73 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= 20.636619 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 = 2. 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". %3D 3 4 5 6 7 8 9 10 Total 1. Result Drop 3. To get a better estimate, combine your data with the class and record the values below. 250 50 Total number of tosses: n = 10 × number of students %3D 123 Total number landing on the line: x = add all successes in the class %3D 2492 4. Use the combined data to obtain the test statistic of our test. Sample proportion: p= x/n 3D 212 74 5. Calculate the p-value of the test. 6. Interpret the p-value. 7. State the conclusion of the test in terms of the problem. 8. Based on the hypothesis test alone, do you expect that the parameter p = 2 0.636619 would be in a 95% confidence interval for p? Explain without computing the interval. %3D 9. This experiment is often used to estimate the value of T. 2 p= – so, with algebra we get T = %3D Plug the class estimate p into the second equation. Do you get a good estimate for 7 2 3.14159265359...?