Inference for a mean

Warm up

Announcements

No class tomorrow
Lab 8 due Thursday
Lab 9 optional
Friday we will talk about final report formatting

Inference for one mean

Today’s focus

Inference for a population mean
Specifically, estimating a population mean via a confidence interval
And generally, what in the world is a confidence interval?!
- Why is it constructed?
- How is it constructed?
- What do the bounds of a confidence interval mean?
- What does the confidence level mean?
- What is the margin of error?

Setup

library(tidyverse)
library(openintro)
library(tidymodels)

Bootstrap confidence interval for the mean

Take a bootstrap sample (sample with replacement) of size n (the original sample size) from the original sample
Record the mean
Repeat steps 1 and 2 many times to build the bootstrap distribution
Find the bootstrap confidence interval using one of two methods:
- Percentile method: The bounds of the middle X% of the bootstrap distribution comprise the X% bootstrap interval.
- Standard error method:
  - Calculate the standard error for the bootstrap distribution
  - Find the critical value associated with the X% confidence interval
  - Calculate the margin of error (ME) as the critical value \(\times\) standard error
  - Construct the confidence interval a the original sample mean \(\pm\) ME

Case study: Length of gestation

Every year, the United States Department of Health and Human Services releases to the public a large dataset containing information on births recorded in the country. This dataset has been of interest to medical researchers who are studying the relation between habits and practices of expectant mothers and the birth of their children. In this case study we work with a random sample of 1,000 cases from the dataset released in 2014. The length of pregnancy, measured in weeks, is commonly referred to as gestation. The histogram below shows the distribution of lengths of gestation from the random sample of 1,000 births.

ggplot(
  openintro::births14, 
  aes(x = weeks)
  ) +
  geom_histogram(binwidth = 1) +
  labs(
    x = "Gestation (weeks)",
    y = "Count",
    title = "Random sample of 1,000 births"
  )

CS: Length of gestation - `boot_dist`

The distribution of bootstrapped means of gestation from 1,500 different bootstrap samples.

set.seed(1234)

boot_dist <- births14 |>
  specify(response = weeks) |>
  generate(
    reps = 1000, 
    type = "bootstrap"
  ) |>
  calculate(stat = "mean")

visualize(boot_dist)

CS: Length of gestation - percentile interval

What does the following code do? How do you adjust it to change the confidence level?

boot_dist |>
  get_ci()

# A tibble: 1 × 2
  lower_ci upper_ci
     <dbl>    <dbl>
1     38.5     38.8

CS: Length of gestation - standard error interval

What does the following code do? Why do we need to provide point_estimate?

obs_stat <- births14 |>
  specify(response = weeks) |>
  calculate(stat = "mean")

obs_stat

Response: weeks (numeric)
# A tibble: 1 × 1
   stat
  <dbl>
1  38.7

boot_dist |>
  get_ci(
    type = "se",
    point_estimate = obs_stat
  )

# A tibble: 1 × 2
  lower_ci upper_ci
     <dbl>    <dbl>
1     38.5     38.8

Interpretations

The interval:
The confidence level:

What does 95% confidence interval mean?

Bootstrap confidence intervals

Exercise 1

Construct and interpret a 95% confidence interval, using the standard error method with the get_ci() function, for the average length of gestation.

# add code here

Add response here.

Exercise 2

Construct the interval without using the get_ci() (or get_confidence_interval()) function.

# add code here

Exercise 3

Would you expect a 90% confidence interval for the average length of gestation to be wider or narrower? Explain your reasoning.