ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 1 of 165
Appeared in Year: 2011
Describe in Detail
Show that the square of the one sample t-statistic has the F-distribution. What are its degrees of freedom?
The t-statistic is defined as the ratio of a standard normal variable X~N (0,1) and the square root of where Y~ and n is the degree of freedom.
Then we show the square of t-statistic follows F-distribution.
We known that X is standard normal distribution, then X 2 follows a chi-square distribution with degree of freedom is 1. Let assume Z=
The joint p. d. f. of Y and Z is
The range of Y and Z is greater than 0
The Jacobian transformation
Then the range of Z > 0 and f > 0
The p. d. f. of f and y is
Integrating it with respect to y, we get
This integral is a gamma distribution by multiple and dividing some constant.
This is a f distribution with degree of freedom is 1 and n.