# load data
library(Ecdat)
data("Airq", package="Ecdat")
# a) estimate mean rainfall
ybar <- mean(Airq$rain)
ybar[1] 36.078# b) standard error
V <- var(Airq$rain)
n <- nrow(Airq)
se <- sqrt(V)/sqrt(n)
se[1] 2.462628# c) t-statistic
h0 <- 25
t <- (ybar-h0)/se
t[1] 4.498446Since \(|t| > 1.96\), we would reject \(H_0\) at the 5% significance level.
# d) p-value
pval <- 2*pnorm(-abs(t))
pval[1] 6.845183e-06There is virtually a 0 percent chance of getting a t-statistic this large in absolute value if the null hypotheses were true.
# e) confidence interval
ciL <- ybar - 1.96*se
ciU <- ybar + 1.96*se
paste0("[",round(ciL,3),", ", round(ciU,3), "]")[1] "[31.251, 40.905]"# f) summary statistics
library(modelsummary)
datasummary_balance(~coas, Airq)| no (N=9) | yes (N=21) | |||||
|---|---|---|---|---|---|---|
| Mean | Std. Dev. | Mean | Std. Dev. | Diff. in Means | Std. Error | |
| airq | 125.3 | 10.5 | 95.9 | 28.7 | -29.5 | 7.2 |
| vala | 4118.2 | 5909.8 | 4218.6 | 4136.7 | 100.4 | 2166.9 |
| rain | 32.3 | 7.6 | 37.7 | 15.2 | 5.4 | 4.2 |
| dens | 1706.4 | 3014.6 | 1738.1 | 2821.2 | 31.7 | 1178.5 |
| medi | 6290.3 | 10065.4 | 10842.2 | 13396.8 | 4551.9 | 4450.1 |
Since \(n\) grows faster than \(\sqrt{n}\), \(n \left(\frac{1}{n}\sum_{i=1}^n (Y_i - \mathbb{E}[Y])\right)\) diverges (i.e., the absolute value goes to infinity as \(n \rightarrow \infty\))
Since \(n^{1/3}\) grows slower than \(\sqrt{n}\), \(n^{1/3} \left(\frac{1}{n}\sum_{i=1}^n (Y_i - \mathbb{E}[Y])\right)\) converges to 0 as \(n \rightarrow \infty\)