11_introduction_to_inference

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# run before presenting to create population
# create virtual box
options(repr.matrix.max.rows = 6)
library(tidyverse)
set.seed(1)
virtual_ugrads <- tibble(student_id = seq(1, 50000, by = 1),
                     phone_type = factor(rbinom(50000, 1, 0.63),
                     labels = c("other", "iphone")))

DSCI 100 - Introduction to Data Science¶

Lecture 11 - Introduction to inference & sampling¶

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

Reminders¶

Final exam (quiz 3) April 26, 19:00PM -- All sections!
- Section 002 - Alphabetical split:
  - AHM-OUY PHRM 1101
  - PAN-ZHU PHRM 1201
- Section 005: LIFE 2201
- Section 006: CIRS 1250
- Section 007: ESB 1013

Be respectful and polite.

What is statistical inference?¶

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

Why?: It's often expensive or not possible to measure the whole population.

E.g.: we want to know the average width of a species of sea star¶

What is statistical inference?¶

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

Why?: It's often expensive or not possible to measure the whole population.

E.g.: we want to know the average width of a species of sea star¶

What is statistical inference?¶

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

Why?: It's often expensive or not possible to measure the whole population.

E.g.: we want to know the average width of a species of sea star¶

What is statistical inference?¶

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

Why?: It's often expensive or not possible to measure the whole population.

E.g.: we want to know the average width of a species of sea star¶

Population: collection of all possible observations + their frequency/count
- e.g. the widths of the entire population of sea stars
Sample: a randomly selected subset of observations
- e.g. I randomly pick 50 sea stars and record their widths
Statistic: something I compute using my sample
- e.g. the mean width of the sea stars in my sample
Inference: using the sample to make a conclusion about the whole population, and knowing how uncertain you are about your conclusion

Inference in the wild: opinion polling¶

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

Inference in the wild: market assessment¶

What proportion of undergraduate students have an iphone?

Inference in the wild: A/B testing¶

(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

This quantitative property is called a population parameter

Question: What proportion (a quantitative population parameter!) of UBC undergrads have an iphone?

Estimation¶

Step 1: randomly select a subset (a sample) and ask them if they have an iPhone

Step 2: calculate the proportion in our sample (a statistic or point estimate) and use it as an estimate of the true population proportion.

Virtual simulation experiment¶

Let's simulate this process to see how well sample estimates reflect the true population parameter!

Question: What proportion of UBC undergrads have an iPhone?

Let's create a virtual group of students (our population) where 63% of the students have iPhones
Then:
- collect a random sample of 40 students,
- calculate a proportion of students with iPhones

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

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

1. Our virtual UBC students (population)¶

Let's examine our population of 50,000 students. Remember that the true proportion of iPhone users is 63%.

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virtual_ugrads

Check that the virtual population contains ~ 63% iPhone users:

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virtual_ugrads %>% 
    group_by(phone_type) %>% 
    summarize(n = n(),
             proportion = n() / 50000)

2. Drawing a single sample of size 40¶

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

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# draw a single sample from the virtual population

student_sample <- rep_sample_n(virtual_ugrads, size = 40)
student_sample

We can tell by the student_id column that R indeed did what we asked - randomly selected 40 students from our virtual population.

What is the proportion of iPhone users in our sample?¶

Is it close to the population proportion?

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# compute the proportion of iPhone users

iphone_estimate_1 <- summarize(student_sample, n = sum(phone_type == "iphone"),
                                    prop = sum(phone_type == "iphone") / 40)
iphone_estimate_1

What happens if we were to take another sample?¶

Would the sample proportion be the same?

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# random sample from virtual population

iphone_estimate_2 <- rep_sample_n(virtual_ugrads, size = 40) %>% 
    summarize(n = sum(phone_type == "iphone"),
              prop = sum(phone_type == "iphone") / 40)
iphone_estimate_2

What about another?¶

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# random sample from virtual student population
iphone_estimate_3 <- rep_sample_n(virtual_ugrads, size = 40) %>% 
    summarize(n = sum(phone_type == "iphone"),
              prop = sum(phone_type == "iphone") / 40)
iphone_estimate_3

iphone_estimate_3 <- rep_sample_n(virtual_ugrads, size = 40) %>% 
    summarize(n = sum(phone_type == "iphone"),
              prop = sum(phone_type == "iphone") / 40)
iphone_estimate_3

What values do we expect when we sample?¶

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iphone_estimate_1
iphone_estimate_2
iphone_estimate_3

Sampling distribution¶

The distribution of estimates you get by taking many samples of a fixed sample size is called the sampling distribution.

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# take many samples from virtual student population

replicates <- rep_sample_n(virtual_ugrads, size = 40, reps = 10000)  %>% 
   group_by(replicate)  %>% 
   summarize(proportion = sum(phone_type == "iphone")/40)

Sampling distribution¶

The distribution of estimates you get by taking many samples of a fixed sample size is called the sampling distribution.

How did we do? How reliable is our sample estimate? Should we do anything to change it?

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ggplot(replicates, aes(x = proportion)) +
    geom_histogram(binwidth = .05) +
    labs(x = "Proportion of iPhone Users", y = "Count") +
    theme(text = element_text(size=20))

Worksheet time - go for it!¶

Go to Canvas, and fill out the survey `iphone_poll` in week 11!¶

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.

Let's vote! Which of these are these proportion values?¶

parameters
statistics
samples
populations

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

DSCI 100 - Introduction to Data Science¶

Lecture 11 - Introduction to inference & sampling¶

Reminders¶

What is statistical inference?¶

E.g.: we want to know the average width of a species of sea star¶

What is statistical inference?¶

E.g.: we want to know the average width of a species of sea star¶

What is statistical inference?¶

E.g.: we want to know the average width of a species of sea star¶

What is statistical inference?¶

E.g.: we want to know the average width of a species of sea star¶

Inference in the wild: opinion polling¶

Inference in the wild: market assessment¶

Inference in the wild: A/B testing¶

Estimation¶

Estimation¶

Virtual simulation experiment¶

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

1. Our virtual UBC students (population)¶

2. Drawing a single sample of size 40¶

What is the proportion of iPhone users in our sample?¶

What happens if we were to take another sample?¶

What about another?¶

What values do we expect when we sample?¶

Sampling distribution¶

Sampling distribution¶

Worksheet time - go for it!¶

Go to Canvas, and fill out the survey iphone_poll in week 11!¶

True or False:¶

Let's vote! Which of these are these proportion values?¶

Acknowledgements¶

Go to Canvas, and fill out the survey `iphone_poll` in week 11!¶