Directions:
- Submit your final answers and all supporting work on Canvas (under
the “Assignments” tab)
- You may submit any file format for this assignment
- Homework is intended to individual work. While you may
discuss the assignment with your peers, you should submit answers that
are uniquely your own.
- Any assistance you receive from resources/materials not on our
course website (such as other websites, course mentors, peer students,
AI, etc.) should be clearly acknowledged
Question #1 (Probability Basics)
A department store manager collected data on a random sample of 200
customers regarding their use of a store credit card and the size of
their purchase. These data are recorded in the table below:
| Large Purchase ($100+) |
50 |
30 |
80 |
| Small Purchase (<$100) |
20 |
100 |
120 |
| Total |
70 |
130 |
200 |
- Part A: Estimate the marginal probability
of a customer paying with a store credit card.
- Part B: Estimate the conditional
probability of a customer paying with a store credit card given
they made a large purchase.
- Part C: Estimate the conditional
probability of a customer making a large purchase given they paid
with a store credit card.
- Part D: In these data 100 of 120 customers who made
a small purchase did not use a store credit card to pay. Use
probability notation, such as \(Pr(\text{outcome})\) or \(Pr(\text{outcome}|\text{condition})\), to
accurately represent the probability that can be estimated by these
data. Be careful about whether your notation denotes a conditional or a
marginal probability.
\(~\)
Question #2 (Statistical Inference using StatKey)
A study
published in 2023 investigated whether dogs could be trained to
detect cancer using their sense of smell. As part of the study,
researchers collected breath samples from healthy controls and cancer
patients, and they then repeated trials where a trained dog was exposed
to five bags containing breath samples, one bag was from a cancer
patient and the other four were from healthy controls.
The dogs had been trained to choose exactly one bag in each trial,
and during training they were given rewards for identifying bags from
cancer patients. In each trial the researchers recorded whether or not
the trained dog correctly identified the breath sample from a cancer
patient.
- Part A: Suppose the dogs are unable to identify
cancer by smelling and are selecting breath samples at random in the
study. Using statistical notation, what is an appropriate
null hypothesis corresponding to this possibility.
- Part B: Suppose one dog from the study correctly
identified the cancer sample in 9 of 13 trials. Use a StatKey
simulation to find the one-sided \(p\)-value measuring how much evidence these
results provide against the null hypothesis from Part A. Use at least
1000 simulated outcomes and report an estimate of the \(p\)-value to 3 decimal places.
- Part C: Provide a one-sentence conclusion based
upon the \(p\)-value you found in Part
B. Remember that an appropriate conclusion in this class should include
scientific context, information about the strength of
evidence, and a description of the type of relationship
found. Do not include phrases like “reject the null” or “conclude the
alternative”.
- Part D: Suppose the researchers had designed this
study differently such that 4 of the 5 bags came from cancer patients.
Assuming the same results were observed (30 of 33 correct
identifications) what would the \(p\)-value now be? Use a StatKey simulation
similar to the one you conducted in Part B to estimate this \(p\)-value.
- Part E: Provide a one-sentence conclusion for the
\(p\)-value you found in Part D.
\(~\)
Question #3 (One-sample Data Analysis in R)
The Transport Security Administration (TSA) is an agency within the
US Department of Homeland Security with authority over the safety and
security of travel in the United States. For this question, you will
analyze data from a random sample of \(n=5000\) claims made by travelers against
the TSA between 2003 and 2008, the first five years that the agency
existed. These data are found at the following link:
https://remiller1450.github.io/data/tsa_small.csv
The relevant variables in this analysis are:
Status - whether the claim was approved (paid in
full), settled (partially paid/negotiated), or denied (not paid at
all)
Claim_Amount - the amount of monetary damages
requested in the initial claim
- Part A: Write
R code that stores these
data in a data frame object named tsa_data. Then find
the average amount of monetary damages claimed by travelers.
Part B: Create a data visualization showing
the distribution of Claim_Amount. Briefly describe 1 or 2
things you can learn from this distribution that you could not have
known using just the average value you calculated in Part A.- Part C: Create a table displaying the frequencies
of each status. Using this table, calculate the proportion of claims
that are denied.
- Part D: Create a data visualization showing the
distribution of claim statuses.
- Part E: According to research
by Weiss Ratings, USAA denies the highest percentage of home
insurance claims, rejecting 48.1% of the claims they receive. Suppose we
use this value to inform a null hypothesis. Does the sample of claims
against the TSA support the conclusion that the TSA rejects a lower
percentage of claims than USAA? Report a \(p\)-value and an appropriate one-sentence
conclusion.
- Part F: Suppose we had used a smaller sized sample
data in Part E, such as a random sample of 500 claims rather than 5000.
Would you expect the \(p\)-value to be
larger or smaller? Try dividing the observed count and sample size by 10
(and rounding to the nearest whole number) then inputting these new
“data” into StatKey to verify your expectation.