Homework Answers
4. Consider the hypothesis test Ho: o rož vs. Hı: 0 <ož. Suppose that the sample...
4. Consider the hypothesis test Ho: o rož vs. Hı: 0 <ož. Suppose that the sample sizes are n1 = 5 and n2 = 10, and that S =23.2 and S2=28. Test this hypothesis using 5% significance.
1. Let X1,...,xn vid N (4,1). Suppose, we are interested in testing the following hypotheses Ho...
1. Let X1,...,xn vid N (4,1). Suppose, we are interested in testing the following hypotheses Ho : M = Mo VS H, :μ# μο The goal of this exercise is for you to understand that, in this setting, we can construct a good test, which is not however UMP. (a) Suppose you were testing Ho :μ = μo Vs H, :μ< μο. Show that the uniformly most powerful a-level test would reject H, if 21-a Χ4 μο vn Call this...
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...
n be a random sample from a Gamma distribution with (a) Show there exists a uniformly most powerful test for testing Ho vs H. Show that the critical region can be expressed as an inequality for Y-:-1...
n be a random sample from a Gamma distribution with (a) Show there exists a uniformly most powerful test for testing Ho vs H. Show that the critical region can be expressed as an inequality for Y-:-1X, that is it will have the form [Y>cor the form Y < c]. Explain which one of the two and why (b) Is there a uniformly most powerful test for testing Ho : θ 1 vs H1 : θメ1? axqplai
n be a...
4. (12 pts) Suppose that (Y, X:) satisfy the three assumptions we made in the regression...
4. (12 pts) Suppose that (Y, X:) satisfy the three assumptions we made in the regression analysis, and in addition, u; is N(0,0%) and is independent of Xi. A random sample of size n = 32 is drawn and yields Y = 43.2 + 61.5 x X, R' = 0.54, SER= 1.52 (10.2) (7.4) where the numbers in parentheses are the homoskedastic-only standard errors for the regression coefficients. (i) Construct a 99% confidence interval for 3. (ii) Test H, :...
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...
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 alternative hypothesis H1 0 2 Denote by C aat the critical region of a test at the significance level of : α-0.05. (f [2 marks] Show that the test is also the uniformly most powerful (UMP) test when the alternative hypothesis is replaced with H1 0 > 1 (g) [2 marks Show...
228. Suppose I test Ho: H = Ho vs. H : H > Ho with a...
228. Suppose I test Ho: H = Ho vs. H : H > Ho with a = 0.2. Write a line of R code to calculate the critical value for my test. 229. Suppose I test Ho: M = Mo vs. H: 4 <Mo with a = 0.15. Write a line of R code to calculate the critical value for my test. 230. Suppose I test H, : : x = 4 vs. Hy : pl > 4. My sample...
4. Exercise Let X, Y be RVs. Denote E[X] = Hy and E[Y] =py. Suppose we...
4. Exercise Let X, Y be RVs. Denote E[X] = Hy and E[Y] =py. Suppose we want to test the null hypothesis Ho : Mx = uy against the alternative hypothesis Hi : 4x > uy. Suppose we have i.i.d. pairs (X1,Yı),...,(Xn, Yn) from the joint distribution of (X,Y). Further assume that we know the X - Y follows a normal distribution. (i) Show that exactly) T:= (X-Y)-(ux-uy) - tn-1), Sin (3) where s2 = n-1 [?-,((X; – Y;) –...
FR2 (4+4+4 12 points) (a) Let XI, X2, X10 be a randoin sample from N(μι,σ?) and Yi, Y2, 10 , Y 15 be a random sample from N (μ2, σ2), where all parameters are unknown. Sup- pose Σ 1 (Xi X 2 0 321 (Y-...
FR2 (4+4+4 12 points) (a) Let XI, X2, X10 be a randoin sample from N(μι,σ?) and Yi, Y2, 10 , Y 15 be a random sample from N (μ2, σ2), where all parameters are unknown. Sup- pose Σ 1 (Xi X 2 0 321 (Y-Y )2-100. obtain a 99% confidence interval for σ of having the form b, 0o) for some number b (No derivation needed). (b) 60 random points are selected from the unit interval (r:0 . We want...
2. Suppose we observe the pairs (X, Y), i-1, , n and fit the simple linear regression (SLR) model...
2. Suppose we observe the pairs (X, Y), i-1, , n and fit the simple linear regression (SLR) model Consider the test H0 : β,-0 vs. Ha : Aメ0. (a) What is the full model? Write the appropriate matrices Y and X. (b) What is the full model SSE? (c) What is the reduced model? Write the appropriate matrix XR. (d) What is the reduced model SSE? (e) Simplify the F statistics of the ANOVA test of Ho B10 vs....
4. Consider the hypothesis test Ho: o rož vs. Hı: 0 <ož. Suppose that the sample sizes are n1 = 5 and n2 = 10, and that S =23.2 and S2=28. Test this hypothesis using 5% significance.
1. Let X1,...,xn vid N (4,1). Suppose, we are interested in testing the following hypotheses Ho : M = Mo VS H, :μ# μο The goal of this exercise is for you to understand that, in this setting, we can construct a good test, which is not however UMP. (a) Suppose you were testing Ho :μ = μo Vs H, :μ< μο. Show that the uniformly most powerful a-level test would reject H, if 21-a Χ4 μο vn Call this...
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...
n be a random sample from a Gamma distribution with (a) Show there exists a uniformly most powerful test for testing Ho vs H. Show that the critical region can be expressed as an inequality for Y-:-1X, that is it will have the form [Y>cor the form Y < c]. Explain which one of the two and why (b) Is there a uniformly most powerful test for testing Ho : θ 1 vs H1 : θメ1? axqplai
n be a...
4. (12 pts) Suppose that (Y, X:) satisfy the three assumptions we made in the regression analysis, and in addition, u; is N(0,0%) and is independent of Xi. A random sample of size n = 32 is drawn and yields Y = 43.2 + 61.5 x X, R' = 0.54, SER= 1.52 (10.2) (7.4) where the numbers in parentheses are the homoskedastic-only standard errors for the regression coefficients. (i) Construct a 99% confidence interval for 3. (ii) Test H, :...
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 alternative hypothesis H1 0 2 Denote by C aat the critical region of a test at the significance level of : α-0.05. (f [2 marks] Show that the test is also the uniformly most powerful (UMP) test when the alternative hypothesis is replaced with H1 0 > 1 (g) [2 marks Show...
228. Suppose I test Ho: H = Ho vs. H : H > Ho with a = 0.2. Write a line of R code to calculate the critical value for my test. 229. Suppose I test Ho: M = Mo vs. H: 4 <Mo with a = 0.15. Write a line of R code to calculate the critical value for my test. 230. Suppose I test H, : : x = 4 vs. Hy : pl > 4. My sample...
4. Exercise Let X, Y be RVs. Denote E[X] = Hy and E[Y] =py. Suppose we want to test the null hypothesis Ho : Mx = uy against the alternative hypothesis Hi : 4x > uy. Suppose we have i.i.d. pairs (X1,Yı),...,(Xn, Yn) from the joint distribution of (X,Y). Further assume that we know the X - Y follows a normal distribution. (i) Show that exactly) T:= (X-Y)-(ux-uy) - tn-1), Sin (3) where s2 = n-1 [?-,((X; – Y;) –...
FR2 (4+4+4 12 points) (a) Let XI, X2, X10 be a randoin sample from N(μι,σ?) and Yi, Y2, 10 , Y 15 be a random sample from N (μ2, σ2), where all parameters are unknown. Sup- pose Σ 1 (Xi X 2 0 321 (Y-Y )2-100. obtain a 99% confidence interval for σ of having the form b, 0o) for some number b (No derivation needed). (b) 60 random points are selected from the unit interval (r:0 . We want...
2. Suppose we observe the pairs (X, Y), i-1, , n and fit the simple linear regression (SLR) model Consider the test H0 : β,-0 vs. Ha : Aメ0. (a) What is the full model? Write the appropriate matrices Y and X. (b) What is the full model SSE? (c) What is the reduced model? Write the appropriate matrix XR. (d) What is the reduced model SSE? (e) Simplify the F statistics of the ANOVA test of Ho B10 vs....
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