TU Wien:Statistik und Wahrscheinlichkeitstheorie UE (Bura)/Übungen 2020W/HW11.5

Regression (part 2)

Continuation of exercise 4: Assume the linear regression model ${\textstyle Y_{i}=\beta _{0}+\beta _{1}x_{i}+\sigma Z_{i}(i=1,...,n)}$, with ${\textstyle Z_{1},...,Z_{n}}$ i.i.d. and ${\textstyle Z1\sim N(0,1)}$, and ${\textstyle (\beta _{0},\beta _{1},\sigma ^{2})\in \mathbb {R} \times \mathbb {R} \times \mathbb {R} ^{+}}$. Test the null hypothesis ${\textstyle H_{0}:\beta _{1}=0}$ (without lm()) at a significance level of 5%.

• a) Compute the standard error of the regression ${\textstyle S_{r}}$.
• b) Compute the standard error ${\textstyle SE_{b}}$, of the slope ${\textstyle \beta _{1}}$.
• c) Compute the value of the ${\textstyle t}$-statistc.
• d) Compute the ${\textstyle t}$-distribution associated with ${\textstyle H_{0}}$.
• e) Compute the ${\textstyle p}$-value.
• f) Formulate a result.

Lösungsvorschlag von Friday

--Friday Sa 30 Jan 2021 19:47:45 CET

5a)

calc_sr <- function(x,y,b0,b1) {
  n <- length(x)
  result <- 0
  # Create the sum
  for (i in 1:n) {
    result <- result + (y[i] - (b0 + b1 * x[i]))^2
  }
  result <- sqrt(result /(n-2))
}
sr <- calc_sr(x, y, b0, b1)   # 8.762359

5b)

n <- length(x)
seb <- sr/(sx * sqrt(n - 1))  # 0.08110709

5c)

t <- (b1-0)/seb   # 2.864922

5d)

alpha <- 0.05
q1 <- qt(1-alpha/2, df=n-2)   # 2.068658
I <- c(b1-q1*seb, b1+q1*seb)  # 0.0645827 0.4001483

5e)

p <- pt(-t, df=n-2) * 2 # 0.0087558512

5f)

The observed postive realation in the data is barely compatible with the null-hypothesis that there is no realation. If ${\textstyle H_{0}}$ holds true we observed a case which occourse under ${\textstyle 1\%}$ of the cases.