**6.4 Activity 17: One Sample Inference for Proportions** **Objective:** The objective of this activity is to gain experience with hypothesis testing for a proportion. We will do this by studying the classic experiment proposed by French naturalist Buffon in 1733. This experiment is popularly known as "Buffon's Needle". **Topics covered:** 1. One sample hypothesis test for a population proportion 2. One sample confidence interval for a population proportion 3. Duality between confidence intervals and hypothesis testing First, we put Buffon's original question from 1733 in our context. We would like to know what is the probability that a standard 2.5-inch toothpick will fall on a line when the lines are parallel. In 1777, Buffon showed that the probability is \( p = \frac{2}{\pi} \approx 0.636619 \) when the lines are also 2.5 inches apart. 1. Suppose we don’t believe Buffon’s proof. That is, we think that the probability of landing on the line is most definitely not \( p = \frac{2}{\pi} \). State the hypotheses for our research claim. 2. Next, we collect data to test our research question. Remove the last page of this activity (Page 75) with parallel lines that are 2.5 inches apart. Drop a standard 2.5-inch toothpick on the page. Record whether it falls on a line or not. Repeat the process ten times and record your answers below using 0 = “not on line” and 1 = “landed on line”. | Drop | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Total | |------|---|---|---|---|---|---|---|---|---|----|-------| | Result | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 5 | 3. To get a better estimate, combine your data with the class and record the values below. Total number of tosses: \( n = 10 \times \text{number of students} = 250 \) Total number landing on the line: \( x = \) add all successes **Educational Text: Understanding P-Values and Pi Estimation** **5. Calculate the p-value of the test.** - This step involves determining the probability of obtaining test results at least as extreme as the results observed, under the assumption that the null hypothesis is correct. **6. Interpret the p-value.** - The interpretation of the p-value helps to understand whether the observed data are consistent with the null hypothesis. A low p-value indicates that the observed data is unlikely under the null hypothesis. **7. State the conclusion of the test in terms of the problem.** - This involves summarizing the results of the hypothesis test. If the p-value is less than a predefined significance level, we reject the null hypothesis in favor of the alternative hypothesis. **8. Based on the hypothesis test alone, do you expect that the parameter \( p = \frac{2}{\pi} \approx 0.636619 \) would be in a 95% confidence interval for \( p \)? Explain without computing the interval.** - This question explores the expectation of whether \( p \) lies within the confidence interval based on the hypothesis testing results, without explicitly calculating the interval. **9. This experiment is often used to estimate the value of \( \pi \).** \[ p = \frac{2}{\pi} \] - Using algebra, we rearrange to find: \[ \pi = \frac{2}{p} \] - To estimate \( \pi \), plug the class estimate \( \hat{p} \) into the equation. This step checks if the calculated value approximates \( \pi \approx 3.14159265359\ldots \). This exercise demonstrates fundamental concepts in statistics and provides a practical method to understand how mathematical constants, like \( \pi \), can be estimated through experimental data.