class: center, middle, inverse, title-slide # STATS 250 Lab 04 ## Probability and Scatterplots ### Nick Seewald<br />nseewald@umich.edu ### Week of 09/21/2020 --- class: center, middle # Reminders 💡 Your tasks for the week running Friday 9/18 - Friday 9/25 (plus an extra): | Task | Due Date | Submission | |:-----|:---------|:-----------| | Quiz 1 | Monday 9/21 11:59PM ET | Canvas | | MWrite 1 Initial Draft | Wednesday 9/23 5PM ET | Canvas | Homework 3 | Friday 9/25 8AM ET | course.work | | Lab 4 | Friday 9/25 8AM ET | Canvas | **Come to office hours! You can attend anyone's office hours you want.** *No office hours or Piazza 9/21 because of the quiz!* --- # Homework 2 Comments 📝 - A causal *statement* is any sentence that is about causation. - "There is not evidence to say that eating chia seeds causes weight loss" **is** a causal statement - "Chia seeds do not cause weight loss" **is** a causal statement - Causal *statements* do not require causal *relationships* - Generalizability to a population is a result of sampling: how are data collected? - Sample size isn't really a big deal - Good (random) sampling = generalizable; bad sampling = not generalizable --- # Homework 2 Comments 📝 - Review your homework! Even questions you got full credit on. - You should have comments on every question you lost points on. - Remember we don't grade every question for correctness. --- class: center, middle # Weekly Advice We're focusing a lot on random sampling this week. ###Your mileage may vary! Your results *will* be different from mine and from your collaborators'. --- # Integer Sequences in R - A **vector** is a way to hold a collection of things in R. Think of it as a pill organizer. - This week, we're going to create a vector that holds a sequence of consecutive integers. ```r 1:6 ``` ``` [1] 1 2 3 4 5 6 ``` - Read the colon `:` as "through", so `1:6` is "1 through 6" --- # Sampling in R - Think of `1:6` as representing a six-sided die. - We can "roll" the die by taking a `sample()` from the vector `1:6` ```r sample(1:6, size = 1) ``` ``` [1] 1 ``` - Run the `sampleDieRoll` chunk (line ~63) and type what you got in the chat! --- # Sampling With vs. Without Replacement - Consider randomly selecting 6 values from the set {1, 2, 3, 4, 5, 6}. - Say our first pick is 3. - What do we do with 3? Do we take 3 out of the set (don't *replace* it), or do we put it back in (*replace* it)? -- ```r sample(1:6, size = 6, replace = F) ``` ``` [1] 3 5 2 4 1 6 ``` -- ```r sample(1:6, size = 6, replace = T) ``` ``` [1] 4 2 5 2 5 3 ``` -- - Which of these strategies represents die-rolling in real life? --- # Law of Large Numbers ♾️ As you collect more data, sample averages will get close to population averages ("*expected values*"). .pull-left[ Roll dice ``` [1] 5 [1] 3 5 [1] 6 6 3 [1] 2 1 1 5 [1] 2 3 3 3 6 ``` ] .pull-right[ Average of rolls ``` [1] 5 [1] 4 [1] 5 [1] 2.25 [1] 3.4 ``` ] -- The mean seems like it's trying to do something, but it's too variable to really see what's happening. --- # Law of Large Numbers ♾️ <img src="lab04-slides-async_files/figure-html/lln-1.png" style="display: block; margin: auto;" /> --- # Expected Value We can compute the value that the sample averages will converge to! $$ \sum_{i=1}^n x_i \cdot p_i $$ - `\(\Sigma\)` means "summation" (addition) - `\(x_i\)` is the value (in our case, 1, 2, 3, 4, 5, or 6) - `\(p_i\)` is the *probability* of observing the value For the six-sided die, the expected value is -- $$ 1 \cdot \left(\frac{1}{6}\right) + 2 \cdot \left(\frac{1}{6}\right) + 3 \cdot \left(\frac{1}{6}\right) + 4 \cdot \left(\frac{1}{6}\right) + 5 \cdot \left(\frac{1}{6}\right) + 6 \cdot \left(\frac{1}{6}\right) = 3.5$$ --- class:inverse ## Penguins! ```r penguins <- read.csv(url("https://raw.githubusercontent.com/STATS250SBI/palmerpenguins/master/inst/extdata/penguins_NArm.csv")) ``` .pull-left[ ```r str(penguins) 'data.frame': 333 obs. of 8 variables: $ species : chr "Adelie" "Adelie" "Adelie" "Adelie" ... $ island : chr "Torgersen" "Torgersen" "Torgersen" "Torgersen" ... $ bill_length_mm : num 39.1 39.5 40.3 36.7 39.3 38.9 39.2 41.1 38.6 34.6 ... $ bill_depth_mm : num 18.7 17.4 18 19.3 20.6 17.8 19.6 17.6 21.2 21.1 ... $ flipper_length_mm: int 181 186 195 193 190 181 195 182 191 198 ... $ body_mass_g : int 3750 3800 3250 3450 3650 3625 4675 3200 3800 4400 ... $ sex : chr "male" "female" "female" "female" ... $ year : int 2007 2007 2007 2007 2007 2007 2007 2007 2007 2007 ... ``` ] .pull-right[ ] --- # Scatterplots A **scatterplot** is a way to display the relationship between quantitative explanatory (x) and response (y) variables. The data are paired (x, y) and then each pair is plotted on a grid. We can use scatterplots to look for **associations** between these quantitative variables. --- # Scatterplots (line ~142) .pull-left[ ```r plot(penguins$bill_length_mm, penguins$body_mass_g, main = "Scatterplot of Bill Length versus Body Mass of Penguins", xlab = "Bill Length in mm", ylab = "Body Mass in g") ``` - positive association - reasonably linear - moderately strong - no apparent unusual points ] .pull-right[ <!-- --> ] --- # Correlation - The **correlation** between two quantitative variables quantifies the strength of the *linear* association. - Denote correlation by `\(r\)` - As `\(\lvert r \rvert\)` gets close to 1, the linear relationship becomes stronger <img src="lab04-slides-async_files/figure-html/corExample-1.png" style="display: block; margin: auto;" /> --- # Correlation (line ~155) We can find the correlation between two quantitative variables in R using the `cor()` function. ```r cor(penguins$bill_length_mm, penguins$body_mass_g) ``` ``` [1] 0.5894511 ``` --- # Lab Project ⌨️ .pull-left[ ### Your tasks - Complete the "Try It!" and "Dive Deeper" portions of the lab assignment by copy/pasting and modifying appropriate code from earlier in the document. ] .pull-right[ ### How to get help - Use the "labs" section of Piazza to ask questions and work with your peers. - If you use Piazza, please note that in the "Collaborators" list at the top of the discussion section. - If you're really stuck, email your lab instructor! ] --- # 📥 Lab Submission: Finding Your Report Hit the Knit button one last time, then: .pull-left[ ## RStudio Cloud 1. In the Files pane, check the box next to `lab04report.html` 2. Click More > Export... 3. Click Download and save the file on your computer in a folder you'll remember and be able to find later. ] .pull-right[ ## RStudio Desktop (local) 1. Locate the `lab04report.html` file on your computer. The file will be saved in the location indicated at the top of the files pane. ] --- # 📥 Lab Submission: Canvas (Due 9/25 8a ET) 1. Click the "Assignments" panel on the left side of the page. Scroll to find "Lab 4", and open the assignment. Click "Submit Assignment". 2. Towards the bottom of the page, you'll be able to choose `lab04report.html` from the folder you saved it in from RStudio Cloud or noted if you're using RStudio Desktop. **You will only be able to upload a .html file -- do not upload any other file type.** 3. Click "Submit Assignment". You're done!