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# Download Introduction to Chance Models (Section 1.1) Introduction A key step

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```c. Would it be appropriate to describe the overall shape of this null distribution as
bell-shaped?
2. As you did in the previous session, use the One Proportion applet to run a simulation
(with 10000 repetitions) to find the approximate p-value for this study. Be sure to click
on âTwo-sided.â Record the p-value: ___________
Theory-based approach (One proportion z test)
In the early 1900s, and even earlier, computers werenât available to do simulations, and as
people didnât want to sit around and flip coins all day long, they focused their attention on
mathematical and probabilistic rules and theories that could predict what would happen if
someone did simulate.
They proved the following key result (often called the Central Limit Theorem):
Central Limit Theorem (CLT): The distribution of sample proportions will be centered at the
long-run probability (ð), with a standard deviation of âð(1 â ð)/ð. If the sample size (n) is
large enough, then the shape of the distribution is bell-shaped.
One bit of ambiguity in the statement is how large is large enough for the sample size? As it
turns out, the larger the sample size is, the better the prediction of bell-shaped behavior in the
null distribution is, but there is not a sample size where all of the sudden the prediction is good.
However, some people have used the convention that you should have at least 10 successes
and at least 10 failures in the sample.
Validity conditions: The normal approximation can be thought of as a prediction of what
would occur if simulation was done. Many times this prediction is valid, but not always,
only when the validity condition (at least 10 successes and at least 10 failures) is met.
3. How well does the CLT predict the center, SD, and shape of the null distribution of
sample proportion of children who choose candy, as displayed in Figure 1.1?
a. Center:
b. SD:
c. Shape:
4. Now go back to the One Proportion applet, and check the box next to âNormal
Approximation." Be sure that the âTwo-sidedâ option is also selected. Record the normal
approximation-based p-value: ___________
5. Compare the simulation-based p-value (from #2) to the normal approximation-based pvalue (from #4). Are the very different or about the same?
June 27, 2014
MAA PREP workshop
36
```
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