intro to probability statistics questions
Please see the attached file for the questions since what I pasted down here may show incorrectly on the website.
1. Consider the following simple linear regression modelyi = Î²0 +Î²1xi +Îµi
with Îµi independent errors with E [Îµi] = 0 and Var[Îµi] = Ïƒ2, for i = 1,…,n.Ë† Ë†âˆ’Ïƒ2âˆ‘ni=1xi
(a) Show that Cov Î²0,Î²1
= nâˆ‘ni=1 xi2âˆ’(âˆ‘ni=1 xi)2 .can be expressed as âˆ‘ni=1(xiâˆ’xÌ„)(yiâˆ’yÌ„).
âˆ‘ ni = 1 ( x i âˆ’ xÌ„ ) 2
(b) Show that the OLS Î²Ë†1
2. Consider the file â€œreading.csvâ€ included with this homework assignment as an attachment. The file contains average reading scores of third-graders from several elementary schools on a standardized test in each of two suc- cessive years (1982 and 1983).
(a) Plot the reading scores in 1983 (this is your outcome or response vari- able) versus the scores in 1982 (this is your covariate or predictor). Does the relationship appear linear? You can use the read.table orread.csv commands in R to read in the data.
(b) Use the following summary statistics to calculate (by hand) the OLS
of Î²0 and Î²1
(c) State the assumption of simple linearthese assumptions are satisfied when fitting a linear regression model to this dataset. Provide the necessary evidence that the assumptions are/are not satisfied.
(d) John took the exam in 1982 but not the exam in 1983. His score in 1982 was 250. Predict his score in 1983 according to the simple linear regression model. If it turns out John took the exam in 1983 and his score was 320, what is the error of the predicted value?
(e) Confirm your results in (b) by fitting the linear regression in R.
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