Question

please show all r code and explain

d. Let's look at different dfs for the 85th percentile of t distributions. If you look at t85,20 , t.85,40 , t.85,60 , etc. (the 85th percentile of t with df = 20, 40, 60, and so on, continuing up by 20 each time), then what is the smallest of these dfs where the 85th percentile of t is less than .02 away from the 85th percentile of z?

Answer 1

The 85th percentile of t with df = 20, 40, 60 and so on is given by: (The R codes are given)
qt(0.85, 20)
qt(0.85, 40)
qt(0.85, 60)
The respective percentile values are 1.064, 1.050 and 1.045 respectively.

The 85th percentile of z is given by: (The R code is given)
qnorm(0.85)
The percentile value is 1.036.

Hence, if we calculate the difference between the 85th percentile of z and the 85th percentile of t with df = 20, 40, 60 and so on, we get the respective differences as: (The R codes are given)
qnorm(0.85)-qt(0.85,20)
qnorm(0.85)-qt(0.85,40)
qnorm(0.85)-qt(0.85,60)
The difference values have come out to be -0.028, -0.014 and - 0.009 respectively. This tells us that as we go on increasing the "df" value, the difference will go on decreasing.

For df = 20, the difference is -0.028 and this means that it is not less than 0.02 away from the 85th percentile of z.
For df = 40, the difference is -0.014 and this means that it is less than 0.02 away from the 85th percentile of z.
Hence, the smallest df for which the condition is satisfied is df = 40.