Using the t-distribution

Packages

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

Data

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.

glimpse(births14)

Rows: 1,000
Columns: 13
$ fage           <int> 34, 36, 37, NA, 32, 32, 37, 29, 30, 29, 30, 34, 28, 28,…
$ mage           <dbl> 34, 31, 36, 16, 31, 26, 36, 24, 32, 26, 34, 27, 22, 31,…
$ mature         <chr> "younger mom", "younger mom", "mature mom", "younger mo…
$ weeks          <dbl> 37, 41, 37, 38, 36, 39, 36, 40, 39, 39, 42, 40, 40, 39,…
$ premie         <chr> "full term", "full term", "full term", "full term", "pr…
$ visits         <dbl> 14, 12, 10, NA, 12, 14, 10, 13, 15, 11, 14, 16, 20, 15,…
$ gained         <dbl> 28, 41, 28, 29, 48, 45, 20, 65, 25, 22, 40, 30, 31, NA,…
$ weight         <dbl> 6.96, 8.86, 7.51, 6.19, 6.75, 6.69, 6.13, 6.74, 8.94, 9…
$ lowbirthweight <chr> "not low", "not low", "not low", "not low", "not low", …
$ sex            <chr> "male", "female", "female", "male", "female", "female",…
$ habit          <chr> "nonsmoker", "nonsmoker", "nonsmoker", "nonsmoker", "no…
$ marital        <chr> "married", "married", "married", "not married", "marrie…
$ whitemom       <chr> "white", "white", "not white", "white", "white", "white…

`t_test`

t_test(births14, 
       response = weeks, 
       alternative = "two-sided",
       mu = 40,
       conf_int = TRUE, 
       conf_level = 0.95)

# A tibble: 1 × 7
  statistic  t_df  p_value alternative estimate lower_ci upper_ci
      <dbl> <dbl>    <dbl> <chr>          <dbl>    <dbl>    <dbl>
1     -16.4   999 5.71e-54 two.sided       38.7     38.5     38.8

births14 |>
  summarize(
    n = n(),
    xbar = mean(weeks),
    s = sd(weeks),
    se = s / sqrt(n),
    alpha = 0.05,
    df = n - 1,
    t_star = qt(1 - (alpha / 2), df = df),
    L = xbar - t_star * se,
    U = xbar + t_star * se
  )

# A tibble: 1 × 9
      n  xbar     s     se alpha    df t_star     L     U
  <int> <dbl> <dbl>  <dbl> <dbl> <dbl>  <dbl> <dbl> <dbl>
1  1000  38.7  2.56 0.0811  0.05   999   1.96  38.5  38.8