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Finish the following question by showing your Code in R.

(μ , σ ). With yield reasonably close result as the R regression model b) Simulate at least 4 different simple linear regression scenarios by modifying the mean, the standard 3. Gven a simulated da taset X, and Y-A) + β, x-c, where X ~ N(3.6.2), and ~ the fixed parameters A,-50, and β1: 6, use the code provided in class to answer the following: a) Verify the explicit derivation of Bo and lm() deviation of distribution of e. Report their effect on g, Bo and B relative to its true value.

math   Statistics-And-Probability   
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X is generated from N( 3.6, ơ-2) taking sample size n-50 The linear model is Y-As + AX + ε, we generate ε from N (Ma-0, sample size 50 σ,-1) taking same a) It is given that β0-50 and β 6 and thus we get 50 observations on Y Now we run linear regression on R to obtain Po 49.47 and ß1 - 6.09 Thus, we see that the least square estimates are very close to that of the original values The r codes and output are as follows: n-50 > beta_0-50 beta 1 6; > x-rnorm(n,3.6,2) ;e rnorm(n,e,1) > y- beta_0+beta_1*xte 1m(yx) Call: lm(formula = y ~ x) Coefficients: (Intercept) 49.471 6.092 b) Now we take X as before but change ε from N(Ma-2, σ,-4) Running the regression on R we get the following least square estimates i. Po-52117 and A-5785 So we see that ß0 has increased by 2 due to the increase of/4 and A has not c hanged much but it has deviated more from the true value B1 6The R codes are given below n 50 > beta 0-50: beta 16 > x=rnorm(n, 3 . 6, 2) ; e=rnorm(n,2,4) y- beta_ 0+beta_1*xte >1m(yx) Call: 1m(formula y x) Coefficients: (Intercept) 52.117 5.785 we take ε from N(Ha-4, σ..8) The OLS estimators are ii. = 53.357 and A = 6.504 Estimate ofßo has even increased and its less accurate also estimate of A accu rateR Codes: > beta_-50 beta 1 6; x-rnorm(n,3.6,2) e rnorm(n,4,8) y= beta 0-beta 11x+e > Call: 1m(formula y x) Coefficients: (Intercept) 53.357 6.504 -0.5) from N (/4-5, ơ The estimates are iii.Taking ε = 55.08 and A = 5.97 Observe that βο is almost 5 more than the original value and A is even closer t o the actual value since ơ has decreased R Codes: n-50 > beta_-50 ; beta 1-6; > x rnorm(n, 3.6,2);e rnorm(n,5,0.5) > v beta 0+beta 1*xte 1m(yx) Call: lm(formula = y ~ x) Coefficients:(Intercept) 55.08 5.97 Taking ε from N(Ha-5, σε-0.1) The OLS estimates are iv. 55032 and A-5993 ^ Here due to low value of σΆ is very accurate and& almost 5 more than origi nal value Here are the R codes: > beta 0-50beta 16 > xrnorm (n, 3.6,2);e-rnorm(n,5,0.1) > y beta_@+beta_1*xte Call: lm(formula -y x) Coefficients: (Intercept) 55.032 5.993

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