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The problem shown in the image involves a simulation where Kerry's observed proportion is 11.11% below the center of the null distribution. It mentions that out of 5000 simulations:

- 7 simulations fell more than 11.11% below the center of the null distribution.
- 9 simulations fell more than 11.11% above the center of the null distribution.

To solve this:

### Step 1: Calculate the p-value
The p-value is determined by calculating the proportion of simulations that provide results as extreme as or more extreme than the observed result. In this case, we are looking at the 7 simulations that fell below and the 9 simulations that fell above the 11.11% threshold.

The total number of extreme cases (both below and above) is:
\[
7 + 9 = 16
\]

The total number of simulations is 5000, so the p-value is:
\[
\text{p-value} = \frac{16}{5000} = 0.0032
\]

### Step 2: Interpret the p-value
With a p-value of 0.0032, Kerry has **strong** evidence against the null hypothesis, suggesting that the null model is **not** a good fit for the observed results.

Would you like more details on any of these steps or explanations?

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Here are 5 related questions to deepen your understanding:

1. What is the significance of a low p-value in hypothesis testing?
2. How can the null distribution be constructed using simulations?
3. Why are both tails (above and below the observed proportion) considered in calculating the p-value?
4. What is the role of the null hypothesis in statistical testing?
5. How can we increase the accuracy of simulation-based p-values?

**Tip:** Always ensure that the number of simulations is large enough to give a reliable estimate of the p-value. A small number of simulations can lead to inaccurate results.

Notice that Kerry's observed proportion is 11.11% below the center of the null distribution. After 5000 simulations, you find that: 7 of the 5000 simulations fell more than 11.11% below the center of the null distribution, and 9 of the 5000 simulations fell more than 11.11% above the center of the null distribution. Use the results of the simulation to calculate the p-value.

Learn how to calculate the p-value in hypothesis testing using 5000 simulations. This example demonstrates how to use simulation results to determine the likelihood of observing a proportion 11.11% below or above the center of the null distribution.

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