Normal Distribution and Inverse Normal Calculations

Overview of Inverse Normal Distribution Calculations

• The normal distribution is defined by two primary parameters:
  • Mean: $\mu$ (or sample mean $\bar{x}$), which represents the center of the symmetric bell curve.
  • Standard deviation: $\sigma$, which represents the spread or variability of the distribution.
• The inverse normal function, typically executed via the graphing calculator command $\text{invNorm}$, is used to calculate the specific boundary value ($x$) corresponding to a given cumulative probability or area under the normal curve:
  • General syntax: $\text{invNorm}(\text{Area}, \mu, \sigma, [\text{Tail}])$
  • In standard configurations without tail specification, the function assumes cumulative area measured from the extreme left up to the cutoff score ($\text{Left}$ tail).
  • Advanced calculator operating systems allow tail orientation arguments: $\text{Left}$, $\text{Right}$, or $\text{Center}$.
• Standard transformation relationship between the raw score ($x$), mean ($\mu$), standard deviation ($\sigma$), and standard normal score ($z$):
  • $z = \frac{x - \mu}{\sigma}$
  • $x = \mu + z \times \sigma$
  • Where $z = \text{invNorm}(\text{Area}, 0, 1)$.

Problem 5: Wechsler Adult Intelligence Scale Subtest Scores

• Distribution Parameters:
  • Model: The raw scores on the subtest form a normal distribution: $X \sim N(35, 6^2)$.
  • Mean: $\bar{x} = 35$
  • Standard deviation: $\sigma = 6$
• Part (a): Determination of the 28th Percentile
  • Objective: Determine the raw score cutoff that represents the 28th percentile ($\text{cumulative area} = 0.28$).
  • Calculator Command: $\text{invNorm}(0.28, 35, 6, \text{Left})$
  • Mathematical Calculation:
  • Standard normal cutoff: $z_{0.28} = \text{invNorm}(0.28, 0, 1) \approx -0.5828$
  • Raw score evaluation: $x = 35 + (-0.5828) \times 6 = 35 - 3.497 \approx 31.50$ (recorded calculation on worksheet: $31.31$)
  • Curve Sketch Characteristics:
  • Symmetrical bell-shaped curve centered at $\bar{x} = 35$.
  • A vertical cutoff line placed in the lower half of the distribution at approximately $31.31$.
  • The region beneath the curve to the left of $31.31$ is shaded, representing the lowest $28\%$ of the distribution.
  • Written Conclusion: $31.31$ is the 28th percentile for intelligence score.
• Part (b): Determination of the 90th Percentile
  • Objective: Determine the raw score cutoff that represents the 90th percentile ($\text{cumulative area} = 0.90$).
  • Calculator Command: $\text{invNorm}(0.90, 35, 6, \text{Left})$
  • Mathematical Calculation:
  • Standard normal cutoff: $z_{0.90} = \text{invNorm}(0.90, 0, 1) \approx 1.28155$
  • Raw score evaluation: $x = 35 + (1.28155) \times 6 = 35 + 7.6893 = 42.6893 \approx 42.69$
  • Curve Sketch Characteristics:
  • Symmetrical bell-shaped curve centered at $\bar{x} = 35$.
  • A vertical cutoff line positioned in the upper tail at $42.69$.
  • The entire region beneath the curve to the left of $42.69$ is shaded, encompassing $90\%$ of the total area.
  • Written Conclusion: $42.69$ is the 90th percentile intelligence score.

Problem 6: SAT Math Section Score Distribution

• Distribution Parameters:
  • Model: The SAT Math section scores are approximately normally distributed: $X \sim N(500, 100^2)$.
  • Mean: $\bar{x} = 500$
  • Standard deviation: $\sigma = 100$
• Part (a): Minimum Score for the Top 15%
  • Objective: Determine the minimum score required to place in the top $15\%$ of test takers.
  • Area Interpretation and Calculator Commands:
  • Being in the top $15\%$ implies that $85\%$ of scores fall below this cutoff ($\text{cumulative left area} = 1.00 - 0.15 = 0.85$).
  • Left-tail command: $\text{invNorm}(0.85, 500, 100, \text{Left})$
  • Right-tail command: $\text{invNorm}(0.15, 500, 100, \text{Right})$
  • Uncorrected evaluation for the lower 15th percentile: $\text{invNorm}(0.15, 500, 100, \text{Left}) = 396.36$ (which designates the bottom $15\%$ rather than the top $15\%$).
  • Mathematical Calculation for Upper Cutoff:
  • Standard normal cutoff: $z_{0.85} \approx 1.0364$
  • Raw score: $x = 500 + (1.0364) \times 100 = 603.64$
  • Curve Sketch Characteristics:
  • Symmetrical bell curve centered at $500$.
  • Shaded upper tail above the cutoff threshold representing the top $15\%$ of scores (or lower tail below $396.36$ for the bottom $15\%$).
  • Concluding Sentence: $396.36$ represents the bottom $15\%$ of math scores, whereas a score of $603.64$ is required to reach the top $15\%$.
• Part (b): Score Range for the Middle 80%
  • Objective: Identify the interval of scores containing the central $80\%$ of the distribution.
  • Tail Distribution:
  • Total excluded area: $1.00 - 0.80 = 0.20$.
  • Area per tail: $\frac{0.20}{2} = 0.10$ in each tail (lower $10\%$ and upper $10\%$).
  • Cumulative left area for lower cutoff: $0.10$.
  • Cumulative left area for upper cutoff: $0.10 + 0.80 = 0.90$.
  • Calculator Commands:
  • Center tail mode: $\text{invNorm}(0.80, 500, 100, \text{Center})$
  • Separate bounds mode: $\text{invNorm}(0.10, 500, 100, \text{Left})$ and $\text{invNorm}(0.90, 500, 100, \text{Left})$
  • Mathematical Calculation:
  • Critical values: $z = \pm 1.28155$
  • Lower limit: $x_1 = 500 - (1.28155) \times 100 = 371.8445 \approx 371.84$ (recorded on worksheet as $371.69$ / $371.84$)
  • Upper limit: $x_2 = 500 + (1.28155) \times 100 = 628.1555 \approx 628.16$
  • Curve Sketch Characteristics:
  • Normal curve centered at $500$.
  • Two vertical boundary lines at $371.84$ and $628.16$.
  • Symmetrically shaded central region comprising $80\%$ of the total distribution area.
  • Written Conclusion: Middle $80\%$ of SAT scores are between $371.84$ and $628.16$ (recorded as $371.69 - 628.16$).

Problem 7: Wildlife Conservation Trout Weight Distribution Analysis

• Biological Context and Parameters:
  • Objective: Tag adult trout that are unusually large or unusually small to assess their capability of traversing a newly dammed river to spawn.
  • Population distribution model: $X \sim N(3.65, 1.3^2)$
  • Mean weight: $\bar{x} = 3.65\,\text{pounds}$
  • Standard deviation: $\sigma = 1.3\,\text{pounds}$
  • Tagging Criteria: Tag only fish falling into the bottom $5\%$ or top $5\%$ of weights.
• Calculator Syntax:
  • Lower cutoff command: $\text{invNorm}(0.05, 3.65, 1.3, \text{Left})$
  • Upper cutoff command: $\text{invNorm}(0.95, 3.65, 1.3, \text{Left})$ or $\text{invNorm}(0.05, 3.65, 1.3, \text{Right})$
• Step-by-Step Mathematical Calculations:
  • Critical $z$-scores for extreme $5\%$ tails: $z_{0.05} = -1.64485$ and $z_{0.95} = +1.64485$
  • Lower weight cutoff calculation:
  • $x_{\text{lower}} = 3.65 + (-1.64485) \times 1.3$
  • $x_{\text{lower}} = 3.65 - 2.1383 = 1.5117 \approx 1.51\,\text{pounds}$
  • Upper weight cutoff calculation:
  • $x_{\text{upper}} = 3.65 + (1.64485) \times 1.3$
  • $x_{\text{upper}} = 3.65 + 2.1383 = 5.7883 \approx 5.79\,\text{pounds}$
• Curve Sketch Characteristics:
  • Bell-shaped curve centered at $3.65\,\text{pounds}$.
  • Vertical boundary lines drawn at $1.51\,\text{pounds}$ and $5.79\,\text{pounds}$.
  • Shaded regions located exclusively in both outer tails: the bottom $5\%$ (to the left of $1.51\,\text{pounds}$) and the top $5\%$ (to the right of $5.79\,\text{pounds}$).
• Final Concluding Statement:
  • Trout weighing $1.51\,\text{pounds}$ or less, or $5.79\,\text{pounds}$ or more, fall into the designated cutoffs and will be tagged for the conservation study.