Data

• Family income and gift aid data from a random sample of fifty students in the freshman class of Elmhurst College in Illinois, USA
• Gift aid is financial aid that does not need to be paid back, as opposed to a loan

Linear model

linear_reg() %>%
  set_engine("lm") %>%
  fit(gift_aid ~ family_income, data = elmhurst) %>%
  tidy()

## # A tibble: 2 × 5
##   term          estimate std.error statistic  p.value
##   <chr>            <dbl>     <dbl>     <dbl>    <dbl>
## 1 (Intercept)    24.3       1.29       18.8  8.28e-24
## 2 family_income  -0.0431    0.0108     -3.98 2.29e- 4

Interpreting the slope

## # A tibble: 2 × 5
##   term          estimate std.error statistic  p.value
##   <chr>            <dbl>     <dbl>     <dbl>    <dbl>
## 1 (Intercept)    24.3       1.29       18.8  8.28e-24
## 2 family_income  -0.0431    0.0108     -3.98 2.29e- 4

For each additional $1,000 of family income, we would expect students to receive a net difference of 1,000 * (-0.0431) = -$43.10 in aid on average, i.e. $43.10 less in gift aid, on average.

Statistical inference

... is the process of using sample data to make conclusions about the underlying population the sample came from

Estimation

So far we have done lots of estimation (mean, median, slope, etc.), i.e.

• used data from samples to calculate sample statistics
• which can then be used as estimates for population parameters

• If we report a point estimate, we probably won’t hit the exact population parameter
• If we report a range of plausible values we have a good shot at capturing the parameter

Confidence intervals

A plausible range of values for the population parameter is a confidence interval.

• In order to construct a confidence interval we need to quantify the variability of our sample statistic
• For example, if we want to construct a confidence interval for a population slope, we need to come up with a plausible range of values around our observed sample slope
• This range will depend on how precise and how accurate our sample mean is as an estimate of the population mean
• Quantifying this requires a measurement of how much we would expect the sample population to vary from sample to sample

Quantifying the variability of slopes

We can quantify the variability of sample statistics using

• simulation: via bootstrapping (now)

or

• theory: via Central Limit Theorem (future stat courses!)

## # A tibble: 2 × 5
##   term          estimate std.error statistic  p.value
##   <chr>            <dbl>     <dbl>     <dbl>    <dbl>
## 1 (Intercept)    24.3       1.29       18.8  8.28e-24
## 2 family_income  -0.0431    0.0108     -3.98 2.29e- 4

Bootstrapping

Observed sample

Bootstrap population

Generated assuming there are more students like the ones in the observed sample...

Bootstrapping scheme

1. Take a bootstrap sample - a random sample taken with replacement from the original sample, of the same size as the original sample
2. Calculate the bootstrap statistic - a statistic such as mean, median, proportion, slope, etc. computed on the bootstrap samples
3. Repeat steps (1) and (2) many times to create a bootstrap distribution - a distribution of bootstrap statistics
4. Calculate the bounds of the XX% confidence interval as the middle XX% of the bootstrap distribution

Bootstrap sample 1

elmhurtst_boot_1 <- elmhurst %>%
  slice_sample(n = 50, replace = TRUE)

Bootstrap sample 2

elmhurtst_boot_2 <- elmhurst %>%
  slice_sample(n = 50, replace = TRUE)

Bootstrap sample 3

elmhurtst_boot_3 <- elmhurst %>%
  slice_sample(n = 50, replace = TRUE)

Bootstrap sample 4

elmhurtst_boot_4 <- elmhurst %>%
  slice_sample(n = 50, replace = TRUE)

Bootstrap samples 1 - 4

Many many samples...

Slopes of bootstrap samples

95% confidence interval

Interpreting the slope, take two

## # A tibble: 1 × 2
##   lower_ci upper_ci
##      <dbl>    <dbl>
## 1  -0.0695  -0.0232

We are 95% confident that for each additional $1,000 of family income, we would expect students to receive $69.5 to $23.24 less in gift aid, on average.