AP Statistics 1.10 The Normal Distribution - MCQs - Exam Style Questions

Use the normal model with mean \(\mu=65{,}200\) and standard deviation \(\sigma=3{,}100\).
Empirical/normal-rule benchmarks: about \(68\%\) within \(\pm1\sigma\), \(95\%\) within \(\pm2\sigma\), and \(99.7\%\) within \(\pm3\sigma\). Roughly \(99\%\) is within about \(\pm2.58\sigma\).
Check each statement quickly with \(z\)-scores:
• (A) \(56{,}700\) is \(z=\dfrac{56{,}700-65{,}200}{3{,}100}\approx -2.74\Rightarrow P\lt 1\%\), not \(5\%\). ✗
• (B) Endpoints: \(57{,}500\Rightarrow z\approx -2.48\); \(73{,}500\Rightarrow z\approx 2.68\). Central area between these \(z\)-values is about \(0.99\) (≈\(99\%\)). ✓
• (C) IQR for a normal is \(Q_3-Q_1\approx(0.674-(-0.674))\sigma\approx1.349\sigma\approx1.349(3{,}100)\approx \$4{,}180\), not \(\$6{,}200\). ✗
• (D) “Maximum” is not determined by a normal model; distribution is unbounded. ✗
• (E) \(57{,}400\Rightarrow z\approx -2.52\Rightarrow P(X\gt 57{,}400)\approx 0.99\), not \(84\%\). ✗
✅ Answer: (B)