Question 3 Suppose that the estimated simple linear regression of a response Y on a predictor...

Question 3 Suppose that the estimated simple linear regression of a response Y on a predictor X based on n = 6 observations produces the following residuals: resid <- c(-0.09, 0.18, -0.27, 0.16, -0.06, 0.09) Note: For this question, all of the computations should be performed “by-hand”.

(a) (1 point) What is the estimate of σ 2?

(b) (2 points) Further, you know that the estimated regression parameters are βˆ 0 = −0.54 and βˆ 1 = 0.08. Additionally, the sample mean of X is 13.5 and the sample variance of X is 15.5. Find the standard errors of the two estimated regression parameters.

(c) (2 points) Use the standard error you calculated in (b) to test the null hypothesis H0 : β1 = 0 that the true population slope has value 0. State your test statistic, two-sided p-value, and what you conclude from the test.

(d) (1 point) What is the mean value of Y we would predict when X = 12? (E.g., what is µˆ(Y |X = 12)?)

(e) (2 points) Calculate the standard error of µˆ(Y |X = 12) and use this value, along with your result from part (d), to find a 95% confidence interval for mean Y when X = 12.

(f) (2 points) Calculate the standard error of P red(Y |X = 12) and use this value, along with your result from part (d), to find a 95% prediction interval for mean Y when X = 12.

(g) (1 point) The sample correlation between X and Y is 0.8831. Find the value of R2 for the regression considered in the previous parts of this question.

Expert Solution

orchestra answered 1 year ago

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Question 3 Suppose that the estimated simple linear regression of a response Y on a predictor...

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