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# visualizingt.r
# Statypus Academy: The Safety Net (t vs. Normal)
# Course Resources: r.statypus.org
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# --- TASK A: THE SAFETY NET AUDIT (LINES 8-53) ---

# 1. SETUP THE X-AXIS AND LIMITS
x <- seq(-5, 5, length = 400)
y_norm <- dnorm(x)
y_t_small <- dt(x, df = 2)

# 2. INITIAL PLOT
plot(x, y_norm, type = "n", # Start with empty plot to layer correctly
     main = "The Safety Net: Statistical Skepticism (n=3)",
     xlab = "Standard Deviations (z or t)",
     ylab = "Density",
     ylim = c(0, 0.5), xlim = c(-5, 5))

# 3. DEFINE CRITICAL VALUES
z_crit <- 1.96
t_crit <- 4.30

# 4. SHADE REJECTION ZONES (Fixed Flat Bottoms)
# Normal Rejection (Gray)
x_z_left  <- seq(-5, -z_crit, length = 100)
polygon(c(-5, x_z_left, -z_crit), c(0, dnorm(x_z_left), 0), col = "gray85", border = NA)
x_z_right <- seq(z_crit, 5, length = 100)
polygon(c(z_crit, x_z_right, 5), c(0, dnorm(x_z_right), 0), col = "gray85", border = NA)

# t (df=2) Rejection (Red)
x_t_left  <- seq(-5, -t_crit, length = 100)
polygon(c(-5, x_t_left, -t_crit), c(0, dt(x_t_left, df=2), 0), col = rgb(0.7, 0, 0, 0.4), border = NA)
x_t_right <- seq(t_crit, 5, length = 100)
polygon(c(t_crit, x_t_right, 5), c(0, dt(x_t_right, df=2), 0), col = rgb(0.7, 0, 0, 0.4), border = NA)

# 5. DRAW CURVES AND THRESHOLDS
lines(x, y_norm, lty = 2, col = "gray40", lwd = 1)
lines(x, y_t_small, col = "black", lwd = 3)
abline(v = c(-z_crit, z_crit), col = "gray40", lty = 2)
abline(v = c(-t_crit, t_crit), col = "darkred", lty = 3, lwd = 2)

# 6. CLEAN SINGLE LEGEND (Fixed Overlap)
legend("topright",
       legend = c("Normal (Steel Rod)", "t (df=2, Rubber Band)",
                  "z-threshold (1.96)", "t-threshold (4.30)"),
       col = c("gray40", "black", "gray40", "darkred"),
       lty = c(2, 1, 2, 3),
       lwd = c(1, 3, 1, 2),
       cex = 0.8, # Slightly smaller text to fit
       inset = 0.02, # Move away from edges
       bty = "n")


# --- TASK B: THE CONVERGENCE (LINES 58-69) ---

plot(x, y_norm, type = "l", lty = 2, col = "gray", lwd = 2,
     main = "The Law of Certainty: Case vs. Pallet",
     xlab = "Standard Deviations (z or t)", ylab = "Density",
     ylim = c(0, 0.5), xlim = c(-5, 5))

lines(x, dt(x, df = 11), col = "firebrick", lwd = 2)
lines(x, dt(x, df = 143), col = "royalblue", lwd = 2)

legend("topright",
       legend = c("Normal", "Single Case (df=11)", "Full Pallet (df=143)"),
       col = c("gray", "firebrick", "royalblue"),
       lty = c(2, 1, 1), lwd = 2, bty = "n", cex = 0.8)