Problem 4: Suppose \( X_1 \) and \( X_2 \) are independent with \( E(X_1) = E(X_2) = \mu \), \( \text{Var}(X_1) = 1 \), and \( \text{Var}(X_2) = 2 \). Your goal is to estimate the unknown mean \(\mu\). Consider the estimator:\[\hat{\mu}_1 = \frac{X_1}{5} + \frac{4X_2}{5}\](a) What is the mean-square error \( \text{MSE}(\hat{\mu}_1) \)?(b) Consider an unbiased estimator of the form: \( \hat{\mu}_2 = aX_1 + (1-a)X_2 \). What value should be chosen for the constant \( a \) in order to minimize the mean squared error of \( \hat{\mu}_2 \)?(c) For the value of \( a \) found above, compute the relative efficiency of \( \hat{\mu}_2 \) with respect to \( \hat{\mu}_1 \). (If you are unsure about the value of \( a \) you found in part (b), you may leave the answer in terms of \( a \).)

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Answered: Problem 4 Suppose X₁ and X₂ are…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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B4
Question 2. Show that, for a simple linear regression, R² = ry, where R² is the coefficient of determination and rxy is the sample correlation between Y and X. Hint: Σ" (Υ - Y)2 Σ1(Y₁-Y)²¹ SxY √SXXSYY' where Sxy = 1 (X; — X)(Y; — Y), Sxx = ₁1(X; - X)², Syy = ₁₁(Y₁ - Y)². R² = "XY =
N= 150 observations were collected on a time series that was identified as a AR(2) time series. The following statistics were computed from the data. Mean - 45.0 Variance 15.6 Autocorrelation function (up to lag 5) r1 = 0.80, r2 = .50, rз = .26, r4 = -.10, rs = 0.08: == Estimate the parameters of the model using the method of moments.
(b) For a given set of trivariate data, the following results were obtained: X = 53, Y = 28, 6 (Y on X) =- 1.5, d (X on Y)=- 0.2. Find : (i) The two regression equations. (ii) the coefficient of correlation. (iii) The most probable value of Y when X=60.
"given a simple regression with slope b=3, s (sub y)=8, and s (sub x)= 2, and n=30. Find the standard error of the estimate."
The following Table summarizes the main estimation results for the below model: C- Bo + B,R + Balet E.where (C) denotes total household Consumption in SL, (R) is the national Incorme, and (1) denotes the Taxes collected from the private sector. The time span goes from 1974, 1st quarter, to 2010, 4th quarter and all the variables are expressed in million Rupees. Dependent variable C Constant R. Coefficient Sd. Eror t-statisties 128 40 3.2 0.90 0.48 0.67 6.0 32 0.15 0.15 R squared Residual Std. 1.35 Deviation Given the information in above table and the null that the Given the information in above table and the null that the Taxes collected from the private sector (B2) is not equal to zero, O a. There is not enough information to test this hypothesis. O b. The test statistic can be computed and the null is rejected in favor of the alternative hypothesis with a 5% significance. O . None cof the these O d. The test statistic can be computed and the null is not rejected in favor of the…
Suppose you obtain the following regression model, E[y]=91+79*x1 +20*x1*x2. What is the impact of a 39 unit change of x1 on the expected value of y when x2 is at its mean of 40?
13
An investigation into the relationship between an adolescent mother's age x in years and the birth weight y of her baby in grams yielded the regression equation y= - 1163.45 + 245.15x as well as r = .88369, r2= .78091, SSE = 337212.45, and s= 205.30844 1) What is the predicted birth weight for a baby brn to a 17 year old woman? 2) What is the propotion of the variability in the weights of babies born to adolescent mothers that is accounted for by the mother's age? 3) For every additional year in the mother's age that mean birth weight of the baby? (a) increases by about 245g (b) decreases by about 245g (c) increases by about 1163g (d) increases by about 1163g (e) changes by an amount that cannot be determined from the information given.
Please help me with this problem using the TI 84 You run a regression analysis on a bivariate set of data (n=85n). With ¯x=21.3 and ¯y=25.4, you obtain the regression equation y=−3.558x+48.067 with a correlation coefficient of r=−0.022. You want to predict what value (on average) for the response variable will be obtained from a value of 150 as the explanatory variable.What is the predicted response value?y =
18. Because ethanol contains less energy than gasoline, a researcher wants to determine if the mileage of a car (y) is affected by the percent ethanol in the gasoline (x). The true population regression line relating the mean mileage y for a given ethanol content x (in percent) is μy|x = β0 + β1x. In a controlled environment, the mileage of a car is recorded when refueled 7 times using between 0% and 10% ethanol. The least squares regression line was found to be yˆ = βˆ0 + βˆ1x = 32.96 − 0.250x. We wish to test H0 : β1 = 0 versus Ha : β1 < 0 at the α = 0.05 significance level (i.e. we want to determine if the addition of ethanol significantly decreases mean gas mileage). What is the p−value for this test? What is your conclusion? Note: s?e(βˆ1) = 0.0485. Which of the following is/are true? (A) In reference to question (18), it would be appropriate to approximate the mean mileage when the car is using 85% ethanol (B) In reference to question (18), a 95% confidence interval for β1…

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