Advanced Normal Distribution: Reverse Table and Raw Score Calculations

Introduction to Reverse Table Problems

• Conceptual Shift: In previous sections, the focus was on looking up a Z-score to find a proportion (either body or tail). Now, the process is reversed: given a proportion, students must find the corresponding Z-score.
• The Algebraic Component: Reverse table problems often incorporate algebra to convert the found Z-score into a raw score ($X$).
• Sign of the Z-score: The biggest challenge in reverse table problems is determining if the Z-score is positive or negative. The statistical table only lists positive Z-scores to avoid redundancy (as negative scores would simply repeat the table with a negative sign).
• The Importance of Drawing: Students must use a visual distribution drawing to determine the sign of the Z-score and whether the given proportion represents a "body" or a "tail."

Analyzing the Anatomy of a Distribution Picture

Every distribution drawing for these problems contains three critical elements:

1. The Sign of Z: Is the score positive or negative?
2. The Size of the Proportion: Is it a body (greater than $50\%$) or a tail (less than $50\%$)?
3. The Direction of Shading: Is the shading to the right (top/above) or to the left (bottom/below)?

Defining Directional Keywords

• Top / Above / Greater Than: These terms indicate shading to the right of the Z-score.
• Bottom / Below / Less Than: These terms indicate shading to the left of the Z-score.
• Misconception Alert: Note that "top" does not automatically mean a positive Z-score, and "bottom" does not automatically mean a negative Z-score. The sign depends on the combination of the direction and the size of the proportion.

Analysis of Specific Problems (Items 8, 9, and 10)

Problem 8: Top 6.68%

• Proportion: $0.0668$ ($6.68\%$). Since this is less than $50\%$, it is a tail.
• Shading: "Top" means shading to the right.
• Determining Z-sign: Shading to the right from a negative Z-score would result in a body. Shading to the right from a positive Z-score results in a tail. Therefore, the Z-score must be positive.
• Finding Z on Table: Looking up $0.0668$ in Column C (tail) yields a Z-score of $1.5$.
• Final Answer: $Z = 1.5$

Problem 9: Bottom 6.68%

• Proportion: $0.0668$ ($6.68\%$), which is a tail.
• Shading: "Bottom" means shading to the left.
• Determining Z-sign: Shading to the left from a positive Z-score would result in a body. Shading to the left from a negative Z-score results in a tail. Therefore, the Z-score must be negative.
• Finding Z on Table: Use the same numerical value from the table ($1.5$) but apply the negative sign identified from the drawing.
• Final Answer: $Z = -1.5$

Problem 10: Top 86.43%

• Proportion: $0.8643$ ($86.43\%$). Since this is greater than $50\%$, it is a body.
• Shading: "Top" means shading to the right.
• Determining Z-sign: A positive Z-score shaded to the right yields a small tail. To get a large body shaded to the right, the Z-score must be located on the left side of the distribution.
• Sign: Negative.
• Finding Z on Table: Looking up $0.8643$ in Column B (body) yields a Z-score of $1.1$.
• Final Answer: $Z = -1.1$

Conversion from Z-Score to Raw Score (Type D Problems)

Type D problems require the extra step of using the Z-score formula to find a raw score ($X$).

$Z = \frac{X - \mu}{\sigma}$

Example 1: Finding X for the top 0.0049

• Data: Mean ($\mu$) = $100$, standard deviation ($\sigma$) = $10$.
• Proportion: $0.0049$ is a very small tail ($0.49\%$).
• Direction: "Top" (shading to the right).
• Sign: A positive Z-score shaded right gives a tail. Sign is positive.
• Table lookup: A tail proportion of $0.0049$ corresponds to a Z-score of $2.58$.
• Algebraic Calculation:
  • $2.58 = \frac{X - 100}{10}$
  • $25.8 = X - 100$
  • $X = 125.8$

Example 2: Finding X for the bottom 0.9332 (Correction Included)

• Data: Mean ($\mu$) = $140$, standard deviation ($\sigma$) = $4$.
• Proportion: $0.9332$ is a body.
• Direction: "Bottom" (shading to the left).
• Sign: A positive Z-score shaded left gives a body. Sign is positive.
• Correction Note: The speaker initially looked up the wrong value ($0.9932$). The correct lookup for $0.9332$ in the body column (Column B) is a Z-score of $1.5$.
• Algebraic Calculation:
  • $1.5 = \frac{X - 140}{4}$
  • $6 = X - 140$
  • $X = 146$

Comparison of Problem Types (A, B, C, D)

The speaker categorizes problems into four types based on the given information and the goal:

• Type A (Z to Proportion): Given a Z-score, find the proportion ($P$). No algebra needed; go straight to the table.
• Type B (Raw Score to Proportion): Given $X$, find $P$. Logic: $X \rightarrow Z \rightarrow P$. Requires algebra first ($Z = \frac{X - \mu}{\sigma}$), then the table.
• Type C (Proportion to Z-score): Given $P$, find $Z$. No algebra needed; reverse table lookup. Identification of sign via drawing is vital.
• Type D (Proportion to Raw Score): Given $P$, find $X$. Logic: $P \rightarrow Z \rightarrow X$. Start with a drawing and table lookup for $Z$, then use algebra to solve for $X$.

Identifying Practice Problems by Type

• Problem 15: "Find $P$" given $X$, $\mu$, and $\sigma$. This is a Type B ($X \rightarrow P$).
• Problem 16: "What Z-score?" given a proportion ($P = 0.2912$). This is a Type C ($P \rightarrow Z$).
• Problem 17: "What proportion?" given a Z-score. This is a Type A ($Z \rightarrow P$).
• Problem 18: "What raw score?" given a proportion ($P = 75\%$). This is a Type D ($P \rightarrow X$).