# topic 11 e   the Student's-t
#
     # for Student's-t with 7 degrees of freedom
     # find P( X < -1.43 )
pt( -1.43, 7 )
     # for Student's-t with 15 degrees of freedom
     #  1) find P( X < -1.3 )
pt( -1.3, 15 )
     #  2) find P( X > 2.6 )
1 - pt( 2.6 ,15 )    # old way
pt( 2.6, 15, lower.tail = FALSE)    # better way
     #  3) find P( X < -1.83 or X > 1.54 )
pt( -1.83, 15 ) + (1 - pt( 1.54, 15) )  #old way
pt( -1.83, 15 ) + pt( 1.54, 15, 
                      lower.tail=FALSE ) #better way
     #  4) find P( X < -2.45 or X > 2.45 )
pt( -2.45, 15 ) + (1 - pt( 2.45, 15) )  #old way
pt( -2.45, 15 ) + pt( 2.45, 15, 
                      lower.tail=FALSE ) #better way
           # or, using symmetry
2 * pt( -2.45, 15 )
     #  5) find P( 0.48 < X < 1.76 )
pt( 1.76, 15) - pt( 0.48, 15 )
     #  6) find P( -2.37 < X < 2.37 )
pt( 2.37, 15) - pt( -2.37, 15 )
         #  or, using the complement 
1 - 2*pt( -2.37, 15 )
     # For Student's-t with 11 degrees of freedom
     # find the value of  y  such that
     #  7) P( X < y ) = 0.33
qt( 0.33, 11 )
     #  8) P( X > y ) = 0.075
qt( 1 - 0.075, 11 )   # the old, ugly way
qt( 0.075, 11, lower.tail = FALSE )    # the better way
     #  9) P( X < -y  or X > y ) = 0.08
-qt( 0.08/2, 11 ) # note the leading - 
qt( 0.08/2, 11, lower.tail=FALSE)  # more clear statement
     # 10) P( -y < X < y ) = 0.033
qt( (1 - 0.033)/2, 11, lower.tail=FALSE)
     # when mean=154.3 and standard deviation = 13.2
     # and for 24 degrees of freedom, find P( X < 140.0 )
t <- (140 -154.3)/13.2
pt( t, 24 )
     # when mean=483.2 and standard deviation=26.4
     # with 27 degrees of freedom, find  y  such 
     # that P( X < y ) = 0.15.
t <- qt( 0.15, 27 )
t
t*26.4 + 483.2
##################################
######     B  O   N   U   S   ####
##################################
#
#   It seems to be a real shame that pt() and qt()
#   do not have the mean= and sd= parameters that we 
#   had in pnorm() and qnorm().  If it is important 
#   for us to have such parameters in pt() and qt() 
#   then we can just define new functions that do 
#   have them.
pt_full <- function( t_val, df_val, mean_val=0, sd_val=1, ...)
{  t_stand <- (t_val  - mean_val)/ sd_val
pt( t_stand, df_val, ...)
}

qt_full <- function( t_area, df_val, mean_val=0, sd_val=1, ...)
{  t_stand <-qt( t_area, df_val, ... )
t_val <- t_stand * sd_val  + mean_val
t_val
}
#  
#   Once we run those statements we can use the 
#   new functions.
#
     # when mean=154.3 and standard deviation = 13.2
     # and for 24 degrees of freedom, find P( X < 140.0 )
pt_full( 140.0, 24, mean=154.3, sd=13.2)
     # when mean=483.2 and standard deviation=26.4
     # with 27 degrees of freedom, find  y  such 
     # that P( X < y ) = 0.15.
qt_full( 0.15, 27, mean=483.2, sd=26.4 )