Extra Exercise 11
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PSLS Confidence Intervals applet.

Sampling 95% confidence intervals and recording the number of intervals
that contain the true parameter (X) corresponds to a binomial setting. For
this exercise, we sample 25 intervals. Therefore X follows B(25,0.95).
By the rules of the binomial distribution, EX = 25*0.95 = 23.75.

We run 30 replications of the experiment: observations X1,...,X30.
These observations are independent and all follow B(25,0.95). The
average, Xmean, is an unbiased estimate for the true mean and its
distribution narrows more down around the true mean if we take
more than 30 samples.

For one run of the experiment (30 replications), the following results were
obtained (for convenience displayed using Minitab). The counts ("Hit") for 
the 30 replications were typed into column 1.

 name c1 'hits'
 Stem-and-Leaf 'hits'.

Stem-and-leaf of hits  N  = 30
Leaf Unit = 0.10

 1    20  0
 2    21  0
 6    22  0000
(11)  23  00000000000
 13   24  000000000
 4    25  0000

 Describe 'hits'.

Variable   N  N*    Mean  SE Mean  StDev  Minimum      Q1  Median      Q3  Maximum
hits      30   0  23.267    0.214  1.172   20.000  23.000  23.000  24.000   25.000


Comments:
--------
The sample mean is somewhat lower than the population mean of 23.75.
The distribution of hits appears clearly left-skewed, as one would
expect for a binomial distribution with p close 1.


Addition:
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In order to compare the stem and leaf plot with the theoretical
B(25,0.95) distribution, we compute the probabilities for counts in the
range 18-25.

 Name c2 "x"
 Set 'x'
  1( 18 : 25 / 1 )1
  End.
 Name c3 "px"
 PDF 'x' 'px';
  Binomial 25 .95.
 Name C4 'expected'
 Let 'expected' = 30*'px'
 Print 'x' 'px' 'expected'.

Data Display 

Row   x        px  expected
  1  18  0.000149    0.0045
  2  19  0.001044    0.0313
  3  20  0.005952    0.1786
  4  21  0.026926    0.8078
  5  22  0.093016    2.7905
  6  23  0.230518    6.9155
  7  24  0.364986   10.9496
  8  25  0.277390    8.3217


Comments:
---------
It is seen that the sample distribution has too many counts of 22 and 23
and too few counts of 24 and 25, thus leading to a lower sample mean than
the correct mean of the distribution. This could however just be random
fluctuation that would smooth out if more than 30 trials were done.

You could also directly generate a plot of the binomial distribution 
using the Graph-Probability Distribution Plot menu:

 DPlot;
  Distribution;
    Binomial 25 .95.