Getting Started

Load packages

In this lab, we will explore and visualize the data using the tidyverse suite of packages, and perform statistical inference using infer. The data can be found in the companion package for OpenIntro resources, openintro.

Let’s load the packages.

library(tidyverse)
library(openintro)
library(infer)

The data

Every two years, the Centers for Disease Control and Prevention conduct the Youth Risk Behavior Surveillance System (YRBSS) survey, where it takes data from high schoolers (9th through 12th grade), to analyze health patterns. You will work with a selected group of variables from a random sample of observations during one of the years the YRBSS was conducted.

Load the yrbss data set into your workspace.

data('yrbss', package='openintro')

There are observations on 13 different variables, some categorical and some numerical. The meaning of each variable can be found by bringing up the help file:

?yrbss
  1. What are the cases in this data set? How many cases are there in our sample?

Theh cases are essentialy observations. There are 13583 cases in this datatset

Remember that you can answer this question by viewing the data in the data viewer or by using the following command:

glimpse(yrbss)
## Rows: 13,583
## Columns: 13
## $ age                      <int> 14, 14, 15, 15, 15, 15, 15, 14, 15, 15, 15, 1~
## $ gender                   <chr> "female", "female", "female", "female", "fema~
## $ grade                    <chr> "9", "9", "9", "9", "9", "9", "9", "9", "9", ~
## $ hispanic                 <chr> "not", "not", "hispanic", "not", "not", "not"~
## $ race                     <chr> "Black or African American", "Black or Africa~
## $ height                   <dbl> NA, NA, 1.73, 1.60, 1.50, 1.57, 1.65, 1.88, 1~
## $ weight                   <dbl> NA, NA, 84.37, 55.79, 46.72, 67.13, 131.54, 7~
## $ helmet_12m               <chr> "never", "never", "never", "never", "did not ~
## $ text_while_driving_30d   <chr> "0", NA, "30", "0", "did not drive", "did not~
## $ physically_active_7d     <int> 4, 2, 7, 0, 2, 1, 4, 4, 5, 0, 0, 0, 4, 7, 7, ~
## $ hours_tv_per_school_day  <chr> "5+", "5+", "5+", "2", "3", "5+", "5+", "5+",~
## $ strength_training_7d     <int> 0, 0, 0, 0, 1, 0, 2, 0, 3, 0, 3, 0, 0, 7, 7, ~
## $ school_night_hours_sleep <chr> "8", "6", "<5", "6", "9", "8", "9", "6", "<5"~

Exploratory data analysis

You will first start with analyzing the weight of the participants in kilograms: weight.

Using visualization and summary statistics, describe the distribution of weights. The summary function can be useful.

summary(yrbss$weight)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
##   29.94   56.25   64.41   67.91   76.20  180.99    1004
  1. How many observations are we missing weights from?

1004

Next, consider the possible relationship between a high schooler’s weight and their physical activity. Plotting the data is a useful first step because it helps us quickly visualize trends, identify strong associations, and develop research questions.

First, let’s create a new variable physical_3plus, which will be coded as either “yes” if they are physically active for at least 3 days a week, and “no” if not.

yrbss <- yrbss %>% 
  mutate(physical_3plus = ifelse(yrbss$physically_active_7d > 2, "yes", "no"))
  1. Make a side-by-side boxplot of physical_3plus and weight. Is there a relationship between these two variables? What did you expect and why?
physical_3plus_weight <- yrbss %>% drop_na()

physical_3plus_weight %>% ggplot(aes(x = physical_3plus, y = weight)) + geom_boxplot()

The boxplots are showing that the cases with 3plus days of physical activity are actually slightly heavier than those with less than 3. Which goes against what i thought the case would be. I expected physical activity and weight to be negatively correlated.

The box plots show how the medians of the two distributions compare, but we can also compare the means of the distributions using the following to first group the data by the physical_3plus variable, and then calculate the mean weight in these groups using the mean function while ignoring missing values by setting the na.rm argument to TRUE.

yrbss %>%
  group_by(physical_3plus) %>%
  summarise(mean_weight = mean(weight, na.rm = TRUE))
## # A tibble: 3 x 2
##   physical_3plus mean_weight
##   <chr>                <dbl>
## 1 no                    66.7
## 2 yes                   68.4
## 3 <NA>                  69.9

There is an observed difference, but is this difference statistically significant? In order to answer this question we will conduct a hypothesis test.

Inference

  1. Are all conditions necessary for inference satisfied? Comment on each. You can compute the group sizes with the summarize command above by defining a new variable with the definition n().

The necessary conditions are met. The data is independent and the smaple size is normal

yrbss %>%
  group_by(physical_3plus) %>%
  summarise(mean_weight = mean(weight, na.rm = TRUE), count=n())
## # A tibble: 3 x 3
##   physical_3plus mean_weight count
##   <chr>                <dbl> <int>
## 1 no                    66.7  4404
## 2 yes                   68.4  8906
## 3 <NA>                  69.9   273
  1. Write the hypotheses for testing if the average weights are different for those who exercise at least times a week and those who don’t.

**H_null = There’s no difference between The average weights for those who exercise at least 3 times per week and Those who don’t.

H_alt = There is a difference between The average weights for those who exercise at least 3 times per week and those who don’t. Students who exercise at least 3 times per week have a different average weight than those who don’t.**

Next, we will introduce a new function, hypothesize, that falls into the infer workflow. You will use this method for conducting hypothesis tests.

But first, we need to initialize the test, which we will save as obs_diff.

obs_diff <- yrbss %>%
  drop_na(physical_3plus) %>%
  specify(weight ~ physical_3plus) %>%
  calculate(stat = "diff in means", order = c("yes", "no"))

Notice how you can use the functions specify and calculate again like you did for calculating confidence intervals. Here, though, the statistic you are searching for is the difference in means, with the order being yes - no != 0.

After you have initialized the test, you need to simulate the test on the null distribution, which we will save as null.

null_dist <- yrbss %>%
  drop_na(physical_3plus) %>%
  specify(weight ~ physical_3plus) %>%
  hypothesize(null = "independence") %>%
  generate(reps = 1000, type = "permute") %>%
  calculate(stat = "diff in means", order = c("yes", "no"))

Here, hypothesize is used to set the null hypothesis as a test for independence. In one sample cases, the null argument can be set to “point” to test a hypothesis relative to a point estimate.

Also, note that the type argument within generate is set to permute, whichis the argument when generating a null distribution for a hypothesis test.

We can visualize this null distribution with the following code:

ggplot(data = null_dist, aes(x = stat)) +
  geom_histogram()

  1. How many of these null permutations have a difference of at least obs_stat?

Insert your answer here

print(paste0('The total null permutations that have a difference of at least obs_stat is: ', nrow(null_dist %>% filter(stat >= obs_diff)),' cases.'))
## [1] "The total null permutations that have a difference of at least obs_stat is: 0 cases."

Now that the test is initialized and the null distribution formed, you can calculate the p-value for your hypothesis test using the function get_p_value.

null_dist %>%
  get_p_value(obs_stat = obs_diff, direction = "two_sided")
## # A tibble: 1 x 1
##   p_value
##     <dbl>
## 1       0

This the standard workflow for performing hypothesis tests.

  1. Construct and record a confidence interval for the difference between the weights of those who exercise at least three times a week and those who don’t, and interpret this interval in context of the data.
yrbss %>% 
  group_by(physical_3plus) %>% 
  summarise(mean_weight = mean(weight, na.rm = TRUE), sd_weight = sd(weight, na.rm = TRUE), count = n())
## # A tibble: 3 x 4
##   physical_3plus mean_weight sd_weight count
##   <chr>                <dbl>     <dbl> <int>
## 1 no                    66.7      17.6  4404
## 2 yes                   68.4      16.5  8906
## 3 <NA>                  69.9      17.6   273
c_weight <- yrbss %>% 
  group_by(physical_3plus) %>% 
  summarise(mean_weight = mean(weight, na.rm = TRUE), sd_weight = sd(weight, na.rm = TRUE), count = n())
t <- 1.96

#Confidence Interval for those who exercise at least three times a week
uci_exercise3plus <- c_weight$mean_weight[2] + t*(c_weight$sd_weight[2]/sqrt(c_weight$count[2]))

lci_exercise3plus <- c_weight$mean_weight[2] - t*(c_weight$sd_weight[2]/sqrt(c_weight$count[2]))

#Confidence Interval for who do not exercise at least three times a week
uci_noexercise <- c_weight$mean_weight[1] + t*(c_weight$sd_weight[1]/sqrt(c_weight$count[1]))

lci_noexercise <- c_weight$mean_weight[1] - t*(c_weight$sd_weight[1]/sqrt(c_weight$count[1]))

the 95% confidence interval of students who exercise 3 times a week is between 68.1062 and 68.7907. The interval for those who do not exercise is 66.1530 and 67.1948 * * *

More Practice

  1. Calculate a 95% confidence interval for the average height in meters (height) and interpret it in context.
data_tb <- as.data.frame(table(yrbss$height))
freqncy <- sum(data_tb$Freq)
m_height  <- mean(yrbss$height, na.rm = TRUE)
sd_height <- sd(yrbss$height, na.rm = TRUE)
mx_height <- max(yrbss$height, na.rm = TRUE)

n_height <- yrbss %>% 
  summarise(freqncy = table(height)) %>%
  summarise(n = sum(freqncy, na.rm = TRUE))

u_height <- m_height  + t * (sd_height/sqrt(n_height))
l_height <- m_height  - t * (sd_height/sqrt(n_height))

the confidence interval for height is an interval of 1.6931 and 1.6894

  1. Calculate a new confidence interval for the same parameter at the 90% confidence level. Comment on the width of this interval versus the one obtained in the previous exercise.
nu_height <- m_height + 1.645 * (sd_height/sqrt(n_height))
nl_height <- m_height - 1.645 * (sd_height/sqrt(n_height))

**90% CI interval is 1.6928 and 1.6897*

  1. Conduct a hypothesis test evaluating whether the average height is different for those who exercise at least three times a week and those who don’t.

exercise average height 1.703, dont exercise av aheight 1.666

exercise3plus <- yrbss %>% 
  filter(physical_3plus == "yes") %>% 
  select(height) %>% 
  na.omit()

noexercise <- yrbss %>% 
  filter(physical_3plus == "no") %>% 
  select(height) %>% 
  na.omit() 

mean_exercise3plus <- mean(exercise3plus$height)
mean_noexercise <- mean(noexercise$height)
boxplot(exercise3plus$height, noexercise$height,
        names = c("exercise", "no_exercise"))

  1. Now, a non-inference task: Determine the number of different options there are in the dataset for the hours_tv_per_school_day there are.

Insert your answer here

yrbss %>% group_by(hours_tv_per_school_day) %>% summarise(n())
## # A tibble: 8 x 2
##   hours_tv_per_school_day `n()`
##   <chr>                   <int>
## 1 1                        1750
## 2 2                        2705
## 3 3                        2139
## 4 4                        1048
## 5 5+                       1595
## 6 <1                       2168
## 7 do not watch             1840
## 8 <NA>                      338
  1. Come up with a research question evaluating the relationship between height or weight and sleep. Formulate the question in a way that it can be answered using a hypothesis test and/or a confidence interval. Report the statistical results, and also provide an explanation in plain language. Be sure to check all assumptions, state your \(\alpha\) level, and conclude in context.

Insert your answer here