MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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**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. \[ H_0: p = 0.636619 \] \[ H_A: p \neq 0.636619 \] 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 Result | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Total | |-------------|---|---|---|---|---|---|---|---|---|----|-------| 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 = \text
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Transcribed Image Text:**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. \[ H_0: p = 0.636619 \] \[ H_A: p \neq 0.636619 \] 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 Result | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Total | |-------------|---|---|---|---|---|---|---|---|---|----|-------| 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 = \text
### Page 74 Transcript for Educational Website --- **Exercise Details:** **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 \( p = \frac{2}{\pi} \approx 0.636619 \) would be in a 95% confidence interval for \( p \)? Explain without computing the interval. **9.** This experiment is often used to estimate the value of \( \pi \). \[ p = \frac{2}{\pi} \] So, with algebra we get: \[ \pi = \frac{2}{p} \] Plug the class estimate \( \hat{p} \) into the second equation. Do you get a good estimate for \( \pi \approx 3.14159265359 \)? Calculation: \[ \pi = \frac{2}{0.596} = 3.3557047 \] **Note:** The estimate is fairly close, but is still off, so this would not be a very good estimate. --- This text is designed to guide students through understanding hypothesis testing, p-value calculation, and estimation processes, particularly in relation to π (pi).
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Transcribed Image Text:### Page 74 Transcript for Educational Website --- **Exercise Details:** **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 \( p = \frac{2}{\pi} \approx 0.636619 \) would be in a 95% confidence interval for \( p \)? Explain without computing the interval. **9.** This experiment is often used to estimate the value of \( \pi \). \[ p = \frac{2}{\pi} \] So, with algebra we get: \[ \pi = \frac{2}{p} \] Plug the class estimate \( \hat{p} \) into the second equation. Do you get a good estimate for \( \pi \approx 3.14159265359 \)? Calculation: \[ \pi = \frac{2}{0.596} = 3.3557047 \] **Note:** The estimate is fairly close, but is still off, so this would not be a very good estimate. --- This text is designed to guide students through understanding hypothesis testing, p-value calculation, and estimation processes, particularly in relation to π (pi).
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