Statistical Inference - Confidence Intervals

Confidence intervals

\(\theta\) is fixed and \(C_n\) is random

\[Pr(\theta \in C_n) \ge 0.95\]

---

p <- .3
X <- rbinom(10, 1, p)
X

##  [1] 0 0 0 0 1 0 1 1 0 1

mean(X)

## [1] 0.4

# computeCI is a user-defined function
computeCI(X)

## [1] 0.09016133 0.70983867

---

# Another trial
X <- rbinom(10, 1, 0.3)
X

##  [1] 0 0 0 0 0 1 0 0 0 0

mean(X)

## [1] 0.1

computeCI(X)

## [1] -0.08973666  0.28973666

---

findInterval(.3, c(.1, 4)) == 1

## [1] TRUE

findInterval(.3, c(.35, 4)) == 1

## [1] FALSE

findInterval(.3, computeCI(rbinom(20, 1, p))) == 1

## [1] TRUE

findInterval(.3, computeCI(rbinom(20, 1, p))) == 1

## [1] TRUE

---

in_int <- replicate(1e5,
  findInterval(.3, computeCI(rbinom(20, 1, p))) == 1)
head(in_int)

## [1] TRUE TRUE TRUE TRUE TRUE TRUE

mean(in_int)

## [1] 0.94742

---

N <- 10
p_hat_dist <- replicate(1e4, mean(rbinom(N, 1, p)))
head(p_hat_dist)

## [1] 0.0 0.4 0.1 0.6 0.4 0.6

---

hist(p_hat_dist, 100, xlim=c(0, 1))

---

N <- 20
p_hat_dist <- replicate(1e4, mean(rbinom(N, 1, p)))
head(p_hat_dist)

## [1] 0.30 0.25 0.25 0.25 0.35 0.25

---

hist(p_hat_dist, 100, xlim=c(0, 1))

---

N <- 100
p_hat_dist <- replicate(1e4, mean(rbinom(N, 1, p)))
head(p_hat_dist)

## [1] 0.35 0.26 0.31 0.29 0.29 0.33

---

hist(p_hat_dist, 100, xlim=c(0, 1))

---

qnorm(.5)

## [1] 0

qnorm(.975)

## [1] 1.959964

pnorm(0)

## [1] 0.5

pnorm(1.96)

## [1] 0.9750021

---

computeCI <- function(X) {
  p_hat <- mean(X)
  n <- length(X)
  p_hat + c(-2, 2) * sqrt(p_hat*(1-p_hat)/n)
}