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

You will be analyzing the same dataset as in the previous lab, where you delved into a sample from the Youth Risk Behavior Surveillance System (YRBSS) survey, which uses data from high schoolers to help discover health patterns. The dataset is called yrbss.

  1. What are the counts within each category for the amount of days these students have texted while driving within the past 30 days?

4792 people reported 0, 925 people reported 1-2 days, 493 people reported 3-5 days, 311 people reported 6-9 days, 373 people reported 10-19 days, 298 people reported 20-29 days and 827 people reported 30 days

yrbss %>%
  count(text_while_driving_30d) %>%
  mutate(p=n / sum(n))
## # A tibble: 9 × 3
##   text_while_driving_30d     n      p
##   <chr>                  <int>  <dbl>
## 1 0                       4792 0.353 
## 2 1-2                      925 0.0681
## 3 10-19                    373 0.0275
## 4 20-29                    298 0.0219
## 5 3-5                      493 0.0363
## 6 30                       827 0.0609
## 7 6-9                      311 0.0229
## 8 did not drive           4646 0.342 
## 9 <NA>                     918 0.0676
  1. What is the proportion of people who have texted while driving every day in the past 30 days and never wear helmets?

6040 people have not texted while driving in 30 days while never wearing helmets, while 463 people or 6.64% of people have texted while driving in the last 30 days while never wearing helmets

Remember that you can use filter to limit the dataset to just non-helmet wearers. Here, we will name the dataset no_helmet.

data('yrbss', package='openintro')
no_helmet <- yrbss %>%
  filter(helmet_12m == "never")

Also, it may be easier to calculate the proportion if you create a new variable that specifies whether the individual has texted every day while driving over the past 30 days or not. We will call this variable text_ind.

no_helmet <- no_helmet %>%
  mutate(text_ind = ifelse(text_while_driving_30d == "30", "yes", "no")) %>%
  filter(!is.na(text_ind))

value <- no_helmet %>%
  count(text_ind) %>%
  mutate( p = n / sum(n))

value
## # A tibble: 2 × 3
##   text_ind     n      p
##   <chr>    <int>  <dbl>
## 1 no        6040 0.929 
## 2 yes        463 0.0712

Inference on proportions

When summarizing the YRBSS, the Centers for Disease Control and Prevention seeks insight into the population parameters. To do this, you can answer the question, “What proportion of people in your sample reported that they have texted while driving each day for the past 30 days?” with a statistic; while the question “What proportion of people on earth have texted while driving each day for the past 30 days?” is answered with an estimate of the parameter.

The inferential tools for estimating population proportion are analogous to those used for means in the last chapter: the confidence interval and the hypothesis test.

no_helmet %>%
  specify(response = text_ind, success = "yes") %>%
  generate(reps = 1000, type = "bootstrap") %>%
  calculate(stat = "prop") %>%
  get_ci(level = 0.95)
## # A tibble: 1 × 2
##   lower_ci upper_ci
##      <dbl>    <dbl>
## 1   0.0649   0.0775

Note that since the goal is to construct an interval estimate for a proportion, it’s necessary to both include the success argument within specify, which accounts for the proportion of non-helmet wearers than have consistently texted while driving the past 30 days, in this example, and that stat within calculate is here “prop”, signaling that you are trying to do some sort of inference on a proportion.

  1. What is the margin of error for the estimate of the proportion of non-helmet wearers that have texted while driving each day for the past 30 days based on this survey?

The margin of error for the estimate of the proportion of non-helmet wearers that have texted while driving each day for the past 30 days based on this survey is 0.065% margin of error

lower_ci <- 0.065
upper_ci <- 0.078


me <- (upper_ci - lower_ci) / 2  

me
## [1] 0.0065
  1. Using the infer package, calculate confidence intervals for two other categorical variables (you’ll need to decide which level to call “success”, and report the associated margins of error. Interpet the interval in context of the data. It may be helpful to create new data sets for each of the two countries first, and then use these data sets to construct the confidence intervals.

we are 95% confident that the proportion of people in the data set that actually watch tv is exactly 86.1% with a 0% margin of error. We are also 95% confident that the proportion of non-helmet wearers that are African American are between 29.3% and 31.7% with a 1.2% margin of error 

## calc confidence interval for hours of tv per school day
hours <- yrbss
hours %>%
  count(hours_tv_per_school_day, sort=TRUE) %>%
  mutate(p = n /sum(n))
## # A tibble: 8 × 3
##   hours_tv_per_school_day     n      p
##   <chr>                   <int>  <dbl>
## 1 2                        2705 0.199 
## 2 <1                       2168 0.160 
## 3 3                        2139 0.157 
## 4 do not watch             1840 0.135 
## 5 1                        1750 0.129 
## 6 5+                       1595 0.117 
## 7 4                        1048 0.0772
## 8 <NA>                      338 0.0249
# Create a new variable "hours_tv_school for those who watch or dont watch tv at all"
yrbss <- yrbss %>%
  mutate(hours_tv_school = ifelse(str_trim(hours_tv_per_school_day) != "do not watch", "yes", "no")) %>%
  filter(!is.na(hours_tv_school))

# Calculate the proportion
hours_proportion <- yrbss %>%
  specify(response = hours_tv_school, success = "yes") %>%
  calculate(stat = "prop") %>%
  get_ci(level = 0.95, type = "percentile")
## calc the margin of error
lower_ci1 <- 0.861
upper_ci1 <- 0.861


me1 <- (upper_ci1 - lower_ci1) / 2  

me1
## [1] 0
# the proportion of non-helmet wearers and are Black or African American

no_helmet <- no_helmet %>%
  mutate(race_or_not = ifelse(str_trim(race) == "Black or African American", "yes", "no")) %>%
  filter(!is.na(race_or_not))

no_helmet %>%
  specify(response = race_or_not, success = "yes") %>%
  generate(reps = 1000, type = "bootstrap") %>%
  calculate(stat = "prop") %>%
  get_ci(level = 0.95)
## # A tibble: 1 × 2
##   lower_ci upper_ci
##      <dbl>    <dbl>
## 1    0.286    0.309
## calc the margin of error
lower_ci2 <- 0.293
upper_ci2 <- 0.317


me2 <- (upper_ci2 - lower_ci2) / 2  # margin of error

me2
## [1] 0.012

How does the proportion affect the margin of error?

Imagine you’ve set out to survey 1000 people on two questions: are you at least 6-feet tall? and are you left-handed? Since both of these sample proportions were calculated from the same sample size, they should have the same margin of error, right? Wrong! While the margin of error does change with sample size, it is also affected by the proportion.

Think back to the formula for the standard error: \(SE = \sqrt{p(1-p)/n}\). This is then used in the formula for the margin of error for a 95% confidence interval:

\[ ME = 1.96\times SE = 1.96\times\sqrt{p(1-p)/n} \,. \] Since the population proportion \(p\) is in this \(ME\) formula, it should make sense that the margin of error is in some way dependent on the population proportion. We can visualize this relationship by creating a plot of \(ME\) vs. \(p\).

Since sample size is irrelevant to this discussion, let’s just set it to some value (\(n = 1000\)) and use this value in the following calculations:

n <- 1000

The first step is to make a variable p that is a sequence from 0 to 1 with each number incremented by 0.01. You can then create a variable of the margin of error (me) associated with each of these values of p using the familiar approximate formula (\(ME = 2 \times SE\)).

p <- seq(from = 0, to = 1, by = 0.01)
me <- 2 * sqrt(p * (1 - p)/n)

Lastly, you can plot the two variables against each other to reveal their relationship. To do so, we need to first put these variables in a data frame that you can call in the ggplot function.

dd <- data.frame(p = p, me = me)
ggplot(data = dd, aes(x = p, y = me)) + 
  geom_line() +
  labs(x = "Population Proportion", y = "Margin of Error")

  1. Describe the relationship between p and me. Include the margin of error vs. population proportion plot you constructed in your answer. For a given sample size, for which value of p is margin of error maximized?

The margin of error is not a constant value but changes depending on the population proportion. As the proportion reaches the median which is 50%, the margin of error reaches its peak which is around 0.0118, and as it reaches either extreme, the margin of error decreases. For a given sample size, the margin of error is maximized when the population proportion is equal to .5 or 50%

Success-failure condition

We have emphasized that you must always check conditions before making inference. For inference on proportions, the sample proportion can be assumed to be nearly normal if it is based upon a random sample of independent observations and if both \(np \geq 10\) and \(n(1 - p) \geq 10\). This rule of thumb is easy enough to follow, but it makes you wonder: what’s so special about the number 10?

The short answer is: nothing. You could argue that you would be fine with 9 or that you really should be using 11. What is the “best” value for such a rule of thumb is, at least to some degree, arbitrary. However, when \(np\) and \(n(1-p)\) reaches 10 the sampling distribution is sufficiently normal to use confidence intervals and hypothesis tests that are based on that approximation.

You can investigate the interplay between \(n\) and \(p\) and the shape of the sampling distribution by using simulations. Play around with the following app to investigate how the shape, center, and spread of the distribution of \(\hat{p}\) changes as \(n\) and \(p\) changes.

  1. Describe the sampling distribution of sample proportions at \(n = 300\) and \(p = 0.1\). Be sure to note the center, spread, and shape.

The distribution looks normal with the shape having a similar look to a bell curve and the center of the graph being around 0.11 and the spread being between 0.05 to 0.12

  1. Keep \(n\) constant and change \(p\). How does the shape, center, and spread of the sampling distribution vary as \(p\) changes. You might want to adjust min and max for the \(x\)-axis for a better view of the distribution.

As I increase the populations proportions towards 0.5, the graph begins to look more and more like a normally distributed graph, and as I increase or decrease the proportion away from 0.5 the graph begins to look less like a normally distributed graph with a bell curve.

  1. Now also change \(n\). How does \(n\) appear to affect the distribution of \(\hat{p}\)?

As I increase the value of n, the graph still remains normal

More Practice

For some of the exercises below, you will conduct inference comparing two proportions. In such cases, you have a response variable that is categorical, and an explanatory variable that is also categorical, and you are comparing the proportions of success of the response variable across the levels of the explanatory variable. This means that when using infer, you need to include both variables within specify.

  1. Is there convincing evidence that those who sleep 10+ hours per day are more likely to strength train every day of the week? As always, write out the hypotheses for any tests you conduct and outline the status of the conditions for inference. If you find a significant difference, also quantify this difference with a confidence interval.

I am 95% confident that the proportion of students who sleep 10+ hours is between 3.25 and 5.03% with a 0.89% margin of error. Therefore there is no convincing evidence that those who sleep 10+ hours per day are more likely to strength train every day of the week

## calc people who train every day of the week
train_7_days_a_week <- yrbss %>%
  filter(strength_training_7d == 7)

train_7_days_a_week <- train_7_days_a_week %>%
  mutate(sleep_over_10 = ifelse(school_night_hours_sleep == "10+", "yes", "no")) %>%
  filter(!is.na(sleep_over_10))

train_7_days_a_week %>%
  specify(response = sleep_over_10, success = "yes") %>%
  generate(reps = 1000, type = "bootstrap") %>%
  calculate(stat = "prop") %>%
  get_ci(level = 0.95)
## # A tibble: 1 × 2
##   lower_ci upper_ci
##      <dbl>    <dbl>
## 1   0.0330   0.0498
## calculate the margin of error
lower_ci3 <- 0.0325
upper_ci3 <- 0.0503


me3 <- (upper_ci3 - lower_ci3) / 2  # margin of error

me3
## [1] 0.0089
  1. Let’s say there has been no difference in likeliness to strength train every day of the week for those who sleep 10+ hours. What is the probability that you could detect a change (at a significance level of 0.05) simply by chance? Hint: Review the definition of the Type 1 error.

a Type 1 error in this situation is equal to the significance level 0.05, which means the probability of the null hypothesis not happening is 5%, which isn’t that significant

  1. Suppose you’re hired by the local government to estimate the proportion of residents that attend a religious service on a weekly basis. According to the guidelines, the estimate must have a margin of error no greater than 1% with 95% confidence. You have no idea what to expect for \(p\). How many people would you have to sample to ensure that you are within the guidelines? Hint: Refer to your plot of the relationship between \(p\) and margin of error. This question does not require using a dataset.

I would need a sample size of about 936 people to, with 95% confidence, to estimate estimate the proportion ofresidents that attend a religious service on a weekly basis

# used previous graph to choose a proportion with a low margin of error
type1 <- 0.05  
p <- 0.025     

z <- qnorm(1 - type1 / 2)  
me <- 0.01 

n <- (z^2 * p * (1 - p)) / me^2

n
## [1] 936.3556