• Fri 5/19 7pm-10pm in Warner 506
• Not a final, but 3rd midterm. Timed at ~1h15m to 1h30m
• Bring your cheatsheets
• Bring a calculator or your smart phone with calculator app

• More conceptual in nature
• Code:
  • Reading/understanding: Fair game
  • Writing: No direct code to write, but pseudocode
• Normal curve of distribution of difficulty

• Lectures 01 through 38 inclusive and cummulative
  • Slides from each lecture
  • Learning Checks
  • Problem set solutions!

• Tidy data. What are the components?
• What is the Grammar of Graphics? How do they tie in with ggplot2?
• What are the first four of the 5NG? What are their distinguishing features?

• All five of the 5NG
• Data manipulation/wrangling
  • 5MV + join
  • Putting it all together: Lec18 - Fri 3/24: The tao of data analysis.
• Sampling, probability, confounding variables, and designed experiments.

• Hypothesis testing
  • Lady tasting tea.
  • There is only one test; it has 5 components.
• Confidence intervals
  • Theory: Sampling distribution and standard errors
  • Interpretation of CI
  • If sampling distribution is normal, the general formula for creating a 95% C.I.

• Regression
  • Regression line is best fitting line in what sense?
  • Interpret ALL regression table outputs
  • Study residuals
  • Categorical variables
  • Multiple Regression

So far we've seen simple linear regression

• Simple means only one predictor/independent variable \(x\)
• Outcome/depedendent variable \(y\)
• \(x\) can be either numerical or categorical

In Lec 36 LC we saw the relationship between \(x =\) dep delay & \(y =\) arr delay for Alaska Airlines flights.

• Since we only have Alaska flights, the variable carrier doesn't vary.
• But now let's also consider Frontier Airlines (carrier == F9)

So we have:

• \(y =\) arrival delay
• \(x_1 =\) departure delay (numerical variable)
• \(x_2 =\) carrier (categorical variable with \(k=2\) levels. In other words, carrier now varies.)

Is there a difference in delays between Alaska and Frontier?

Is there a difference in delays between Alaska and Frontier?

• Continuing Regression Outputs: Lec36 Learning Check
• Categorical Predictors

What does "best fitting line"" mean?

Consider ANY point (in blue).

Now consider this point's deviation from the regression line.

Do this for another point…

Do this for another point…

Regression line minimizes the sum of squared arrow lengths.

• Residuals
• Review of Lec36 Learning Check outputs
• Regression viewed through the lens of sampling

• In Lec35 LC you all created your own 95% C.I. to estimate the proportion \(p\) of the OkCupid dataset which is female
• We took a single sample of size n=100
• We pretended we didn't know the true \(p = 0.4023\) = 40.23%

Here are your 12 resulting \(\widehat{p}\)'s…

Let me add 8 of my own so we have 20…

Let's compute \(\mbox{SE} = \sqrt{\frac{\widehat{p}(1-\widehat{p})}{n}}\)…

p_hat <- p_hat %>% 
  mutate(
    n = 100,
    SE = sqrt(p_hat*(1-p_hat)/n)
  )

Finally the left and right end points of the 95% confidence interval. Whose CI's captured the true \(p=0.4023\)?

p_hat <- p_hat %>% 
  mutate(
    left = p_hat - 1.96*SE,
    right = p_hat + 1.96*SE
  )

• Dots are \(\widehat{p}\)
• Dashed line is true \(p=0.4023\)

• Final topic for this course!
• Correlation Coefficient

Recall the nycflights data set. For Alaska Air flights, let's explore the relationship between

• Departure delay
• Arrival delay

The correlation coefficient is computed as follows:

cor(alaska_flights$dep_delay, alaska_flights$arr_delay)

## [1] 0.8373792

83.7% is fairly strongly positively associated!

Play Guess the Correlation

Chalk talk

For large \(n\), the sampling distribution for these point estimates are bell-shaped, thus a 95% C.I. is \(\mbox{PE} \pm 1.96\times \mbox{SE}\).

NPR report on Obama from 2013. Chalk talk…

We are estimating a population parameter using a point estimate based on a sample. Example: Mean (Chalk Talk)

Imagine the \(\mu\) is a fish:

Point Estimate \(\overline{x}\)	Confidence Interval

• Lec33 Learning Check Discussion
• Chalk Talk.

Age example:

1. I picked a random sample of n=3 students
2. I computed sample mean age \(\overline{x}\)
3. I did this three times

Note:

1. They are not the same because of sampling variability
2. What quantifies how much these point estimates vary?

From the OkCupid population:

1. Take samples of size n
2. Compute sample mean height \(\overline{x}\)
3. Do this many, many, many times (10000)
4. Visualize distribution of these sample means

Taking a sample in order to infer about a population:

Let's Google "define infer"…

library(lubridate)
library(mosaic)
library(dplyr)

# Randomly sample three people:
students <- 
  c("Arthur", "Caroline", "Claire", "Clare", "Conor", "Daniel", 
    "Dylan", "Elana", "Jacob", "Jay", "Joe", "Julian", "Kelsie", 
    "Lisa", "Maya", "Naing", "Parker", "Rebecca", "Ry", "Theodora", 
    "Zebediah", "Albert")
resample(students, size=3, replace=FALSE)

# Get average age:
birthdays <- c("1980-11-05", "2000-01-01", "1955-08-05")
ages <- as.numeric(as.Date("2017-04-27") - as.Date(birthdays))/365.25
ages
mean(ages)

• We randomly sample 3 students and get mean age
• We randomly sample 3 students and get mean age
• We randomly sample 3 students and get mean age…

Questions:

1. Why is the mean (AKA) age different each time?
2. What numerical summary quantifies how these means vary?

Chalk talk…

• Hypothesis testing in general
• Background statistical theory

• View Lec29 Learning Check
• Chalk talk

If we assume \(H_0\) is true (there is no difference in test scores between evens and odds) then:

1. Whether you have an even number of letters or odd is irrelevant
2. Hence the categorical variable even_vs_odd is irrelevant
3. Hence we can permute/shuffle it to no consequence

From last lecture: How do we construct null distribution?

In this case, the null distribution is barplot:

• Analytically/Mathematically: Necessitates probability background. Covered in MATH 310.
• Simulation: Necessitates random number generator. We take this approach.

• Lady Tasting Tea: Assuming she is guessing at random, we simulated many, many, many instances of "the number she got right".
• Odds vs Evens Test Score: Chalk talk

Only chalk talk today, based on Learning Checks for Lec26.

1. Null hypothesis \(H_0\): she is guessing at random
2. Alternative hypothesis \(H_A\): she is not. i.e. she can really tell which came first
3. Test statistic: # of correct guesses out of 8
4. Observed test statistic: 8 correct
5. Null distribution: the bar chart
6. Decision: compare observed test statistic to null distribution

• We going to assume \(H_0\) is true and see how likely the observed test statistic was as compared to null distribution.
• How likely was the observed test statistic?

Not very! Only occurs 0.34% of the time

p-value: Chalk Talk

• Lady Tasting Tea claims she can tell if tea or milk was poured first into a cup
• You run an experiment with 8 cups if she can tell or if she is bullshitting
• Let's assume a hypothetical world where she is guessing at random.

If guessing at random, here are hypothetical outcomes:

She got 8/8 right!

• In our hypothetical world of guessing at random, 8/8 occured 34 times out of 10000. i.e. 0.34% of the time.
• Can she tell, or is she bullshitting?

Critical chalk talk.

• Evening Exam: Wed 4/7 7pm in Warner 506
• Closed book, no calculators
• Bring your cheatsheets

• More conceptual in nature
• Code:
  • Reading/understanding: Fair game
  • Writing: No direct code to write, but pseudocode
• Normal curve of distribution of difficulty

• Lectures 01 through 21 inclusive and cummulative
  • Slides from each lecture
  • Corresponding textbook material (if any)
  • Learning Checks
  • Problem Sets

• Tidy data. What are the components?
• What is the Grammar of Graphics? How do they tie in with ggplot2?
• What are the first four of the 5NG? What are their distinguishing features?

• All five of the 5NG
• Data manipulation/wrangling
  • 5MV + join
  • Putting it all together: Lec18 - Fri 3/24: The tao of data analysis.
• Sampling, probability, confounding variables, and designed experiments.

• Disclaimer, disclaimer, disclaimer
  • Do not overly interpret the content of this midterm.
  • Rather, view it to get a rough sense of my exam philosophy.
• Note: There was no probability in last year's Midterm II

• Say you are a statistician and you meet someone called the "Lady Tasting Tea."
• She claims she can tell by tasting whether the tea or the milk was added first to a cup.
• You want to test whether
  • She can actually tell which came first OR
  • She's lying and is only guessing at random
• Say you have just enough tea/milk to pour into 8 cups.

Binary situations, like

• True vs False
• Correct vs Incorrect
• Yes vs No

are often coded as 1 vs 0 in many programming languages.

1. Random sampling: To obtain a representative sample from a population.
2. Random assignment: To design an experiement.

• Correlation is not necessarily causation
• Spurious correlations

• Confounding variables
• Two types of studies
• Principles of designing experiments

Ezell's Fried Chicken is a famous chicken restaurant in Seattle. Oprah Winfrey has it flown into to Chicago.

One day I was raving about Ezell's Chicken, but my friend accused me of "buying into the hype".

So what did we do?

Fried Chicken Face Off:

Do people prefer this?	Or this?

How would you design a taste test to ascertain, independent of hype, which fried chicken tastes better?

Use the relevant principles of designing experiements from above.

The mosaic package has functions for the random simulation.

1. rflip(): Flip a coin
2. shuffle(): Shuffle a set of values
3. do(): Do the same thing many, many, many times
4. resample(): the swiss army knife for sampling

Run the following in your console:

library(mosaic)
# Define a vector fruit
fruit <- c("apple", "orange", "mango")

# Do this multiple times:
shuffle(fruit)

Two types of sampling:

1. Sampling with replacement
2. Sampling without replacement

resample() by default samples with replacement. Run this in the console multiple times:

resample(fruit)

Chalk Talk

Chalk Talk 1

• In short: Probability is the study of randomness.
• Its roots lie in one historical constant
• It is the theoretical backbone of statistics.

There are two approaches to studying probability:

Mathematically (MATH 310)	Via Simulations

• The mathematical approach requires A LOT of math background, whereas the simulation approach does not.
• To do simulations, we need a computer's random number generator. Why?

Doing this repeatedly by hand is tiring:

All hail the mosaic package: library(mosaic).

Chalk Talk 2

Best viewed in HTML mode, not slide deck mode:

1. Define your target
2. Clean up your workspace
3. Have a plan

You should draw out what your end data frame should look like in tidy format:

• How many columns and what are their names i.e. the variables
• How many rows will I have?
• Fill in elements of the table the best you can.

Why? If you don't clearly identify this, not only will your work not be focused, but more importantly, how would you know when you're done?

Before starting any substantive data wrangling using mutate, summarise, arrange, or _join, I like to pare down the necessary data sets to the minimum of what I need by

• filter only the absolutely necessary rows
• Optional: select only the absolutely neccesary columns

Why? This has several benefits:

1. It will make it much easier to digest View()s of your work as you progress.
2. It will minimize the chance of weird errors creeping in.
3. Most importantly: It will force you to think
   • "What variables do I need?" and hence
   • "Where are they located?" and hence
   • "What data sets do I need?"

1. Explicitly draw out on paper/blackboard what the end data frame should look like.
2. Based on the variables in this table, reverse-engineer what variables you will need from the five nycflights13 data sets: flights, planes, airlines, weather, and airports.
3. Explicitly write out pseudocode on paper/blackboard all data wrangling steps you need to get to your goal. i.e. What am I doing?
4. Only after you've done 1-3, then start coding. i.e. How am I going to do it?

Why? If you confuse the what and the how, you'll only get doubly lost. Separate them out!

Done with "Tidy" and "Transform", start with "Model":

Growing up I used to only eat white rice, but now I only eat multigrain rice.

White Rice	Multigrain Rice