Supplemental Material
Everything in here is meant to help you succeed at the exams in the course. There’s no due date attached to any of it because you aren’t meant to turn any of it in.
If you do want feedback on the work you’ve done feel free to send your work to me as a picture/document, just don’t expect the \(\approx\) 72 hour response time.
The format of this material will mostly be large form word problems with real or simulated data. If this is your last/only math class during your college career this next concept isn’t super helpful to understand, but anyone with more than one math class in their future should pay attention.
Mathematics/Computational courses operate off of a system of exercises and problems
These are gross oversimplifications, many proper mathematicians would argue these lack rigor.
Axioms are logical proofs of concepts, operations, and theorems. They are meant to leverage common sense and intuition. Meaning they have to have a core that is understandable in any context.
Algorithms are a system of steps for solving a problem/yielding a result. Think of them as mathematical/computational recipes. If you follow them exactly you should yield the appropriate result.
Exercises are simplified, in-class examples led by the instructor. Their purpose is to introduce the axioms and algorithms that make up the foundation of the course topics.
Problems are higher complexity, generally difficult, student directed puzzles. The axioms will still apply, the algorithms may not. The student is expected to flex their critical thinking and logic skills to either piece together which axioms and algorithms resolve the puzzle, or construct a new algorithm out of pieces of previously learned ones. They are never unsolvable, but you should have never seen them before.
Household Income (Chapter 1-3)
The Census Bureau keeps track of a larger variety of interesting U.S. Household/citizen metrics and an even larger variety of uninteresting metrics.
In order to calculate metrics from Census data, it’s a common practice to split states into counties, counties into districts, residents of the districts into groups based on similarities, and randomly select from those groups in even batches. The result is a sample that is ideally homogeneous.
Below is a table with a sample of \(n=50\) from the original sample size of 320. The sample was taken by evenly dividing the data by year and selecting 5 rows of data at random from each year:
| Income (in 1000s of USD) | Year | Share Proportion | State |
|---|---|---|---|
| 53 | 2013 | 0.0476919 | Kansas |
| 6232 | 2013 | 0.0539027 | Kansas |
| 5410 | 2013 | 0.0467958 | Kansas |
| 8380 | 2013 | 0.0724881 | Kansas |
| 6215 | 2013 | 0.0537543 | Kansas |
| 89 | 2014 | 0.0796244 | Kansas |
| 57 | 2014 | 0.0509154 | Kansas |
| 11540 | 2014 | 0.0993029 | Kansas |
| 5714 | 2014 | 0.0491657 | Kansas |
| 57 | 2014 | 0.0513577 | Kansas |
| 119 | 2015 | 0.1068316 | Kansas |
| 57 | 2015 | 0.0513888 | Kansas |
| 8421 | 2015 | 0.0720238 | Kansas |
| 61 | 2015 | 0.0551635 | Kansas |
| 91 | 2015 | 0.0817416 | Kansas |
| 9200 | 2016 | 0.0781513 | Kansas |
| 60 | 2016 | 0.0536152 | Kansas |
| 6029 | 2016 | 0.0512156 | Kansas |
| 45 | 2016 | 0.0398922 | Kansas |
| 5404 | 2016 | 0.0459074 | Kansas |
| 4725 | 2017 | 0.0397657 | Kansas |
| 6931 | 2017 | 0.0583302 | Kansas |
| 58 | 2017 | 0.0521292 | Kansas |
| 5619 | 2017 | 0.0472877 | Kansas |
| 49 | 2017 | 0.0438035 | Kansas |
| 48 | 2018 | 0.0429701 | Kansas |
| 6932 | 2018 | 0.0578984 | Kansas |
| 5558 | 2018 | 0.0464243 | Kansas |
| 59 | 2018 | 0.0524877 | Kansas |
| 5507 | 2018 | 0.0459955 | Kansas |
| 9090 | 2019 | 0.0752758 | Kansas |
| 63 | 2019 | 0.0553547 | Kansas |
| 48 | 2019 | 0.0427709 | Kansas |
| 15375 | 2019 | 0.1273196 | Kansas |
| 8174 | 2019 | 0.0676866 | Kansas |
| 7146 | 2020 | 0.0584022 | Kansas |
| 51 | 2020 | 0.0444936 | Kansas |
| 5030 | 2020 | 0.0411120 | Kansas |
| 5382 | 2020 | 0.0439888 | Kansas |
| 9123 | 2020 | 0.0745613 | Kansas |
| 8982 | 2021 | 0.0724300 | Kansas |
| 5097 | 2021 | 0.0410990 | Kansas |
| 9695 | 2021 | 0.0781784 | Kansas |
| 50 | 2021 | 0.0437530 | Kansas |
| 155 | 2021 | 0.1363234 | Kansas |
| 44 | 2022 | 0.0378789 | Kansas |
| 91 | 2022 | 0.0795614 | Kansas |
| 16085 | 2022 | 0.1279288 | Kansas |
| 48 | 2022 | 0.0418941 | Kansas |
| 44 | 2022 | 0.0386058 | Kansas |
The original sampling method for the Census data was a multi-step process using multiple different sampling methods. Three to be exact. What are those three nested sampling methods?
What was the sampling method I used to create the second, smaller sample set in the table above?
Name the type and subtype of each variable in the table.
Compute the following for Household Income:
Mean
Median
Mode
Range
Variance
Standard Deviation
Justify which measure of center (Mean/Median/Mode) is most representative of the data set.
Why would someone prefer Standard Deviation over Variance as a measure of spread for this data?
The below histogram represents the original sample of \(n=320\):
Describe the shape of the histogram
Identify the median and mode
Identify where the mean you calculated falls on this histogram
- Does your calculation feel reasonable?
Does this histogram follow our rules of histograms?