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. A 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 1 2 3 4 5 6 7 8 9 10 Total Drop Result 3. To get a better estimate, combine your data with the class and record the values below. Total number of tosses: n = 10 x number of students = 30 =300 %3D 191 Total number landing on the line: x= add all successes in the class %3D %3D %3D Sample proportion: p= x/n=\93003D0.637 4. Use the combined data to obtain the test statistic of our test. 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 = = - 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 T. p= - so, with algebra we get T = = Plug the class estimate p into the second equation. Do you get a good estimate for T N 3.14159265359...?
Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
number 9
![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.
A 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
1 2 3 4 5 6 7 8 9
10
Total
Drop
Result
3. To get a better estimate, combine your data with the class and record the values below.
Total number of tosses: n = 10 x number of students = 30 =300
%3D
191
Total number landing on the line: x= add all successes in the class
%3D
%3D
%3D
Sample proportion: p= x/n=\93003D0.637
4. Use the combined data to obtain the test statistic of our test.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F430d3a46-8feb-4de2-92ab-a1767f45d8ef%2Fbfcafabe-b271-4406-ac2c-0e7c447d2284%2Fhmc11yf.jpeg&w=3840&q=75)
![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 = = - 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 T.
p= - so, with algebra we get T = =
Plug the class estimate p into the second equation. Do you get a good estimate for T
N
3.14159265359...?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F430d3a46-8feb-4de2-92ab-a1767f45d8ef%2Fbfcafabe-b271-4406-ac2c-0e7c447d2284%2F2rw98hx.jpeg&w=3840&q=75)
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