worksheet_inference1

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 parameters (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 parameter, point estimate, sampling distribution). Explain the difference between a population parameter 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. This worksheet covers parts of the Inference chapter of the online textbook. You should read this chapter before attempting the worksheet. ### Run this cell before continuing. library ( tidyverse ) library ( repr ) library ( infer ) library ( cowplot ) options ( repr.matrix.max.rows = 6 ) source ( 'tests.R' ) source ( 'cleanup.R' ) 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 In [ ]:

Terms Definitions 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 parameter 5. a collection of observations from a population sample 6. a single number calculated from a random sample that estimates an unknown population parameter 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_parameter <- NULL sample <- NULL observation <- NULL sampling_distribution <- NULL ### BEGIN SOLUTION point_estimate <- 6 population <- 1 random_sampling <- 2 representative_sampling <- 7 population_parameter <- 3 sample <- 5 observation <- 8 sampling_distribution <- 4 ### END SOLUTION 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 to 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 we make a simulated dataset of ages for our population (all Canadian seniors) bounded by realistic values ( 65 and 117): In [ ]: In [ ]:

pop_sd = sd ( age )) ### END SOLUTION pop_parameters 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! # ... <- ... |> # rep_sample_n(...) ### BEGIN SOLUTION sample_1 <- can_seniors |> rep_sample_n ( 40 ) ### END SOLUTION 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 the title "Sample 1 Distribution" (using ggtitle ) and the x-axis a descriptive label. options ( repr.plot.width = 8 , repr.plot.height = 7 ) ### BEGIN SOLUTION sample_1_dist <- ggplot ( sample_1 , aes ( age )) + geom_histogram ( binwidth = 1 ) + xlab ( "Age (years)" ) + ggtitle ( "Sample 1 Distribution" ) + theme ( text = element_text ( size = 20 )) ### END SOLUTION sample_1_dist test_1.5 () Question 1.6 {points: 1} Use summarize to calculate the following point estimates from the random sample you just took ( sample_1 ): mean median In [ ]: In [ ]: In [ ]: In [ ]: In [ ]:

standard deviation Name this data frame sample_1_estimates which has the column names sample_1_mean , sample_1_med and sample_1_sd . ### BEGIN SOLUTION sample_1_estimates <- sample_1 |> summarize ( sample_1_mean = mean ( age ), sample_1_med = median ( age ), sample_1_sd = sd ( age )) ### END SOLUTION sample_1_estimates test_1.6 () Let's now compare our random sample to the population from which it was drawn. In ggplot , it is possible to display multiple charts together by using the function plot_grid from a separate package called cowplot . We can use the ncol parameter to control how many columns of plots the grid contains. Since we want to compare the distributions' shape and position on the x-axis, it is most effective to concatenate these charts vertically in a single column. # run this code cell options ( repr.plot.width = 7 , repr.plot.height = 7 ) plot_grid ( pop_dist , sample_1_dist , ncol = 1 ) And now let's compare the point estimates (mean, median and standard deviation) with the true population parameters we were trying to estimate: # run this cell pop_parameters sample_1_estimates |> select ( - replicate ) Question 1.7 Multiple Choice {points: 1} After comparing the population and sample distributions above, and the true population parameters 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 parameters 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 parameters we are trying to estimate Assign your answer to an object called answer1.7 . Your answer should be a single character surrounded by quotes. In [ ]: In [ ]: In [ ]: In [ ]:

B. The sample point estimates from different random samples are close to the values for the true population parameters 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 identical set of values 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. ### BEGIN SOLUTION answer1.8.1 <- "C" ### END SOLUTION 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 . Here we use the functions head() , tail() and dim() to view the first few rows, the last few rows and the dimension of the data set respectively. set.seed ( 4321 ) # DO NOT CHANGE! # ... <- rep_sample_n(..., size = ..., reps = ...) ### BEGIN SOLUTION samples <- rep_sample_n ( can_seniors , size = 40 , reps = 1500 ) ### END SOLUTION head ( samples ) tail ( samples ) dim ( samples ) test_1.9 () Question 2.0 {points: 1} In [ ]: In [ ]: In [ ]: In [ ]:

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 mean_age . ### BEGIN SOLUTION sample_estimates <- samples |> group_by ( replicate ) |> summarize ( mean_age = mean ( age )) ### END SOLUTION 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 . Give the plot the title "Sampling Distribution of the Sample Means" using ggtitle , and give the x-axis a descriptive label. options ( repr.plot.width = 8 , repr.plot.height = 7 ) ### BEGIN SOLUTION sampling_distribution <- ggplot ( sample_estimates , aes ( x = mean_age )) + geom_histogram ( binwidth = 1 ) + xlab ( "Sample mean (age in years)" ) + ggtitle ( "Sampling Distribution of the Sample Means" ) + theme ( text = element_text ( size = 20 )) ### END SOLUTION sampling_distribution test_2.1 () Question 2.2 {points: 1} Let's refresh our memories: what is the mean age of the whole population (we calculated this above)? Assign your answer to an object called answer2.2 . Your answer should be a single number reported to two decimal places. ### BEGIN SOLUTION answer2.2 <- 79.30 ### END SOLUTION answer2.2 test_2.2 () Question 2.3 Multiple Choice {points: 1} In [ ]: In [ ]: In [ ]: In [ ]: In [ ]: In [ ]:

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 and assign this data to a column called 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 . Give the plot the title "Sampling Distribution (n=20)" using ggtitle , and give the x-axis a descriptive label. 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! options ( repr.plot.width = 8 , repr.plot.height = 7 ) ### BEGIN SOLUTION sample_estimates_20 <- rep_sample_n ( can_seniors , size = 20 , reps = 1500 ) |> group_by ( replicate ) |> summarise ( mean_age = mean ( age )) sampling_distribution_20 <- ggplot ( sample_estimates_20 , aes ( x = mean_age )) + geom_histogram ( binwidth = 1 ) + xlab ( "Sample mean (age in years)" ) + xlim ( c ( 65 , 95 )) + ggtitle ( "Sampling Distribution (n=20)" ) + theme ( text = element_text ( size = 20 )) ### END SOLUTION 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 and assign this data to a column called 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 . Give the plot the title "Sampling Distribution (n=100)" using ggtitle , and give the x axis a descriptive label. 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! options ( repr.plot.width = 8 , repr.plot.height = 7 ) ### BEGIN SOLUTION sample_estimates_100 <- rep_sample_n ( can_seniors , size = 100 , reps = 1500 ) |> In [ ]: In [ ]: In [ ]:

group_by ( replicate ) |> summarise ( mean_age = mean ( age )) sampling_distribution_100 <- ggplot ( sample_estimates_100 , aes ( x = mean_age )) + geom_histogram ( binwidth = 1 ) + xlab ( "Sample mean (age in years)" ) + xlim ( c ( 65 , 95 )) + ggtitle ( "Sampling Distribution (n=100)" ) + theme ( text = element_text ( size = 20 )) ### END SOLUTION 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 <- "Sampling Distribution (n=40)" Question 2.7 {points: 1} Fill in the blanks in the code below to use plot_grid to concatenate the three sampling distributions vertically. Order them from smallest sample size on the on the top, to largest sample size on the bottom. Name the final panel figure sampling_distribution_panel . options ( repr.plot.width = 6 ) # sampling_distribution_panel <- plot_grid( # ..., # ..., # ..., # ncol = 1 # ) ### BEGIN SOLUTION sampling_distribution_panel <- plot_grid ( sampling_distribution_20 , sampling_distribution , sampling_distribution_100 , ncol = 1 ) sampling_distribution_panel ### END SOLUTION test_2.7 () Question 2.8 Multiple Choice {points: 1} Considering the panel figure you created above in question 2.7 , which of the following statements below is not correct: In [ ]: In [ ]: In [ ]: In [ ]: