**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| | | | | | | | | | | | 3. To get a better estimate, combine your data with the class and record the values below. - Total number of tosses: \( n = 10 \times \) number of students = 250 - Total number landing on the line: \( x = \) add all successes in the class = 149 - Sample proportion