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Introduction to the Practice of Statistics

Chapter 4

The following example uses theheadsprogram. If you don't have theheadsprogram, you can download it from within Stata by typingfindit heads(see How can I use the findit command to search for programs and get additional help? for more information about usingfindit).

Theheadsprogram can be used to produce results simulating those shown in Figure 4.1, as shown below. First start theheadsprogram by typing

heads

You can then change the number of trials to 1000. When we ran this, our simulated 1000 coin flips looked like the graph below. Every time you run theheadsprogram, it generates a new set of 1000 flips, so the results come out different every time. However if you do a large number of coin flips, you will notice that in the long run the proportion of heads always converges on .5. Try this on your own and see for yourself.

Go ahead and click onquit.The

headsprogram can be used to simulate the results in Example 4.2. We now typeheads, saveto save the coin tosses in a Stata file. In the first example we simulate the 4040 coin tosses by Buffon (who got 2048 heads). Here is the graph we get.

We then click quit. Then we use the

tabulatecommand to see that our simulation, we got 2051 heads, or a proportion of .492. Every time we do this, we get a different proportion of heads. Try this for yourself and see how many heads you get.

tabulate heads# heads out | of 1 tossed | Freq. Percent Cum. ------------+----------------------------------- 0 | 2051 50.77 50.77 1 | 1989 49.23 100.00 ------------+----------------------------------- Total | 4040 100.00summ headsVariable | Obs Mean Std. Dev. Min Max ---------+----------------------------------------------------- heads | 4040 .4923267 .500003 0 1

Next, we simulate the 24,000 coin tosses by Karl Pearson, and we got 12058 heads (compared to the 12,012 heads that Pearson got).

heads , save tabulate heads

# heads out | of 1 tossed | Freq. Percent Cum. ------------+----------------------------------- 0 | 11,942 49.76 49.76 1 | 12,058 50.24 100.00 ------------+----------------------------------- Total | 24,000 100.00summ heads

Variable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- heads | 24000 .5024167 .5000046 0 1

Next, we simulate the 10,000 coin tosses by John Kerrich, and we got 4972 heads (compared to the 5067 heads that Kerrich got).

heads2 1000, save tab heads# heads out | of 1 tossed | Freq. Percent Cum. ------------+----------------------------------- 0 | 5028 50.28 50.28 1 | 4972 49.72 100.00 ------------+----------------------------------- Total | 10000 100.00summarize headsVariable | Obs Mean Std. Dev. Min Max ---------+----------------------------------------------------- heads | 10000 .4972 .5000172 0 1

We can use theheadsprogram to simulate the results shown in Figure 4.8 and Example 4.20, which shows the probability histogram for 4 tosses of a coin. We can do 1000 coin tosses and graph the probabilities and compare those results to Figure 4.8, as illustrated below. When you try this, you will probably get similar (but not exactly the same) results.

Note that 1000 is the number of tosses, and .5 is the probability of a head, and 4 is the number of coins tossed. We use thesaveoption to save the results.

heads , save

We then quit and graph the results below.

histogram heads, discrete

We can show the probability distribution using thetabulatecommand and compare this to example 4.20. As you can see, the probabilities we found (from the Percent column) are quite similar to the expected percentages shown in example 4.20.

tab heads

# heads out | of 4 tossed | Freq. Percent Cum. ------------+----------------------------------- 0 | 54 5.40 5.40 1 | 235 23.50 28.90 2 | 397 39.70 68.60 3 | 244 24.40 93.00 4 | 70 7.00 100.00 ------------+----------------------------------- Total | 1,000 100.00

We can get the mean of the distribution with thesummarizecommand. The expected mean is 2.0, and ours is very close and 2.041.

summarize heads

Variable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- heads | 1000 2.041 .987063 0 4

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