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 L₁ (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 L₂. These will be the y-coordinates of points in the unit square.

In the header of list L₃ 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 L₃ will be less that some value c (it’s up to you to choose an appropriate value for c). In the header of list L₄ insert L₃ c. I’ll let you decide what the resulting output is telling you.

Next, go to Stat > Calc > 1-Var Stats and enter L₄ to see summary statistics for list L₄. 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 L₄ or your use of the summary statistics for list L₄ in estimating ?.

Briefly describe below your choice of c in the test used in list L₄ and how you used the summary statistics for list L₄ 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