STAT 332 – Q4 Solutions

# part a)

y <- d$books
ybar <- mean(y, na.rm = TRUE)
s2   <- var(y,  na.rm = TRUE)
SE_ybar <- sqrt((1 - n/N) * s2 / n)
tcrit   <- qt(0.975, df = n - 1)
CI_ybar <- c(ybar - tcrit * SE_ybar, ybar + tcrit * SE_ybar)

cat(
  "n =", n, "\n",
  "Mean (ybar) =", round(ybar, 4), "\n",
  "SE(ybar)    =", round(SE_ybar, 4), "\n",
  "95% CI      = [", round(CI_ybar[1], 4), ", ", round(CI_ybar[2], 4), "]\n", sep = ""
)

## n =91
## Mean (ybar) =8.912
## SE(ybar)    =1.583
## 95% CI      = [5.768, 12.06]

# part b)
p_hat <- mean(d$sn == 1, na.rm = TRUE)
SE_p  <- sqrt((1 - n/N) * p_hat * (1 - p_hat) / n)
z     <- qnorm(0.975)
CI_p  <- c(p_hat - z * SE_p, p_hat + z * SE_p)

cat(
  "p-hat =", round(p_hat, 4), "\n",
  "SE(p) =", round(SE_p, 4), "\n",
  "95% CI = [", round(CI_p[1], 4), ", ", round(CI_p[2], 4), "]\n", sep = ""
)

## p-hat =0.2857
## SE(p) =0.0474
## 95% CI = [0.1929, 0.3785]

# part c)
h <- 0.005
s <- sqrt(s2)
z <- qnorm(0.975)


n_req_fpc <- (N * z^2 * s2) / (N * h^2 + z^2 * s2)


cat("respondants needed:", ceiling(n_req_fpc))

## respondants needed: 14018714