Exercise 1: A Two-Proportion Z-Test, n=500 per Group — Possible Solution
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GIVEN
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Control: n1=500, x1=60 conversions -> p1 = 60/500 = 0.12 (12%)
Treatment: n2=500, x2=85 conversions -> p2 = 85/500 = 0.17 (17%)

STEP 1: THE VALIDITY CHECK
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Control:   n1*p1 = 500(0.12) = 60,  n1*(1-p1) = 500(0.88) = 440
Treatment: n2*p2 = 500(0.17) = 85,  n2*(1-p2) = 500(0.83) = 415

All four values are comfortably above 5, so per this chapter's own
validity rule, the normal approximation is safe to use here.

STEP 2: THE POOLED PROPORTION
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p_pool = (x1+x2)/(n1+n2) = (60+85)/(500+500) = 145/1000 = 0.145

STEP 3: THE POOLED STANDARD ERROR
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SE_pooled = sqrt(p_pool(1-p_pool)(1/n1 + 1/n2))
          = sqrt(0.145 x 0.855 x (1/500 + 1/500))
          = sqrt(0.123975 x 0.004)
          ~= 0.02227

STEP 4: THE Z-STATISTIC
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z = (p2 - p1) / SE_pooled = (0.17 - 0.12) / 0.02227 ~= 2.245

STEP 5: THE P-VALUE AND DECISION
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p-value = 2 x (1 - Phi(2.245)) ~= 0.0247

Since 0.0247 < 0.05, per this chapter's own decision rule, we REJECT
H0 - this difference (12% vs 17%) is statistically significant at
alpha=0.05, unlike this chapter's own main worked example.

WHY THIS WORKS AS AN ANSWER
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The validity check is performed first per this chapter's own rule of
thumb before trusting the normal approximation, and every subsequent
quantity is computed in the same sequence this chapter's own worked
example used - pooled proportion, pooled SE, z-statistic, p-value -
with the final decision made by direct comparison against alpha.