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e. Consider the multiple regression model y X 3+E. with E(e)-0 and var (e) ơ21 Assume that ε ~ N(0 σ21), when we test the hypothesis Ho : βί-0 against Ha : βί 0 we use the t statistic with n-k-1 degr...
For the following problems assume y-Xß + ε and assume E(e)-0 and var(e)-σ21 1. We are...
For the following problems assume y-Xß + ε and assume E(e)-0 and var(e)-σ21 1. We are often interested in estimating where r. is a (p+ 1)x 1 vector of predictors. We will find that a reasonable estimator is Find var(y*).
5. We wish to conduct a hypothesis test of the form Ho : μ1-2-0 vs Ha...
5. We wish to conduct a hypothesis test of the form Ho : μ1-2-0 vs Ha : μί-μ2 > 0. Both populations are assumed normal with equal variance and samples are assumed independent. We draw 15 observations for the first sample and 17 observations for the second sample. State the distribution of the test statistic for this test. (a) t-distribution with 32 degrees of freedom (b) t-distribution with 30 degrees of freedom (e) Normal distribution with 30 degrees of freedom...
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...
Exercise 1 Answer the following questions: a. Consider the multiple regression model y-Xe subject to a set of linear constraints of the form Cß-γ, where C is mx (k + 1) matrix. The Gauss-Markov condi...
Exercise 1 Answer the following questions: a. Consider the multiple regression model y-Xe subject to a set of linear constraints of the form Cß-γ, where C is mx (k + 1) matrix. The Gauss-Markov conditions hold and also ε ~ N(0, σ21) Is it true that we can test the hypothesis C9-γ using a k¿SSRfillm d SKreducedmodel Please explain b. Refer to question (a). Let H and Hi be the hat matrices of the full and reduced model respectively. Show...
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...
4. Suppose we run a regression model Y = β0+AX+U when the true model is Y-a0+ α1X2 + V. Assume th...
a,b,c,d
4. Suppose we run a regression model Y = β0+AX+U when the true model is Y-a0+ α1X2 + V. Assume that the true model satisfies all five standard assumptions of a simple regression model discussed in class. (a) Does the regression model we are running satisfy the zero conditional mean assumption? (b) Find the expected value of A (given X values). (e) Does the regression model we are running satisfy homoscedasticity? d) Find the variance of pi (given X...
Consider the hypothesis test Ho : 41 – 42 = 0 against Hp : H1 -...
Consider the hypothesis test Ho : 41 – 42 = 0 against Hp : H1 - 12 € 0 samples below: 1 36 40 31 33 33 30 31 30 39 38 30 37 36 30 39 32 36 39 11 34 29 33 33 32 29 30 38 32 34 30 30 29 33 34 35 Variances: 01 = 3.8, 02 = 2.9. Use a = 0.05. (a) Test the hypothesis and find the P-value. Find the test statistic....
Consider the hypothesis test Ho: 41-42 = 0 against H1: H1-H2 + 0 samples below: I...
Consider the hypothesis test Ho: 41-42 = 0 against H1: H1-H2 + 0 samples below: I 36 38 32 32 33 30 31 29393830373730 3930 34 40 II 32 3035 32 31 29 30 3731 34 30 30 30 32 33 34 Variances: 01 - 3.9, 02 = 2.4. Use a = 0.05. (a) Test the hypothesis and find the P-value. Find the test statistic. Round your answers to four decimal places (e.g. 98.7654). 2.0877 P-value = 0.0368 (b) Explain...
Consider the hypothesis test Ho: N1-N2 = 0 against Hi: 11-42 + 0 samples below: 1...
Consider the hypothesis test Ho: N1-N2 = 0 against Hi: 11-42 + 0 samples below: 1 37 38 31 33 33 30 31 29 39 38 30 38 37 30 39 31 35 38 II 32 29 35 32 32 29 30 37 31 34 30 30 30 32 33 34 Variances: 01 = 4.0, 02 = -0.5. Use 0 = 0.05. (a) Test the hypothesis and find the P-value. Find the test statistic. Round your answers to four decimal...
For the following problems assume y-Xß + ε and assume E(e)-0 and var(e)-σ21 1. We are often interested in estimating where r. is a (p+ 1)x 1 vector of predictors. We will find that a reasonable estimator is Find var(y*).
5. We wish to conduct a hypothesis test of the form Ho : μ1-2-0 vs Ha : μί-μ2 > 0. Both populations are assumed normal with equal variance and samples are assumed independent. We draw 15 observations for the first sample and 17 observations for the second sample. State the distribution of the test statistic for this test. (a) t-distribution with 32 degrees of freedom (b) t-distribution with 30 degrees of freedom (e) Normal distribution with 30 degrees of freedom...
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...
Exercise 1 Answer the following questions: a. Consider the multiple regression model y-Xe subject to a set of linear constraints of the form Cß-γ, where C is mx (k + 1) matrix. The Gauss-Markov conditions hold and also ε ~ N(0, σ21) Is it true that we can test the hypothesis C9-γ using a k¿SSRfillm d SKreducedmodel Please explain b. Refer to question (a). Let H and Hi be the hat matrices of the full and reduced model respectively. Show...
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...
a,b,c,d
4. Suppose we run a regression model Y = β0+AX+U when the true model is Y-a0+ α1X2 + V. Assume that the true model satisfies all five standard assumptions of a simple regression model discussed in class. (a) Does the regression model we are running satisfy the zero conditional mean assumption? (b) Find the expected value of A (given X values). (e) Does the regression model we are running satisfy homoscedasticity? d) Find the variance of pi (given X...
Consider the hypothesis test Ho : 41 – 42 = 0 against Hp : H1 - 12 € 0 samples below: 1 36 40 31 33 33 30 31 30 39 38 30 37 36 30 39 32 36 39 11 34 29 33 33 32 29 30 38 32 34 30 30 29 33 34 35 Variances: 01 = 3.8, 02 = 2.9. Use a = 0.05. (a) Test the hypothesis and find the P-value. Find the test statistic....
Consider the hypothesis test Ho: 41-42 = 0 against H1: H1-H2 + 0 samples below: I 36 38 32 32 33 30 31 29393830373730 3930 34 40 II 32 3035 32 31 29 30 3731 34 30 30 30 32 33 34 Variances: 01 - 3.9, 02 = 2.4. Use a = 0.05. (a) Test the hypothesis and find the P-value. Find the test statistic. Round your answers to four decimal places (e.g. 98.7654). 2.0877 P-value = 0.0368 (b) Explain...
Consider the hypothesis test Ho: N1-N2 = 0 against Hi: 11-42 + 0 samples below: 1 37 38 31 33 33 30 31 29 39 38 30 38 37 30 39 31 35 38 II 32 29 35 32 32 29 30 37 31 34 30 30 30 32 33 34 Variances: 01 = 4.0, 02 = -0.5. Use 0 = 0.05. (a) Test the hypothesis and find the P-value. Find the test statistic. Round your answers to four decimal...
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