Exercise 2 (20 minutes)

You practice: standard errors, t- and p-values, and confidence intervals. When you're done, assist some of the others …

Now the same logic, on trust in the legal system (trstlgl). Keep the prepared ESS from Exercise 1 (its "Stuck?" box still applies).

1. Write it yourself: regress trust in the legal system on years of education, with weights, and look at the full results with tidy().

R solution -> don't peek too early ;) !

mod <- lm_robust(trstlgl ~ eduyrs, data = ESS, weights = pspwght)
tidy(mod) # Estimate, standard error, t-value, p-value and 95% CI
#          term estimate std.error statistic p.value conf.low conf.high   df outcome
# 1 (Intercept)    6.135     0.287      21.4 5.2e-89    5.573      6.70 1509 trstlgl
# 2      eduyrs    0.099     0.019       5.2 2.0e-07    0.062      0.14 1509 trstlgl

2. Read off the estimated slope \(\hat\beta\): one more year of education goes with a change in legal-system trust of (three digits; point or comma is fine).

3. What is its estimated standard error \(\hat\sigma\)?

4. The \(t\)-value here is about 5.2. What does it tell you? The estimate is too large to be significantThe probability that H0 is true is about 5%The estimate is that many standard errors away from 0The share of variance the model explains

5. And the \(p\)-value? Which reading is correct? The probability that H0 is trueThe probability that education really predicts trustIf H0 (β = 0) were true, the probability of drawing a sample with an estimate this far from 0 (or further)The share of respondents who trust the legal system

6. Predict, then check. With \(p =\) 2e-07, at the conventional 5% threshold we … reject H0 — the association is statistically significantcannot reject H0 — the association could be chancehave proven that education causes trust

7. Write two sentences (as # comments): (a) can we reject \(H_0\), and (b) does that make the effect causal? Then compare:

Model answer

(a) Yes: p is well below 0.05, so we reject the null of no association — better-educated Danes trust the legal system significantly more. (b) No — this is a significant association in observational survey data. Significance rules out chance, not confounding: more-educated people differ in many other ways too. Causal claims need the designs we meet later in the course.

8. Visualise it: make a scatter plot of trstlgl on eduyrs with a weighted OLS line and its 95% confidence band, and map the weights to dot size.

R solution -> don't peek too early ;) !

ggplot(ESS, aes(y = trstlgl, x = eduyrs)) +
  geom_jitter(aes(size = pspwght), alpha = 1/3, height = 0.1, width = 0.1) +
  geom_smooth(aes(weight = pspwght), method = "lm", color = "#901A1E") +
  labs(y = "Trust in the legal system", x = "Years of education") +
  theme_minimal() +
  theme(legend.position = "none")

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Bonus (for the fast): re-estimate the model on a random sample of 50 respondents (ESS %>% sample_n(50)). Is the slope still significant? Run it a few times (without set.seed) — what does that teach you about sample size and sampling error?

R solution + takeaway

mod_small <- lm_robust(trstlgl ~ eduyrs, data = ESS %>% sample_n(50))
tidy(mod_small) %>% filter(term == "eduyrs") %>%
  select(estimate, std.error, statistic, p.value)
#   estimate std.error statistic p.value
# 1    0.083      0.05       1.6    0.11

With only 50 people the standard error is large, so the estimate usually is not significant — and it jumps around from run to run. The full-sample effect is real, but a tiny sample rarely has the power to detect it. More data → smaller standard error → sharper conclusions.