Problem 1 and 2 Assistance

The t distribution

For problems 1 and 2, there is no data. Only t.

What is t?

• The mean of t is zero.
• The metric of t is standard errors.
• t is entirely defined by degrees of freedom [\(n-1\) for the mean]

For these two problems, we are given the characteristics of a t distribution – the degrees of freedom – and a probability. With those two things, we should be able to query R to find the required t values.

par(mfrow=c(1,2))
  .x <- seq(-3.745, 3.745, length.out=1000)  
  plotDistr(.x, dt(.x, df=24), cdf=FALSE, xlab="x", 
  ylab="Density", 
  main=paste("t:  Degrees of freedom=24"))
  plotDistr(.x, pt(.x, df=24), cdf=TRUE, xlab="x", 
  ylab="Cumulative Probability", 
  main=paste("t:  Degrees of freedom=24"))
  abline(v=0, col="red")
  abline(h=0.5, col="red")

Question 2

Here, the two inputs change, we have different degrees of freedom and a different probability.

par(mfrow=c(1,2))
  .x <- seq(-3.745, 3.745, length.out=1000)  
  plotDistr(.x, dt(.x, df=15), cdf=FALSE, xlab="x", 
  ylab="Density", 
  main=paste("t:  Degrees of freedom=15"))
  plotDistr(.x, pt(.x, df=15), cdf=TRUE, xlab="x", 
  ylab="Cumulative Probability", 
  main=paste("t:  Degrees of freedom=15"))
  abline(v=0, col="red")
  abline(h=0.5, col="red")