Standard Normal Distribution – Detailed Study Notes

Lesson Goals

• Determine areas (probabilities) under the normal curve with the aid of published tables.
• Work backward from a specified probability to identify the corresponding $z$‐score.

Background & Key Definitions

• Normal distribution
  • Family of continuous distributions indexed by mean $\mu$ and standard deviation $\sigma$.
  • Because $\mu$ and $\sigma$ can take infinitely many values, there are infinitely many distinct normal curves.
• Standard normal distribution
  • Special case with $\mu = 0$ and $\sigma = 1$.
  • Denoted by the random variable $Z$.
• Standardizing ("z‐transformation")
  • Any normal variable $X \sim N(\mu,\sigma)$ can be converted to $Z$ with
    $Z = \frac{X-\mu}{\sigma}.$
  • Allows every normal problem to be solved with a single table.

Types of Standard Normal Tables Mentioned

• Table: $(-\infty,\,-z)$ → Cumulative probability P(Z < -z).
• Table: $(-\infty,\,z)$ → Cumulative probability P(Z < z).
• Table: $(0,\,z)$ → Probability from the mean up to $z$, i.e. P(0 < Z < z).

How to Read a Two-Decimal Standard Normal Table

• $z$ values are rounded to two decimals.
  • First decimal place = left-hand column (rows list 0.0, 0.1, 0.2, …).
  • Second decimal place = top header row (0.00, 0.01, 0.02, …, 0.09).
• Intersection of chosen row and column gives the area to the left of that $z$ (for the cumulative tables).

Finding Specific Areas (Probabilities)

1. Area to the Left of a Given $z$

• Direct: look the value up in the $( -\infty, z )$ table.
• Example workflow (not in transcript but implied):
  • Want P(Z < 1.23) → row 1.2, column 0.03 → read area (e.g. $0.8907$).

2. Area to the Right of a Given $z$

Two equivalent methods:

1. Complement Rule
   • Total area under a normal curve = $1$.
   • P(Z > z) = 1 - P(Z < z).
2. Symmetry Rule
   • Normal curve is symmetric about $0$.
   • P(Z > z) = P(Z < -z).

3. Area Between Two $z$ Values (say $z<em>1 < z</em>2$)

• Steps:
  1. Obtain P(Z < z_2) from the table.
  2. Obtain P(Z < z_1) from the table.
  3. Subtract: P(z1 < Z < z2) = P(Z < z2) - P(Z < z1).
• Guarantees a positive result because $z_2$ is larger.

Determining $z$ for a Given Area (Inverse Use of the Table)

A. When Area is to the Left

• Locate the desired probability in the body of the cumulative table.
• Read corresponding row (1st decimal) and column (2nd decimal) to reconstruct $z$.
• Example (hypothetical):
  • Want $z$ such that P(Z < z) = 0.9750
  • Table entry $0.9750$ sits at row 1.9, col 0.06 → $z \approx 1.96$.

B. When Area is to the Right

• First convert to a left-side probability: P(Z < z) = 1 - \text{(area to the right)}.
• Use same lookup procedure as above.
• By symmetry, you could equivalently search for the left‐tail probability associated with $-z$.

Key Properties & Connections

• Total Probability under any density curve = $1$. Used repeatedly for complement calculations.
• Symmetry about $0$ makes P(Z > z) fast to compute via P(Z < -z), halving table work.
• Earlier Chapter Connection: z-scores previously introduced to compare observations from different normal populations—here extended to compute exact probabilities.

Practical & Ethical Considerations

• Accurate probability lookup supports decision-making in fields such as quality control, finance, and biomedical research.
• Misusing tail probabilities (e.g.
  p-hacking) can lead to ethical breaches; ensure correct tail direction and area interpretation.
• Round-off awareness: tables rounded to two decimals introduce small approximation error—important for critical applications.

Numerical & Formula Recap

• Standardization: $Z = \frac{X-\mu}{\sigma}$.
• Complement rule: P(Z > z) = 1 - P(Z < z).
• Symmetry: P(Z > z) = P(Z < -z).
• Region between two points: P(z1 < Z < z2) = P(Z < z2) - P(Z < z1).

Study Tips & Takeaways

• Memorize the meaning of each table type: full left tail vs. half-tail $0 \rightarrow z$.
• Always sketch a quick normal curve marking the area of interest; prevents left/right mix-ups.
• For inverse problems, scan table values systematically; probability values grow monotonically so search is straightforward.
• Keep the symmetry rule in mind to avoid unnecessary subtraction; sometimes P(Z < -z) is tabulated directly.
• Practice by converting at least five real-world normal-variable questions into standard normal form to build fluency.