Portfolio Project 3

In this portfolio project, you or your team will have three deliverables, each requiring you to conduct a diferent type of hypothesis test. Below, we’ll refer to "you", but remember, you’re always working in a group!

Deliverable 1: Connecting Hypothesis Testing with Confidence Intervals

In this deliverable, you will collect data on something that you’re interested in. Based on that data, you will then

(i)         construct a confidence interval

(ii)        conduct a two-sided hypothesis test.

Because the choice of parameter is left to you, this means that you will have to be the one to collect data. Our intent is to encourage you to – momentarily – tear your eyes from your screens and go out into the world to gather your evidence!

Your work will be a composition of learning goals 1-4, stated below, and will culminate in a slide deck consisting of 6 total pages. Please reference the format below for your slide deck.

Formatting of Deliverable 1:

Please submit one slide deck; you can create the slide deck in any program, but it should be submitted as a pdf file on GradeScope.

o Learning Goal 1: (30 points) Slides 1-3 of your slide deck.

o Learning Goal 2: (10 points) Slide 4 of your slide deck.

o Learning Goal 3: (15 points) Slide 5 of your slide deck.

o Learning Goal 4: (15 points) Slide 6 of your slide deck.

In what follows, we will state the learning goals for this deliverable.

Learning Goal 1: Defining an objective, gathering evidence, and constructing a point estimate

1. Defining your objective: provide the following on Slide 1

Here, you need to come up with a parameter that you can collect data about in person and hypothesize what it may be for the population. It is important that this is something that you could actually collect data on! This parameter could be a mean or a proportion. Some examples:

Mean:

The average number of students in study rooms of JFF is 3.

The average number of Teslas on the floor of McCarthy Parking Garage is 2.5.

The average height of students on campus is 5’7”.

Proportion:

The fraction of cars that pass USC on Expo which are red is .1.

The fraction of students wearing USC swag on campus is .8.

The fraction of spots empty in the McCarthy parking garage at noon is .05.

a.   (2.5 points) State your parameter of interest

i.   e.g. “I am interested in the average number of students in study rooms in JFF.”

b.   (2.5 points) State your hypothesized value for the parameter

i.   E.g. “I believe the average number of students in study rooms in JFF is 3.”

c.   (2.5 points) State your choice of α ∈ (0,1), which is 1 minus your confidence level.

2. Gathering evidence: provide the following on Slide 2

a.   (10 points) Summarize your data and display a picture of yourself whilst on your data collection adventure

i.   (5 points) provide a seven number summary.

ii.   (2 points) describe in a few sentences how you collected the data.

iii.   (3 points) picture of you collecting your data.

b.   (2.5 points) Report your sample size n

3. Point estimation: provide the following on Slide 3

a.   (5 points) Based on your data, report a point estimate for your parameter

b.   (5 points) Based on your point estimate, report its sampling distribution Learning Goal 2: Constructing a confidence interval

1.   Provide the following on Slide 4

a.   (10 points) Based on your results from Learning Goal 1, report your 1 一 α confidence interval for your parameter.

Learning Goal 3: Conducting a two-sided hypothesis test

1.   Provide the following on Slide 5

a.   (5 points) Based on your results from Learning Goal 1, state your null and alternative hypotheses for a two-sided hypothesis test

b.   (10 points) Report your p-value

Learning Goal 4: Putting it all together

1.   Provide the following on Slide 6

a.   (2.5  points) Based on your results from Learning Goal 2, report on whether your hypothesized parameter value falls within your 1 一 α confidence interval

b.   (2.5 points) Based on your results from Learning Goal 3, report on your decision to reject your null hypothesis

c.   (5 points) Answer the following: what is the connection between your responses to questions (a) and (b)? More generally, what is the connection between a confidence interval and a two-sided hypothesis test?

d.   (5 points) Answer the following: what are the pros / cons between (i) reporting the entire confidence interval or p-value versus (ii) simply reporting a “yes” / “no” on whether your  hypothesized value falls within the  interval or a  hard decision on whether you reject your null hypothesis?

Deliverable 2 & 3: Benford’s Law

When interviewing for quantitative positions, you'll often meet with professionals who are just a few years ahead of you in their careers and may believe they have a much stronger grasp of math than you do. If you can share something intriguing about mathematics that they’re unfamiliar with, you’ll demonstrate both your knowledge and quantitative expertise.

In deliverable 2 & 3, you will learn about “Benford’s Law”, a seeming paradox / unexpected fact about the distribution of leading numbers in datasets which draw from distributions that span multiple orders of magnitude. (We actually touched on this in class! Can you recall the session?) You will show that Benford’s Law holds in all sorts of surprising datasets. Fortunately, few people know about this interesting law, so it’s a perfect way to surprise an interviewer with how smart you are!

Formatting / Deliverables:

• Learning Goal 5: Nothing needs to be submitted.

• Learning Goal 6: (60 points available) Submit answers to your questions in the attached worksheet as a .pdf.

• Learning Goal 7: (20 points available) Submit a short video of you explaining Benford’s Law, why it is so surprising, and how it can be used to detect fraudulent business records.

Assignment:

Learning goal 5: independently learning about complex statistical subjects.

Learn about Benford’s Law. Here are places where you can learn about it. You’ll know you’ve learned about it when you can say what it is, when it can be applied, and why it’s useful.

1.   Readings:

a. https://en.wikipedia.org/wiki/Benford%27s_law

b. https://www.isaca.org/resources/isaca-journal/past-issues/2011/understanding-and-applying-benfords-law

c. https://www.scientificamerican.com/article/what-is-benfords-law-why-this- unexpected-pattern-of-numbers-is-everywhere/

2.   On YouTube, you can watch some of the following:

a.   (Great video) Numberphile: https://www.youtube.com/watch?v=XXjlR2OK1kM

b.   (About politics) Stand-up Maths:

https://www.youtube.com/watch?v=etx0k1nLn78

c.   (Short and sweet) Minding The Data:

https://www.youtube.com/watch?v=42fGFDNs-0A

3.   If you have Netflix, watch Season 1 Episode 4 “Digits” of the show “Connected: The Hidden Science of Everything”. This is a little pop-sciencey, but it’s entertaining.

4.   If you don’t understand Benford’s Law after these resources, it is a good time to practice hunting for information you don’t know. Google Benford’s law and see if people have discussed what you don’t understand.

Due: Nothing needs to be submitted.

Learning Goal 6: Conducting Chi-Square Goodness of Fit tests.

Go to https://testingbenfordslaw.com/ and select 3 of the available datasets that interest you.

Then, run hypothesis tests on each to see if they follow Benford’s Law.

If you need help testing Benford’s Law in Excel, read this article which describes it:

https://www.isaca.org/resources/isaca-journal/past-issues/2010/using-spreadsheets-and- benfords-law-to-test-accounting-data

Due: Fill out the Learning Goal 6 worksheet with the following information.

1.   (10 points each) Go to https://testingbenfordslaw.com/ and select 3 of the available datasets that interest you.

You can switch between datasets using the following drop-down menu in the top right:

2.   For each dataset:

a.   (1 point) Record the number of records.

b.   (1 point) Record the orders of magnitude.

c.   (1 point) Record the frequency of each leading digit.

d.   (2 points) Find the expected count and the actual count of each leading digit.

e.   (5 points) Using a Chi-square goodness of fit test, test whether the observed distribution matched the expected distribution of Benford’s Law at the 5% significance level.

3.   (10 points) In 4-5 sentences, why is it surprising that Benford’s Law held (or nearly held) across your three datasets?

Learning Goal 7: Practice describing Benford’s Law to potential recruiters.

The goal if this assignment is to practice explaining Benford's Law in a compelling way, that way you can show it o; to recruiters. You’ll record a 60 second video of you doing just that. If you work in groups, make sure all group members show up and speak out in the video.

Due:

1.   (20 points) Record a video of yourself succinctly explaining Benford's Law. The video should be a minimum of 60 seconds long. You should cover three topics:

a.   What is Benford’s Law

b.   What makes it interesting (using one of your datasets as an example)

c.   Describe how it can be used to detect fraud in business settings.

Get extra practice by recording you telling a friend or family member about Benford’s Law!