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?

Here are the counts within each category for the amount of days these students were texting while driving within the past 30 days:

yrbss |>
  count(text_while_driving_30d, sort = TRUE)
## # A tibble: 9 × 2
##   text_while_driving_30d     n
##   <chr>                  <int>
## 1 0                       4792
## 2 did not drive           4646
## 3 1-2                      925
## 4 <NA>                     918
## 5 30                       827
## 6 3-5                      493
## 7 10-19                    373
## 8 6-9                      311
## 9 20-29                    298
  1. What is the proportion of people who have texted while driving every day in the past 30 days and never wear helmets?

The proportion of people who have texted while driving everyday in the past 30 days and never wear helmets is 7.12%

# Filter the data to only include those who never wear helmets
no_helmet <- yrbss %>%
  filter(helmet_12m == "never")

# Create a new variable that specifies whether the individual has texted every day while driving over the past 30 days or not
no_helmet <- no_helmet %>%
  mutate(text_ind = ifelse(text_while_driving_30d == "30", "yes", "no"))

# Calculate the proportion of people who have texted while driving every day in the past 30 days and never wear helmets
no_helmet |>
  drop_na(text_ind) |>
  count(text_ind) |>
  mutate(proportion = n / sum(n))
## # A tibble: 2 × 3
##   text_ind     n proportion
##   <chr>    <int>      <dbl>
## 1 no        6040     0.929 
## 2 yes        463     0.0712

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"))

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 %>%
  drop_na(text_ind) %>% # Drop missing values
  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   0.0650   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 the survey is approximately .6% .

set.seed(1)

# Calculate the proportion
no_helmet_prop <- no_helmet |>
  drop_na(text_ind) |>
  count(text_ind) |>
  mutate(proportion = n / sum(n))

# Construct an interval estimate for a proportion of non-helmet wearers that have texted while driving each day for the past 30 days
no_helmet %>%
  drop_na(text_ind) %>% # Drop missing values
  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   0.0652   0.0767
# Create a margin of error function
marg_of_error <- function(p, n, z = 1.96) {
  round(z * sqrt(p * (1 - p) / n), 3)
}

no_helmet_prop$proportion[2] |>
  marg_of_error(nrow(no_helmet))
## [1] 0.006
  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. Interpret 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.

From only the females, what proportion are physically active for at least 5 days a week?

From the females, the proportion that are physically active for at least 5 days a week is 35.7%

We can assert with 95% confidence that the estimate of the proportion of females that are physically active at least 5 days a week is between 34.5% and 36.8% with a 1.2% margin of error

set.seed(2)
data('yrbss', package = 'openintro')

# Filter the females
females <- yrbss |>
  filter(gender == "female")

# Create a new variable that specifies whether the individual is physically active for at least 5 days a week
females <- females |>
  mutate(active_ind = ifelse(physically_active_7d >= 5, "yes", "no"))

# Calculate the proportions
female_prop <- females |>
  drop_na(active_ind) |>
  count(active_ind) |>
  mutate(proportion = n / sum(n))
female_prop
## # A tibble: 2 × 3
##   active_ind     n proportion
##   <chr>      <int>      <dbl>
## 1 no          4198      0.643
## 2 yes         2328      0.357
# Construct an interval estimate
females |>
  drop_na(active_ind) |>  # Drop missing values
  specify(response = active_ind, success = "yes") |>
  generate(reps = 1000, type = "bootstrap") |>
  calculate(stat = "prop") |>
  get_ci(level = .95)
## # A tibble: 1 × 2
##   lower_ci upper_ci
##      <dbl>    <dbl>
## 1    0.345    0.368
# Call the margin of error function
female_prop$proportion[2] |>
  marg_of_error(nrow(females))
## [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 relationship is the closer the population proportion is to 50% then the closer we get to the highest margin of error. Margin of error is maximized at at population proportion 50%

# Plot the relationship between proportion and margin of error
ggplot(data = dd, aes(x = p, y = me)) +
  geom_line() +
  geom_vline(xintercept = dd$p[which.max(dd$me)],
             linetype = "dashed",
             color = "red") +
  labs(x = "Population Proportion", y = "Margin of Error")

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.

It seems to reflect a normal distribution. with the max towards the middle.

  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.

The mean (center) of the sampling distribution will always be equal to the population proportion (p). This holds true regardless of the sample size or the value of p itself. In this case, with n = 300 and p varying, the mean of the sampling distribution will always be at the p value on the x-axis.As p moves away from 0.5, towards either 0 or 1, the standard error (SE) decreases. The closer p gets to 0 or 1, the smaller the standard error becomes. This makes the sampling distribution more concentrated around the mean. When p is very close to 0 (less than 0.1) or very close to 1 (greater than 0.9), the sampling distribution becomes skewed. When p is close to 0.5 (around 0.25 to 0.75), the sampling distribution of sample proportions (p̂) will tend towards a normal distribution.

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

As n increases the sampling distribution gets narrower. The shape tends to approach normal distribution even for extreme values of p.

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.

There is no convincing evidence that those who sleep 10+ hours per day are more likely to strength train every day of the week.

sleep_10_strength <- yrbss |>
  mutate(
    sleep_10_ind = ifelse(school_night_hours_sleep == "10+", "yes", "no"),
    strength_ind = ifelse(strength_training_7d == 7, "yes", "no")
  ) |>
  drop_na(sleep_10_ind, strength_ind) |>
  specify(response = strength_ind,
          explanatory = sleep_10_ind,
          success = "yes") |>
  generate(reps = 1000, type = "bootstrap") |>
  calculate(stat = "diff in props", order = c("yes", "no")) |>
  get_ci(level = 0.95)

sleep_10_strength
## # A tibble: 1 × 2
##   lower_ci upper_ci
##      <dbl>    <dbl>
## 1   0.0619    0.155
  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 I error, also known as a false positive, occurs in statistical hypothesis testing. It happens when you reject a null hypothesis that is actually true. In simpler terms, it means you find a difference between things being compared when there really isn’t one. The probability of detecting a change in this scenario simply by chance is the significance level - 5%

  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.

The Sample size would need to be 9604 or more surveys are needed to have a confidence level of 95% that a real value is within 1% of the surveyed value. Sample Size Calculator