worksheet_11.ipynb

Kernel: R

Worksheet 11 - Introduction to statistical inference

Lecture and Tutorial Learning Goals:

After completing this week's lecture and tutorial work, you will be able to:

• Describe real world examples of questions that can be answered with the statistical inference methods.

• Name common population quantities (e.g., mean, proportion, median, variance, standard deviation) that are often estimated using sample data, and use computation to estimate these.

• Define the following statistical sampling terms (population, sample, population quantity, point estimate, sampling distribution).

• Explain the difference between a population quantity and sample point estimate.

• Use computation to draw random samples from a finite population.

• Use computation to create a sampling distribution from a finite population.

• Describe how sample size influences the sampling distribution.

### Run this cell before continuing.
library(tidyverse)
library(repr)
library(digest)
library(infer)
library(gridExtra)
source('tests_worksheet_11.R')

Question 1.0 Multiple Choice:
{points: 1}

In which of the following questions would inferential methods (e.g., estimation or hypothesis testing) be appropriate?

A. Does treating a corn crop with Roundup cause greater yields compared to corn crops that are not treated with pesticides in Saskatchewan?

B. Are yields of corn crops which are treated with Roundup different than corn crops which are not treated with pesticides in Saskatchewan?

C. What will be the yield of a corn crop in Saskatchewan if we treat it with Roundup next year?

D. Are yields of corn crops which are treated with Roundup different than corn crops which are not treated with pesticides in the data set collected from the Rural Municipality of Cymri No. 36 in Saskatchewan?

Assign your answer to an object called answer1.0. Your answer should be a single character surrounded by quotes.

# your code here
fail() # No Answer - remove if you provide an answer

test_1.0()

Question 1.1 Matching:
{points: 1}

Read the mixed up table below and assign the variables in the code cell below a number to match the the term to it's correct definition. Do not put quotations around the number or include words in the answer, we are expecting the assigned values to be numbers.

Terms	Definitions
point estimate	1. the entire set of entities/objects of interest
population	2. selecting a subset of observations from a population where each observation is equally likely to be selected at any point during the selection process
random sampling	3. a numerical summary value about the population
representative sampling	4. a distribution of point estimates, where each point estimate was calculated from a different random sample from the same population
population quantity	5. a collection of observations from a population
sample	6. a single number calculated from a random sample that estimates an unknown population quantity of interest
observation	7. selecting a subset of observations from a population where the sample’s characteristics are a good representation of the population’s characteristics
sampling distribution	8. a quantity or a quality (or set of these) we collect from a given entity/object

point_estimate <- NULL
population <- NULL
random_sampling <- NULL
representative_sampling <- NULL
population_quantity <- NULL
sample <- NULL
observation <- NULL
sampling_distribution <- NULL

# your code here
fail() # No Answer - remove if you provide an answer
digest(population)

test_1.1()

Virtual sampling simulation

In real life, we rarely, if ever, have measurements for our entire population. Here, however we will pretend that we somehow were able ask every single Candian senior what their age is. We will do this so that we can experiment to learn about sampling and how this relates to estimation.

Here's a human readable scenario that we will create a population for:

Here we make a make a population (ages of all Canadian seniors) bounded by realistic values ($\geq$ 65 and $\leq$ 118):

# run this cell to simulate a finite population
set.seed(4321) # DO NOT CHANGE
can_seniors <- tibble(age = (rexp(2000000, rate = 0.1)^2) + 65) %>% 
    filter(age <= 117, age >= 65)
head(can_seniors)

Question 1.2
{points: 1}

A distribution defines all the possible values (or intervals) of the data and how often they occur. Visualize the distribution of the population (can_seniors) that was just created by plotting a histogram using binwidth = 1 in the geom_histogram argument. Name the plot pop_dist and give x axis a descriptive label.

options(repr.plot.width = 4, repr.plot.height = 3)
#pop_dist <- ggplot(..., ...) + 
#    geom_...(...) +
#    ... +
#    ggtitle("Population distribution")

# your code here
fail() # No Answer - remove if you provide an answer
pop_dist

test_1.2()

Question 1.3
{points: 1}

Distributions are complicated to communicate, thus we often want to represent them by a single value or small number of values. Common values used for this include the mean, the mean, median, standard deviation, etc).

Use summarise to calculate the following population quantities from the can_seniors population:

• mean (use the mean function)

• median (use the median function)

• standard deviation (use the sd function)

Name this data frame pop_quantities which has the column names pop_mean, pop_med and pop_sd.

# your code here
fail() # No Answer - remove if you provide an answer
pop_quantities

test_1.3()

Question 1.4
{points: 1}

In real life, we usually are able to only collect a single sample from the population. We use that sample to try to infer what the population looks like.

Take a single random sample of 40 observations from the Canadian seniors population (can_seniors). Name it sample_1. Use 4321 as your seed.

set.seed(4321) # DO NOT CHANGE!
#sample_1 <- ... %>% 
#    rep_sample_n(...)
# your code here
fail() # No Answer - remove if you provide an answer
head(sample_1)

test_1.4()

Question 1.5
{points: 1}

Visualize the distribution of the random sample you just took (sample_1) that was just created by plotting a histogram using binwidth = 1 in the geom_histogram argument. Name the plot sample_1_dist and give the plot (using ggtitle) and the x axis a descriptive label.

# your code here
fail() # No Answer - remove if you provide an answer
sample_1_dist

test_1.5()

Question 1.6
{points: 1}

Use summarise to calculate the following point estimates from the random sample you just took (sample_1):

• mean

• median

• standard deviation

Name this data frame sample_1_estimates which has the column names sample_1_mean, sample_1_med and sample_1_sd.

# your code here
fail() # No Answer - remove if you provide an answer
sample_1_estimates

test_1.6()

Let's now compare our random sample to the population from where it was drawn:

# run this code cell
options(repr.plot.width = 6)
grid.arrange(pop_dist, sample_1_dist, ncol = 2)

And now let's compare the point estimates (mean, median and standard deviation) with the true population quantities we were trying to estimate:

# run this cell
pop_quantities
sample_1_estimates %>% select(-replicate)

Question 1.7 Multiple Choice
{points: 1}

After comparing the population and sample distributions above, and the true population quantities and the sample point estimates, which statement below is not correct:

A. The sample point estimates are close to the values for the true population quantities we are trying to estimate

B. The sample distribution is of a similar shape to the population distribution

C. The sample point estimates are identical to the values for the true population quantities we are trying to estimate

Assign your answer to an object called answer1.7. Your answer should be a single character surrounded by quotes.

# your code here
fail() # No Answer - remove if you provide an answer

test_1.7()

Question 1.8.0
{points: 0}

What if we took another sample? What would we expect? Let's try! Take another random sample from population (use a different random seed this time so that you get a different sample), and visualize its distribution and calculate the point estimates for the sample mean, median and standard deviation:

# your code here
fail() # No Answer - remove if you provide an answer

Question 1.8.1
{points: 1}

After comparing the distribution and point estimates of this second random sample from the population with that of the first random sample and the population, which statements below is not correct:

A. The sample distributions from different random samples are of a similar shape to the population distribution, but they vary a bit depending which values are captured in the sample

B. The sample point estimates from different random samples are close to the values for the true population quantities we are trying to estimate, but they vary a bit depending which values are captured in the sample

C. Every random sample from the same population should have an indentical distribution and yield identical point estimates.

Assign your answer to an object called answer1.8.1. Your answer should be a single character surrounded by quotes.

# your code here
fail() # No Answer - remove if you provide an answer

test_1.8.1()

Exploring the sampling distribution of an estimate

Just how much should we expect the point estimates of our random samples to vary? To build an intuition for this, let's experiment a little more with our population of Canadian seniors. To do this we will take 1500 random samples, and then calculate the point estimate we are interested in (let's choose the mean for this example) for each sample. Finally, we will visualize the distribution of the sample point estimates. This distribution will tell us how much we would expect the point estimates of our random samples to vary for this population for samples of size 40 (the size of our samples).

Question 1.9
{points: 1}

Draw 1500 random samples from our population of Canadian seniors (can_seniors). Each sample should have 40 observations. Name the data frame samples and use the seed 4321.

#samples <- rep_sample_n(..., size = ..., reps = ...)
set.seed(4321) # DO NOT CHANGE!
# your code here
fail() # No Answer - remove if you provide an answer
head(samples)
tail(samples)
dim(samples)

test_1.9()

Question 2.0
{points: 1}

Group by the sample replicate number, and then for each sample, calculate the mean as the point estimate. Name the data frame sample_estimates. The data frame should have the column names replicate and sample_mean.

# your code here
fail() # No Answer - remove if you provide an answer
head(sample_estimates)
tail(sample_estimates)

test_2.0()

Question 2.1
{points: 1}

Visualize the distribution of the sample estimates (sample_estimates) you just calculated by plotting a histogram using binwidth = 1 in the geom_histogram argument. Name the plot sampling_distribution and give the plot (using ggtitle) and the x axis a descriptive label.

# your code here
fail() # No Answer - remove if you provide an answer
sampling_distribution

test_2.1()

Question 2.2
{points: 1}

Let's refresh our memories, what is the mean age of the population (we calculated this above). Assign your answer to an object called answer2.2. Your answer should be a single number.

# your code here
fail() # No Answer - remove if you provide an answer
answer2.2

test_2.2()

Question 2.3 Multiple Choice
{points: 1}

Considering the true value for the population mean, and the sampling distribution you created and visualized in question 2.1, which statement below is not correct:

A. The sampling distribution is centered at the true population mean

B. All the sample means are the same value as the true population mean

C. Most sample means are at or very near the same value as the true population mean

D. Few sample means are far away from the same value as the true population mean

Assign your answer to an object called answer2.3. Your answer should be a single character surrounded by quotes.

# your code here
fail() # No Answer - remove if you provide an answer
answer2.3

test_2.3()

Question 2.4 True/False
{points: 1}

Taking a random sample and calculating a point estimate is a good way to get a "best guess" of the population quantity you are interested in. True or False?

Assign your answer to an object called answer2.4. Your answer should be a either "True" or "False", surrounded by quotes.

# your code here
fail() # No Answer - remove if you provide an answer
answer2.4

test_2.4()

The influence of sample size on the sampling distribution

What happens to our point estimate when we change the sample size? Let's answer this question by experimenting! We will create 3 different sampling distributions of sample means, each using a different sample size. As we did above, we will draw samples from our Canadian seniors population. We will visualize these sampling distributions and see if we can see a pattern when we vary the sample size.

Question 2.5
{points: 1}

Using the same strategy as you did above, draw 1500 random samples from the Canadian seniors population (can_seniors), each of size 20. For each sample, calculate the mean age. Then visualize the distribution of the sample estimates (means) you just calculated by plotting a histogram using binwidth = 1 in the geom_histogram argument. Name the plot sampling_distribution_20 and give the x axis a descriptive label. Give the plot the title "n = 20". Also specify the x-axis limits to be 65 and 95 using xlim(c(65, 95)).

Set the seed as 4321 when you collect your samples.

set.seed(4321) # DO NOT CHANGE THIS!
# your code here
fail() # No Answer - remove if you provide an answer
sampling_distribution_20

test_2.5()

Question 2.6
{points: 1}

Using the same strategy as you did above, draw 1500 random samples from the Canadian seniors population (can_seniors), each of size 100. For each sample, calculate the mean age. Then visualize the distribution of the sample estimates (means) you just calculated by plotting a histogram using binwidth = 1 in the geom_histogram argument. Name the plot sampling_distribution_100 and give the x axis a descriptive label. Give the plot the title "n = 100". Also specify the x-axis limits to be 65 and 95 using xlim(c(65, 95)).

Set the seed as 4321 when you collect your samples.

set.seed(4321) # DO NOT CHANGE THIS!
# your code here
fail() # No Answer - remove if you provide an answer
sampling_distribution_100

test_2.6()

# run this cell to change the sampling distribution plot created
# earlier in the notebook so that the x-axis is the same dimensions
# as the other two plots you just made, and so that the title is "n = 40"
sampling_distribution <- sampling_distribution + 
    xlim(c(65, 95))
sampling_distribution$labels$title <- "n = 40"

Question 2.7
{points: 0}

Fill in the blanks in the code below to use grid.arrange to plot the three sampling distributions side-by-side. Order them from smallest sample size on the left, to largest sample size on the right. Name the final panel figure sampling_distribution_panel.

options(repr.plot.width = 8)
# sampling_distribution_panel <- grid.arrange(...,
#                                           ...,
#                                           ...,
#                                           ncol = 3)
# your code here
fail() # No Answer - remove if you provide an answer
sampling_distribution_panel

Question 2.8 Multiple Choice
{points: 1}

Considering the panel figure you created above in question 2.7, which statement below is not correct:

A. As the sample size increases, the sampling distribution of the point estimate becomes narrower.

B. As the sample size increases, more sample point estimates closer to the true population mean

C. As the sample size decreses, the sample point estimates become more variable.

D. As the sample size increases, the sample point estimates become more variable.

Assign your answer to an object called answer2.8. Your answer should be a single character surrounded by quotes.

# your code here
fail() # No Answer - remove if you provide an answer
answer2.8

test_2.8()

Question 2.9 True/False
{points: 1}

Given what you observed above, and considering the real life scenario where you will only have one sample, answer the True/False question below:

The smaller your random sample, the better your sample point estimate reflect the true population quantity you are trying to estimate. True or False?

Assign your answer to an object called answer2.9. Your answer should be a either "True" or "False", surrounded by quotes.

# your code here
fail() # No Answer - remove if you provide an answer
answer2.9

test_2.9()