2. Suppose we observe the pairs (X, Y), i-1, , n and fit the simple linear regression (SLR) model...
Suppose you fit the multiple regression model y = β0 + β1x1 + β2x2 + ϵ to n = 30 data points and obtain the following result: y ̂=3.4-4.6x_1+2.7x_2+0.93x_3 The estimated standard errors of β ̂_2 and β ̂_3 are 1.86 and .29, respectively. Test the null hypothesis H0: β2 = 0 against the alternative hypothesis Ha: β2 ≠0. Use α = .05. Test the null hypothesis H0: β3 = 0 against the alternative hypothesis Ha: β3 ≠0. Use α...
31. Suppose you fit a multiple linear regression model y = β0 + β1x1 + β2x2 + β3x3 + β4x4 + ε to n = 30 data points and obtain SSE = 282 and R^2 = 0.8266 a.) Find an estimate of s^2 for the multiple regression model (a) s^2 ≈ 30.9856 (b) s^2 ≈ 28.6021 (c) s^2 ≈ 1.3111 (d) s^2 ≈ 29.7938 (d) b.) Based on the data information given in a.), you use F-test to test H0...
Suppose you fit the multiple regression model y = β0 + β1x1 + β2x2 + ϵ to n = 30 data points and obtain the following result: y ̂=3.4-4.6x_1+2.7x_2+0.93x_3 The estimated standard errors of β ̂_2 and β ̂_3 are 1.86 and .29, respectively. Test the null hypothesis H0: β2 = 0 against the alternative hypothesis Ha: β2 ≠0. Use α = .05. Test the null hypothesis H0: β3 = 0 against the alternative hypothesis Ha: β3 ≠0. Use α...
1.The following tables give the results for the full model, as well as a reduced model, containing only experience Test Ho: ß,-Bs-0 vs HA: β2 and/or β3 # 0 Complete Model: Y-βο + β1X1 + β2X2 + β3Xs + ε ANOVA MS P-value df 76.9 Regression Residual Total 2470.4 823.5 224.7 2695.1 .0000 10.7 21 24 Reduced Model: Y = β0 + β X + ε MS df 1 23 24 value 2394.9 2394.9 183.5 0.0000 300.2 13.1 2695.1 Regression...
Question 2: Suppose that we wish to fit a regression model for which the true regression line passes through the origin (0,0). The appropriate model is Y = Bx + €. Assume that we have n pairs of data (x1.yı) ... (Xn,yn). a) From first principle, derive the least square estimate of B. (write the loss function then take first derivative W.r.t coefficient etc) b) Assume that e is normally distributed what is the distribution of Y? Explain your answer...
2) Suppose the regression model y = B0 + B1x1 + B2x2 + B3x3 + B4x1x2 + B5x1x3 + B6x2x3 was fit to n = 27 data points with SSE = 2000.0. a) Set up the null and alternative hypotheses for testing whether the interaction terms are significant. b) Give the reduced model necessary to test the significance of the interaction terms. c) The reduced model resulted in SSE = 2800. Calculate the value of the test statistic appropriate for...
Suppose we fit the simple linear regression model (with the usual assumptions) Y = Bo+B1X+ € and get the estimated regression model ♡ = bo+bix What aspect or characteristic of the distribution of Y does o estimate? the value of Y for a given value of X the total variability in Y that is explained by X the population mean number of Y values above the mean of Y when X = 0 the increase in the mean of Y...
2) Suppose the regression model y = B0 + B1x1 + B2x2 + B3x3 + B4x1x2 + B5x1x3 + B6x2x3 was fit to n = 27 data points with SSE = 2000.0. a) Set up the null and alternative hypotheses for testing whether the interaction terms are significant. b) Give the reduced model necessary to test the significance of the interaction terms. c) The reduced model resulted in SSE = 2800. Calculate the value of the test statistic appropriate for...
(Do this problem without using R) Consider the simple linear regression model y =β0 + β1x + ε, where the errors are independent and normally distributed, with mean zero and constant variance σ2. Suppose we observe 4 observations x = (1, 1, −1, −1) and y = (5, 3, 4, 0). (a) Fit the simple linear regression model to this data and report the fitted regression line. (b) Carry out a test of hypotheses using α = 0.05 to determine...
2. Consider a simple linear regression i ion model for a response variable Y, a single predictor variable ,i1.., n, and having Gaussian (i.e. normally distributed) errors: This model is often called "regression through the origin" since E(X) = 0 if xi = 0 (a) Write down the likelihood function for the parameters β and σ2 (b) Find the MLEs for β and σ2, explicitly showing that they are unique maximizers of the likelihood function Hint: The function g(x)log(x) +1-x...