math statistics

Math 125, Paper Two Spring 2023

(5% of Final Grade)

Overview

Pick a subject that interests you and design an experiment that has two outcomes: success and

failure. You want to pick something that you can easily replicate because you will do it 70 times.

Then you will investigate your results: could it be a binomial experiment?

Topics to Cover in the Paper

Experimental Design

Why did you choose the experiment you did? Why did it interest you?

What counts as a success/failure? (Try to pick something where the probability of a success is

between 25% and 75%. Definitely do not pick something where you will always succeed or always

fail.)

Data Gathering

Perform the 70 trials. List the results in order and include them in your

paper.

I would suggest putting this in the form of a table, as shown here.

Examples are included in grey throughout this assignment.

Calculations using all trials

Based on all 70 trials, calculate the empirical probability of success in one trial.

Assuming that you actually do have a binomial experiment, with the value of 𝑝 that you just

found, answer the following questions. Show your work.

1) What is the probability that you’d get exactly 35 successes in your 70 trials?

2) Find the mean number of successes you’d expect to get in your 70 trials.

Trial Results

1. S 2. S

3. F 4. S

5. F 6. Etc.

2

Independence

Do the results of one trial influence the next? Consider some

conditional probabilities (you will not be able to use the first trial,

since it does not have a previous trial).

Figure out 𝑃(𝑆|𝑆) and 𝑃(𝑆|𝐹). Below is an example:

Find the 90% confidence intervals for 𝑃(𝑆|𝑆) and 𝑃(𝑆|𝐹).

Consider the 𝑃(𝑆) that you found earlier for all 70 trials. Is it in either (or both) of the

confidence intervals you found?

Based on this, do you think your trials were independent?1 Or does the result of a trial change

the likelihood of success/failure on the next trial? Explain, referencing your calculations. If you

have other reasons to think the trials are independent or dependent, please include that as part

of your reasoning, too.

1 This calculation is a big simplification, since we do not allow for the possibility that, say, a trial influences

a trial that does not come immediately after it or influences the next two trials, etc.

Trial Number & Result

Conditional Situation

1. S n/a 2. S 𝑆|𝑆 3. F 𝐹|𝑆 4. S 𝑆|𝐹 5. F 𝐹|𝑆

Current Trial

Success Failure

Tr ia

l

Su cc

es s Success | Previous

Trial is a Success 20

Failure | Previous Trial is a Success

17

𝑃(𝑆|𝑆) = 20

20 + 17

≈ 𝟎. 𝟓𝟒

The 90% confidence interval for 𝑃(𝑆|𝑆) is (𝟎. 𝟒𝟏, 𝟎. 𝟔𝟖).

P re

vi o

u s

Fa ilu

re Success | Previous

Trial is a Failure 8

Failure | Previous Trial is a Failure

24

𝑃(𝑆|𝐹) = 8

8 + 24 = 𝟎. 𝟐𝟓

The 90% confidence interval for 𝑃(𝑆|𝐹) is (𝟎. 𝟏𝟐, 𝟎. 𝟑𝟖).

3

Constant Probability

[This is a completely different way of looking at the data than the previous section – ignore

conditional probability here. Just split your data into the first and second 35 trials.]

Using only the first half of your trials, find the point estimate for p. Use this to create a 90%

confidence interval for p.

Using only the second half of your trials, find �̂�. Use this to find a 90% confidence interval for p.

TGraph both intervals on a number line and compare them. Do they overlap?

You can use Wolfram|Alpha for this. To get the graph here, I typed “plot interval

(0.5,0.68) and (0.4,0.58).”

Do you think the probability of a success was constant throughout your trials? Explain why or

why not, based on the statistical results you found and any other reasoning you have (as before,

it is impossible to prove the two are equal, but you may be able to conclude that they are not

equal).

Note that this is only one way at looking at constant probability – we could also choose to split

up the trials by even/odd, for instance – so it’s certainly incomplete. If you have other reasons to

think the probability isn’t constant, you should include them.

Extra Credit (up to 3 points):

Perform a hypothesis test, with 𝛼 = 0.10, comparing the probabilities from your

first and second 35 trials (call these 𝑝1 and 𝑝2). Use as your null hypothesis 𝑝1 =

𝑝2.

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Conclusion

Do you think your experiment was binomial? Why or why not? Justify your answer using your

work from all of the previous parts. Either result is fine, as long as your data support it. Which of

the criteria for a binomial experiment does it fulfill? Which, if any, does it fail to fulfill?

Other things I will look for

Spelling/Grammar/Formatting

Your paper should be typed & double-spaced. There is no minimum or maximum length,

provided you include all parts of the assignment.

This is a short paper; use full sentences and proper grammar. If you have any concerns about

these, I suggest you work with the writing tutors, who are available at

https://prepare.ccc.edu/mxvss/.

If you use any outside resources/information, you should cite them. Otherwise you are

technically stealing and may earn a zero on the paper. More information regarding citations can

be found here Plagiarism_one_pager.pdf (purdue.edu) and here Plagiarism Overview // Purdue

Writing Lab.

Timeliness/Feedback

You will upload a draft and a final version of this paper.

Please upload to Brightspace. They are linked in Module Three or you can go to Assessments →

Assignments →

Second Paper draft due, 4/21

Second Paper comments returned, 4/28

Revised Second Paper due, 5/5

Upload it by 11:59 PM, but you are welcome to turn it in earlier.

5

Late work will be penalized 5 points per day, but I also will not be grading any papers that are

submitted after the final exam is due.

Paper Two Rubric

Sections Missing Not There

Yet Almost There

Completely There

Experimental design

Explanation of why you chose this experiment. Description of what counts as a success & failure.

0 3 7 10

Data Raw data for all 70 trials, in order. (Indicate if this is at the end of the paper.)

0 0 0 15

Calculations for all data, assuming

Binomial

Correct calculation of empirical probability of success in one trial.

0 1 3 5

Correct calculation for 𝑃(𝑋 = 35). 0 1 3 5

Correct calculation for 𝜇. 0 1 3 5

Independence Correct 90% confidence interval for 𝑃(𝑆|𝑆). 0 1 3 5

Correct 90% confidence interval for 𝑃(𝑆|𝐹). 0 1 3 5

Identifying if 𝑃(𝑆) is in either or both intervals. 0 1 3 5

Explanation for why you think the trials were/were not independent, which references your previous work.

0 1 3 5

Constant Probability

First half of data: correct point estimate for p and confidence interval.

0 1 3 5

Second half of data: correct �̂� and confidence interval.

0 1 3 5

Graph of intervals & comparison. 0 1 3 5

Explanation for why you think the probability was/was not constant, which references previous calculations.

0 1 3 5

Hypothesis test (optional) (1) (3)

Conclusion

Determination of what aspects of a binomial experiment are/are not satisfied by your experiment. All the requirements are correctly identified. Your answer references the

0 3 7 10

6

calculations from the paper and is consistent with it.

Draft

Submitted draft on time.

Incorporated comments from draft into final version.

0 3 7 10

Late work penalty Loss of 5 point per day for every day late.