Question

This worksheet is about doing simulations on a TI-83/84, but feel free to do the work on a computer if you prefer.

You are going to estimate the value of ? through a simulation. A
circle inscribed in a 1x1 square has area ?/4 (you may want to draw
a picture to convince yourself of this). Now, we can simulate
picking points inside the square as follows: randomly select a
value between 0 and 1 to be the *x*-coordinate, and repeat
the process to select a value to be the *y*-coordinate. Then
the proportion of the (*x*, *y*) points thus selected
that lie in the inscribed circle can be used to estimate ?. The
accuracy of the estimate depends on the number of points simulated,
so if your calculator limits the size of lists to less than 999 you
shouldn’t expect great results, but it’s the process that counts
here.

Go to *Stat* > *Edit* and select list
L_{1} (you’ll need at least four lists for this exercise).
In the header of the list insert *Math* > *Prob*
and select the Rand utility, which randomly selects numbers between
0 and 1, and enter it as *rand*(*n*), where
*n* is the maximum list size your calculator will allow. Hit
*Enter* to see the calculator generate the values you will
use for the *x*-coordinate of a point in the unit
square.

Repeat the process above in list L_{2}. These will be
the *y*-coordinates of points in the unit square.

In the header of list L_{3} compute the distance of each
point generated from the previous lists from the center of the
square (actually, it will suffice to find the squared distance.)
Note: the center of the square is also the center of the circle
inscribed *in* the square. I’ll let you do the math, but a
picture should help. In fact, I WANT TO SEE SUCH A PICTURE IN THE
DESCRIPTION OF WHAT YOU DID TO ESTIMATE ?.

The last list will test whether a simulated point lies within
the inscribed circle (you don’t want to try counting the simulated
points that lie within the circle by hand!). If the point is within
the circle, then the distance computed in list L_{3} will
be less that some value *c* (it’s up to you to choose an
appropriate value for *c*). In the header of list
L_{4} insert L_{3} *c*. I’ll let you decide
what the resulting output is telling you.

Next, go to *Stat* > *Calc* > *1-Var
Stats* and enter L_{4} to see summary statistics for
list L_{4}. From here you should be able to come up with an
estimate of ?. Don’t be disappointed if it’s off by as much as
0**.**05 or more for a simulation of 999 points (or
more if your calculator severely restricted the size of your
lists). However, if your estimate displays unreasonable error, you
may need to review your choice of *c* in the test used in
list L_{4} or your use of the summary statistics for list
L_{4} in estimating ?.

Briefly describe below your choice of *c* in the test
used in list L_{4} and how you used the summary statistics
for list L_{4} to estimate ?. Your description should
include the picture discussed in the third bullet above, and how
you chose the value for *c*. While your description should
be brief (the picture and a few sentences should suffice), it
should be neatly presented and use complete sentences! If your
handwriting is illegible, you will need to type up your response!
Poor presentation will affect your score on this worksheet.
Finally, what was your final estimate for ??

Answer #1

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