Exercise 1: A Full Z-Test, n=36 — Possible Solution
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GIVEN
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H0: mu = 50, H1: mu != 50
n = 36, x-bar = 54, sigma = 12

STEP 1: THE STANDARD ERROR
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SE = sigma / sqrt(n) = 12 / sqrt(36) = 12 / 6 = 2

STEP 2: THE Z-STATISTIC
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z = (x-bar - mu0) / SE = (54 - 50) / 2 = 2.0

STEP 3: THE P-VALUE
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p-value = 2 x (1 - Phi(|z|)) = 2 x (1 - Phi(2.0))

Phi(2.0) ~= 0.9772 (the standard normal CDF at z=2, per Probability &
Statistics Fundamentals Chapter 8's own empirical-rule cross-check)

p-value = 2 x (1 - 0.9772) = 2 x 0.0228 ~= 0.0455 (about 4.55%)

STEP 4: THE DECISION AT alpha=0.05
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0.0455 < 0.05

Since the p-value is less than alpha, per this chapter's own decision
rule, we REJECT H0 - though only just; the p-value sits close enough
to the 0.05 threshold that this is a borderline result, not an
overwhelming one.

WHY THIS WORKS AS AN ANSWER
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Each quantity is computed in sequence directly from this chapter's own
formulas - standard error, then z-statistic, then p-value via the
standard normal CDF - and the final decision is made by comparing the
p-value against alpha exactly per this chapter's own stated rule,
noting how close the result sits to the threshold rather than treating
"reject" as an unambiguously strong conclusion.