Standard Normal Distribution: Unit Normal Table and Z-Score Calculations

Basic Problem Types for Standard Normal Distribution

• There are two fundamental types of problems encountered when working with the standard normal distribution:
  • Type A: Looking up a proportion ($p$) given a specific $z$-score.
  • Type B: Looking up a $z$-score given a specific proportion ($p$).
• These two types represent the inverse of each other; one moves from the axis value to area, while the other moves from area to the axis value.

Fundamental Definitions and Interchangeable Terminology

• Within the context of a normal distribution, several terms are functionally interchangeable as they all refer to the shared area under the density curve:
  • Proportion: Expressed as a decimal between $0$ and $1$.
  • Percent: The proportion multiplied by $100$ (moving the decimal two places to the right).
  • Probability: The likelihood of randomly selecting a specific score or range of scores.
  • Area: The physical region shaded under the bell curve.
• Conversion between proportions and percentages is a core skill:
  • To convert a proportion to a percent: Move the decimal point two spots to the right.
  • To convert a percent to a proportion: Move the decimal point two spots to the left.

The Essential Tools: The Unit Normal Table and The Distribution Drawing

• Two tools are required to solve these problems. It is taught that one cannot solve these problems accurately using only one tool; both must be used in conjunction.
• The Unit Normal Table (or $z$-table):
  • Column A: Contains the $z$-scores.
  • Column B: Contains the proportion in the Body (the larger portion).
  • Column C: Contains the proportion in the Tail (the smaller portion).
  • Column D: Not focused on in this specific session.
• The Drawing:
  • A normal distribution curve (bell curve) must be drawn for every problem.
  • The center of the distribution is always marked as $z = 0$. This point represent the mean, where the score is zero standard deviations away from the mean.
  • Positive $z$-scores are placed to the right of zero; negative $z$-scores are placed to the left.

Determining Shading Direction through Keywords

• Keywords within a word problem indicate the direction of shading relative to the $z$-score. This is critical for identifying whether the objective is a "body" or a "tail."
• Shading to the Right:
  • Keywords: "Above," "Greater than," "Top," "To the right."
  • Notation: $p(z > \text{value})$.
• Shading to the Left:
  • Keywords: "Below," "Less than," "Bottom," "To the left."
  • Notation: $p(z < \text{value})$.
• Important caveat: Directional keywords like "top" or "below" refer to the direction of shading, not the sign of the $z$-score itself.

Understanding the "Body" versus the "Tail"

• Slicing a normal distribution at any point other than the exact center ($z = 0$) creates two unequal sections:
  • The Body: The larger portion of the distribution (greater than $0.5$ or $50\%$" ").
  • The Tail: The smaller portion of the distribution (less than $0.5$ or $50\%$" ").
• If the distribution is sliced exactly at the middle ($z = 0$), the area is perfectly divided into two halves ($0.5$ and $0.5$), meaning there is no distinct body or tail.

Finding Proportions from Z-Scores (Type A Problems)

• Example 1: Proportion above $z = 1.3$
  • Step 1: Draw the curve and mark $z = 0$ at the center.
  • Step 2: Place $z = 1.3$ to the right of zero.
  • Step 3: Shade to the right (keyword: "above").
  • Step 4: Identify the shaded area as the Tail.
  • Step 5: Look up $z = 1.3$ in Column A and find the value in Column C.
  • Result: Proportion = $0.0968$ ($9.68\%$" ").
• Example 2: Proportion below $z = 1.3$
  • Step 1: Re-use the drawing for $z = 1.3$, but shade to the left.
  • Step 2: Identify the shaded area as the Body.
  • Step 3: Look up $z = 1.3$ in Column A and find the value in Column B.
  • Result: Proportion = $0.9032$ ($90.32\%$" ").

Handling Negative Z-Scores and the Principle of Symmetry

• The Unit Normal Table typically only lists positive $z$-scores because the normal distribution is perfectly symmetrical.
• Example 3: Percent above $z = -1$
  • Step 1: Mark $z = -1$ to the left of zero.
  • Step 2: Shade to the right (keyword: "above").
  • Step 3: Identify the shaded area as the Body.
  • Step 4: Look up the positive version ($z = 1$) in the table and find the Body column.
  • Result: Proportion = $0.8413$ ($84.13\%$" ").
• Example 4: Probability of selecting a $z$-score below $z = -1$
  • Notation: $p(z < -1)$.
  • Step 1: Mark $z = -1$ to the left of center and shade to the left.
  • Step 2: Identify the shaded area as the Tail.
  • Step 3: Look up $z = 1$ in the table and find the Tail column.
  • Result: Proportion = $0.1587$ ($15.87\%$" ").

Finding Z-Scores from Proportions or Percentages (Type B Problems)

• In these problems, the area (proportion/percent) is given, and the goal is to find the corresponding $z$-score on the axis.
• The logic of the drawing changes:
  • In Type A, the drawing identifies Body/Tail.
  • In Type B, the drawing primarily identifies the sign of the $z$-score (positive or negative).
• Heuristic for sign determination:
  • If you need a tail ($p < 0.5$) and are shading top/above, the $z$-score is positive.
  • If you need a tail ($p < 0.5$) and are shading bottom/below, the $z$-score is negative.
  • If you need a body ($p > 0.5$) and are shading top/above, the $z$-score is negative.
  • If you need a body ($p > 0.5$) and are shading bottom/below, the $z$-score is positive.

Case Studies for Finding Z-Scores from Area

• Example 5: The top (above) $5.05\%$ of scores
  • Step 1: $5.05\% = 0.0505$. This is a Tail (less than $0.5$).
  • Step 2: Shade the "top" (right side). To get a small tail by shading right, the line must be on the right side of zero (positive $z$).
  • Step 3: Find $0.0505$ in Column C (Tail column).
  • Step 4: Find the corresponding $z$-score in Column A.
  • Result: $z = +1.64$.
• Example 6: The bottom (below) $5.05\%$ of scores
  • Step 1: $0.0505$ is a Tail.
  • Step 2: Shade the "bottom" (left side). To get a small tail by shading left, the line must be on the left side of zero (negative $z$).
  • Step 3: Sign is negative.
  • Result: $z = -1.64$.
• Example 7: The top (above) $90\%$ of scores
  • Step 1: $90\% = 0.9000$. This is a Body (more than $0.5$).
  • Step 2: Shade the "top" (right side). To get a massive body by shading right, the cut must start deep in the left side of the distribution (negative $z$).
  • Step 3: Look for $0.9000$ in the Body column (Column B).
  • Step 4: The values nearest on the table are $0.8997$ ($z = 1.28$) and $0.9015$ ($z = 1.29$). Both are acceptable approximations.
  • Step 5: Apply the negative sign identified in Step 2.
  • Result: $z = -1.28$ or $z = -1.29$.

Special Cases and Final Reminders

• The 50% Threshold: If the problem asks for the bottom $50\%$" " or top $50\%$" ", the $z$-score is always exactly $0$. This is because the mean splits the normal distribution into two equal halves.
• The Importance of the Sign: Missing a negative sign is considered a major error, not a small oversight. A $z$-score of $+1.64$ (way above average) is qualitatively different from $-1.64$ (way below average). Correct identification of the sign proves a correct drawing was made.
• Approximations: Often, the exact proportion requested (like $0.9000$) is not on the table. Choosing the closest value available is the standard procedure for these exercises.
• Laboratory Practice: The speaker advises rewatching this specific instructional video while completing the laboratory assignments, as the procedural steps are the core aspect of the record.