= Solution
\b[TODO: Complete the data-specific numerical results and diagnostics when the original `Iris_data.csv` is available.]
The identity
$$
\sum_i(x_i-\mu)^2=\sum_i(x_i-\bar x)^2+n(\mu-\bar x)^2
$$
shows that the <likelihood function> is a bell-shaped curve in $\mu$, maximized at the <sample mean>, with width $1/\sqrt n$. Its actual vertical scale also depends on the within-sample sum of squares.
Continue the R script with:
``
sse <- sum((x - xbar)^2)
loglik <- function(mu) {
-n/2 * log(2*pi) - 0.5 * (sse + n * (mu - xbar)^2)
}
grid <- seq(3, 6, length.out = 2001L)
ell <- loglik(grid)
png("q2-likelihood.png", width = 900, height = 600)
par(mar = c(4.5, 4.5, 4, 1))
if (max(ell) > log(.Machine$double.xmin)) {
plot(grid, exp(ell), type = "l", xlab = expression(mu),
ylab = expression(L(mu)), main = "Likelihood on [3,6]")
} else {
# Explicitly label a proportional likelihood if the true scale underflows.
plot(grid, exp(ell - max(ell)), type = "l", xlab = expression(mu),
ylab = "Likelihood relative to maximum on plotted grid",
main = "Likelihood on [3,6] (rescaled)")
}
abline(v = xbar, col = "steelblue", lty = 2)
dev.off()
``
Working first with the <log-likelihood> avoids premature numerical underflow. The fallback preserves the shape on the requested interval and labels the altered scale. \b[A data-specific plot requires the missing CSV file]; no fabricated observation set is used here.
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