β 0.05. Then do (c)...
Suppose that X ~ POI(μ), where μ > 0. You will need to use the following fact: when μ is not too close to 0, VR ape x N(VF,1/4). (a) Suppose that we wish to test Ho : μ-710 against Ha : μ μί a...
Suppose that X ~ POI(μ), where μ > 0. You will need to use the following fact: when μ is not too close to 0, VR ape x N(VF,1/4). (a) Suppose that we wish to test Ho : μ-710 against Ha : μ μί are given and 10 < μι. m, where 140 and Using 2 (Vx-VHo) as the test statistic, find a critical region (rejection region) with level approximately a (b) Now suppose that we wish to test Ho...
2. A randon sample XI, X. is drawn frotn Normal(μ, σ2), where-oo < μ < oo and 0 < σ2 < x. To test the null hypothesis Ho : σ2-1 against the alternative H1: σ2 > 1, we have designed the...
2. A randon sample XI, X. is drawn frotn Normal(μ, σ2), where-oo < μ < oo and 0 < σ2 < x. To test the null hypothesis Ho : σ2-1 against the alternative H1: σ2 > 1, we have designed the following test Reject Ho if S>k where S2 = "LE:-1(x,-X)2, k ís a constant. Noticed that (n-1) distribution with degree of freedom 1 has a (a) Determine k so that the test will have size a. (b) Use k...
12. Consider a statistical inference that test the null hypothesis be Ho: c against H : esuch that c is a positive...
12. Consider a statistical inference that test the null hypothesis be Ho: c against H : esuch that c is a positive value. The test statistic associated with this mull hypothesis is given by t(b-c)/se(b) At significance level a, the test statistic is smaller than the critical value te(a/2, N - 2), that is iste(a/2, N- 2). Mark the correct alternative: (a) The test p-value increases if we increase c. (b) c does not belong to the estimated confidence interval...
P1 To test the hypothesis Ho: p = 1/2 against H :p < 1/2, we take...
P1 To test the hypothesis Ho: p = 1/2 against H :p < 1/2, we take a random sample of Bernoulli trials, X,X,..., X.,, and use for our test statistic Y = ¿x, which has a binomial distribution b(n, p). Let the critical region be defined by C ={y:y sc}. Find the values of n and c so that (approximately) a = 0.05 and B = 0.10 when p = 1/4. 1=1
Let X1,X be a random sample from an EXP(0) distribution (0 > 0) You will use the following facts for this question:...
Let X1,X be a random sample from an EXP(0) distribution (0 > 0) You will use the following facts for this question: Fact 1: If X EXP(0) then 2X/0~x(2). Fact 2: If V V, are a random sample from a x2(k) distribution then V V (nk) (a) Suppose that we wish to test Ho : 0 against H : 0 = 0, where 01 is specified and 0, > Oo. Show that the likelihood ratio statistic AE, O0,0)f(E)/ f (x;0,)...
Alejandra is using a one-sample t-test to test the null hypothesis Ho: u = 10.0 against...
Alejandra is using a one-sample t-test to test the null hypothesis Ho: u = 10.0 against the alternative H1: 4 < 10.0 using a simple random sample of size n = 10. She requires her results to be statistically significant at level a = 0.10. Determine the maximum value of t that will reject this null hypothesis. You may find this table of t-critical values useful. If you are using software, you may find this catalog of software guides useful....
---------------------------------------------------------------------------------------------------------------------------------------------------------- Reference: 11.2.3 Suppose that X. , X are...
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Reference:
11.2.3 Suppose that X. , X are iid MI, σ2) where σ (E 9t*) is the unknown parameter but μ(€ 9) is assumed known. With preassigned α ε (0. 1), derive a level α LR test for a null hypothesis Ho : σ.-a> 0) against an alternative hypothesis H, : σ2 σ1 in the implementable form. {Note: Recall from the Exercise 8.5.5 that no UMP level a test exists for testing Ho versus 8.5.5 Let X, X, be...
18 marks] Suppose X~N(0,0). We wish to use a single value X hypothesis to test the null against the alternative hypothesis Denote by C aa) the critical region of a test at the significance level of -0.05 (a) 2 marks] What is the sample space S, the parameter space 9 space Θο of the test? and the null parameter (b) 12 marks) Computea (c) 12 marks Compute the power of the test (i.e., at 2) (d) [2 marks] Compute the...
Exercise 4.8: Suppose that X1, X2,..., Xn is a random sample of observations on a r.v. X, which takes values only in the range (0, 1). Under the null hypothesis Ho, the distribution of X is uniform on (0, 1), whereas under an alternative hypothesis, њ, the distribution is the truncated exponential with p.d.f. 0e8 where 6 is unknown. Show that there is a UMP test of Ho vs Hi and find, roximately, the critical region for such a test...
5. Let X ~ N( ,o2) and assume ơ--3. We are to test the null hypothesis Ho: 10 against e) Let a -0.05. Design a test to accept or reject He (b) Determine and the power of your test. (c) Plot the probability density functions under the two hypotheses in the same coordinate systern and locate the critical value, acceptance and critical regions, and designate the quantities α and β graphically. revised hypothesis we have β 0.05. Then do (c)...
Suppose that X ~ POI(μ), where μ > 0. You will need to use the following fact: when μ is not too close to 0, VR ape x N(VF,1/4). (a) Suppose that we wish to test Ho : μ-710 against Ha : μ μί are given and 10 < μι. m, where 140 and Using 2 (Vx-VHo) as the test statistic, find a critical region (rejection region) with level approximately a (b) Now suppose that we wish to test Ho...
2. A randon sample XI, X. is drawn frotn Normal(μ, σ2), where-oo < μ < oo and 0 < σ2 < x. To test the null hypothesis Ho : σ2-1 against the alternative H1: σ2 > 1, we have designed the following test Reject Ho if S>k where S2 = "LE:-1(x,-X)2, k ís a constant. Noticed that (n-1) distribution with degree of freedom 1 has a (a) Determine k so that the test will have size a. (b) Use k...
12. Consider a statistical inference that test the null hypothesis be Ho: c against H : esuch that c is a positive value. The test statistic associated with this mull hypothesis is given by t(b-c)/se(b) At significance level a, the test statistic is smaller than the critical value te(a/2, N - 2), that is iste(a/2, N- 2). Mark the correct alternative: (a) The test p-value increases if we increase c. (b) c does not belong to the estimated confidence interval...
P1 To test the hypothesis Ho: p = 1/2 against H :p < 1/2, we take a random sample of Bernoulli trials, X,X,..., X.,, and use for our test statistic Y = ¿x, which has a binomial distribution b(n, p). Let the critical region be defined by C ={y:y sc}. Find the values of n and c so that (approximately) a = 0.05 and B = 0.10 when p = 1/4. 1=1
Let X1,X be a random sample from an EXP(0) distribution (0 > 0) You will use the following facts for this question: Fact 1: If X EXP(0) then 2X/0~x(2). Fact 2: If V V, are a random sample from a x2(k) distribution then V V (nk) (a) Suppose that we wish to test Ho : 0 against H : 0 = 0, where 01 is specified and 0, > Oo. Show that the likelihood ratio statistic AE, O0,0)f(E)/ f (x;0,)...
Alejandra is using a one-sample t-test to test the null hypothesis Ho: u = 10.0 against the alternative H1: 4 < 10.0 using a simple random sample of size n = 10. She requires her results to be statistically significant at level a = 0.10. Determine the maximum value of t that will reject this null hypothesis. You may find this table of t-critical values useful. If you are using software, you may find this catalog of software guides useful....
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Reference:
11.2.3 Suppose that X. , X are iid MI, σ2) where σ (E 9t*) is the unknown parameter but μ(€ 9) is assumed known. With preassigned α ε (0. 1), derive a level α LR test for a null hypothesis Ho : σ.-a> 0) against an alternative hypothesis H, : σ2 σ1 in the implementable form. {Note: Recall from the Exercise 8.5.5 that no UMP level a test exists for testing Ho versus 8.5.5 Let X, X, be...
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