378 CHAPTER 6 Inferential Statistics
There are two important observations to make here. First of all, even
though we haven’t sampled from a normal distribution, the sample
means appear to be somewhat normally distributed (more so in the
n= 50 case). Next, notice that the range of the samples of the mean
forn= 50 is much less than forn= 5. This is because the standard
deviations for these two sample means are respectively √σ 5 and √σ 50 ,
whereσ is the standard deviation of the given uniform distribution.^22
Simulation 3. Let’s take 100 samples of the mean (where each mean
is computed from 5 observations) from the distribution having density
function
f(x) =
2 x if 0≤x≤ 1 ,
0 otherwise.(Recall that this is the density function for
√
rand.) We display the
corresponding histogram.
(^22) The variance of this distribution was to be computed in Exercise 1 on page 371; the result is
σ^2 = 121.