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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 âTwo-sided.â Record the p-value: ___________ Theory-based approach (One proportion z test) In the early 1900s, and even earlier, computers werenât available to do simulations, and as people didnât 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 âð(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 âNormal Approximation." Be sure that the âTwo-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