# run before presenting to create population
# create virtual box
options(repr.matrix.max.rows = 6)
library(tidyverse)
set.seed(1234)
virtual_box <- tibble(timbit_id = seq(1, 10000, by = 1),
                     color = factor(rbinom(10000, 1, 0.63),
                     labels = c("old fashioned", "chocolate")))

DSCI 100 - Introduction to Data Science¶

Lecture 11 - Introduction to inference & sampling¶

Source: Desirée De Lyon's Tea cup Giraffe's statistics course

Housekeeping¶

Today: lecture 11 & 12
Thursday: last tutorial & project work session
Final project and teamwork document due Sat Aug 14
Final exam (quiz 3): Wed Aug 18 at 8:30am (PT)
- check UBC Exam Schedule to be sure!
- Quiz 3 will cover textbook linear regression (only sections 8.1 to 8.6), clustering and statistical inference
- 45 minutes + 5 minutes for connectivity issues
- I will be available in the Zoom course room during the exam if you have questions
Regular office hours end this week
- My office hour before exam: Monday, Aug 16 at 9 AM
Remember to download material before then
- Last chapter of the textbook has instructions
Course evaluation is live (left hand side of Canvas)

Survey results¶

Thank you for your responses!

What is working?¶

Worksheets & tutorials
Open book for quizzes
Piazza for asking questions
Recorded lectures

What is NOT working¶

Tight timing of the quizzes
When you are alone in class with no group members
Quiet Zoom rooms
want more coding demos related to worksheets/tutorials code
Tutorial feedback takes time
Large workload

What is statistical inference?¶

Statistical inference is the process of using a sample to make conclusions about a wider population the sample came from.

For example, what if we wanted to know the width of a species of sea star that lives in the pacific ocean?¶

What is statistical inference?¶

Statistical inference is the process of using a sample to make conclusions about a wider population the sample came from.

For example, what if we wanted to know the width of a species of sea star that lives in the pacific ocean?¶

What is statistical inference?¶

Statistical inference is the process of using a sample to make conclusions about a wider population the sample came from.

For example, what if we wanted to know the width of a species of sea star that lives in the pacific ocean?¶

What is statistical inference?¶

Statistical inference is the process of using a sample to make conclusions about a wider population the sample came from.

For example, what if we wanted to know the width of a species of sea star that lives in the pacific ocean?¶

Population: collection of all possible observations + their frequency/count
- e.g. the widths of the entire population of all sea stars (from the species we are interested in)
Sample: a randomly selected subset of observations
- e.g. I randomly pick 50 sea stars (from the species we are interested in) and record their widths
Inference: Using the sample to make a conclusion about the population
- e.g. use those 50 measurements to estimate the average width of all sea stars (from the species we are interested in)

The process of inferring something about the population based on what is measured in the sample is called statistical inference
We want to use the information from our sample to say something concrete about the larger population.
for instance, we may was to estimate the average price of all one bedroom houses in Vancouver

Things we can do with inference¶

1. Use poll data to make conclusions about the population¶

Based on the results of a recent Reuters poll, we estimate that 47.2% of Americans think that firearms should have strong regulations or restrictions when thinking about gun ownership rights and gun laws.

source: http://polling.reuters.com/#!response/PV20/type/smallest/dates/20180505-20181002/collapsed/true

2. Answer marketing questions such as:¶

What proportion of undergraduate students have an iphone?

3. Answer A/B testing questions:¶

(not in DSCI100 - you'll need to take a more advanced DSCI or STAT class for this!)

Which of the 2 website designs will lead to more customer engagement (measured by click-through-rate, for example)?

source: https://images.ctfassets.net/zw48pl1isxmc/4QYN7VubAAgEAGs0EuWguw/165749ef2fa01c1c004b6a167fd27835/ab-testing.png

Estimation¶

A particular inferential problem where we try to estimate a quantitative property of the population.

we give that quantitative property a special name: parameter

Marketing example revisited¶

Question: What proportion (a quantitative population property!) of all undergraduates have an iphone?

Marketing example revisited¶

What if we randomly selected a subset (a sample) and then asked them if they have an iphone?

We could then calculate the proportion in our sample and use it as an estimate of the true population proportion.

Virtual sampling simulation¶

Let's experiment and see how well sample estimates reflect the true population parameter we are interested in measuring!

Question: What proportion of Timbits are chocolate?

Let's create a virtual box of timbits (our population)
Then:
- collect a random sample of 40 timbits,
- calculate a proportion of chocolate timbits

As always, load the libraries we'll be using:¶

# load libraries for wrangling and plotting
library(dplyr)
library(infer)

1. Our a virtual box of timbits (population)¶

Let's checkout our population of 10000 timbits where the proportion of chocolate is 0.63 and the proportion of old fashioned is 0.37.

virtual_box

Check that the virtual box contains ~ 63% chocolate timbits:

virtual_box %>% 
    group_by(color) %>% 
    summarize(n = n(),
             proportion = n() / 10000)

2. Drawing a single sample of size 40¶

Let's simulate taking one random sample from our virtual timbits box. We will use the rep_sample_n function from the infer package:

# draw a single sample from the virtual box

samples_1 <- rep_sample_n(virtual_box, size = 40)
samples_1

We can tell by the timbit_id column that R indeed did what we asked - randomly selected 40 timbits from our virtual box.

What is the proportion of chocolate in our single sample?¶

Is it close to the population proportion?

choc_sample_1 <- summarize(samples_1, n = sum(color == "chocolate"),
                                    prop = sum(color == "chocolate") / 40)
choc_sample_1

What happens if we were to take another sample?¶

Would the sample proportion be the same?

# random sample from virtual box

choc_sample_2 <- rep_sample_n(virtual_box, size = 40) %>% 
    summarize(n = sum(color == "chocolate"),
              prop = sum(color == "chocolate") / 40)
choc_sample_2

What about another?¶

Would the sample proportion be the same?

# random sample from virtual box
choc_sample_3 <- rep_sample_n(virtual_box, size = 40) %>% 
    summarize(n = sum(color == "chocolate"),
              prop = sum(color == "chocolate") / 40)
choc_sample_3

choc_sample_3 <- rep_sample_n(virtual_box, size = 40) %>% 
    summarize(n = sum(color == "chocolate"),
              prop = sum(color == "chocolate") / 40)
choc_sample_3

What values do we expect when we sample?¶

choc_sample_1
choc_sample_2
choc_sample_3

Let's vote! What are these proportion values?¶

parameters
statistics
samples
populations

We are going to explore this in the worksheet, with the goal of working towards how understanding this will allow us to describe how accurate our sample statistics are.

# take many random samples from virtual box

replicates <- rep_sample_n(virtual_box, size = 40, reps = 1000)  %>% 
   group_by(replicate)  %>% 
   summarize(proportion = sum(color == "chocolate")/40)

Back to our marketing example¶

Is randomly selecting a subset of the students (taking a single sample) and then asking them if they have an iphone a good way to estimate the true proportion of all undergraduates who have iphones (population parameter we are interested in)?

Worksheet time - go for it!¶

True or False:

A larger sample will give a more accurate sample statistic.
If our population is only size 20, we should just measure all the observations instead of sampling.
In order to understand how reliable our sample estimate is, we can take many replicates and see how the estimate varies.
UBC has implanted a secret microchip in all of your brains to know whether or not you have an iPhone, and so doesn't need to sample.

Acknowledgements¶

Data Science in a box by Mine Cetinkaya-Rundel
Inference in 3 hours by Allan Downey
Modern Dive: An Introduction to Statistical and Data Sciences via R by Chester Ismay and Albert Y. Kim