Normal Distribution, Z-Scores, and Area Calculations

Foundations of the Normal Distribution and Hypothesis Testing

• Transition from Randomization and Simulation:

  • Statistical inference previously relied on randomization tests and computer simulation to calculate empirical $p$-values.

  • The normal model serves as an analytical, mathematical framework to evaluate hypotheses and calculate $p$-values without requiring repeated random simulations.

• Defining the $p$-Value under the Normal Model:

  • The $p$-value is defined geometrically as an area under the continuous normal distribution curve relative to an observed test statistic (observed value or cutoff value).

  • Right-Tailed Test: The $p$-value corresponds to the area under the normal curve located to the extreme right of the observed value.

  • Left-Tailed Test: The $p$-value corresponds to the area under the normal curve located to the extreme left of the observed value.

  • Two-Sided (Two-Tailed) Test: The $p$-value comprises the combined area in both the extreme left and extreme right tails.

• Observed Value and Cutoff Value Terminology:

  • In hypothesis testing contexts, the point marking the boundary of the tail area is termed the observed value.

  • In normal distribution modeling problems, this same boundary point is frequently termed the cutoff value.

• Academic Strategy and Course Expectations:

  • Mastery of Section 1.5 requires computing areas under the normal curve and identifying corresponding cutoff values.

  • Exam 2 is structured as the most rigorous examination because statistical concepts and terminology are tightly interconnected.

Standardized Scores ($Z$-Scores) and Relative Standing

• Purpose of Standardization:

  • Raw scores from differing distributions or different grading scales cannot be compared directly on face value.

  • An apparent score disparity (such as a $95\%$ score on a Statistics exam versus an $85\%$ score on a Finance 101 exam) may reflect different exam difficulties, grading standards, or formats rather than superior relative mastery.

  • Converting raw scores into standardized $z$-scores normalizes disparate distributions to a common metric for valid comparative analysis.

• The $Z$-Score Formula:

  • For population data:     $z = \frac{x - \mu}{\sigma}$

  • For sample data:     $z = \frac{x - \bar{x}}{s}$

  • Where:

  • $x$ represents the raw data point or observed score.

  • $\mu$ represents the population mean; $\bar{x}$ represents the sample mean.

  • $\sigma$ represents the population standard deviation; $s$ represents the sample standard deviation.

• Interpretation of $Z$-Scores:

  • A $z$-score quantifies the exact number of standard deviations a given raw observation $x$ lies away from the distribution mean.

  • A positive $z$-score indicates that the observation falls strictly above the mean.

  • A negative $z$-score indicates that the observation falls strictly below the mean.

  • The absolute value $|z|$ reflects distance from the mean (zero on a standard normal curve); observations farther from zero are more extreme and less probable under the distribution.

• Worked Example: Exam Score Standard Deviations:

  • Assume an examination possesses a mean of $\mu = 80$ and a standard deviation of $\sigma = 10$.

  • A student scoring a raw $x = 90\%$ yields:     $z = \frac{90 - 80}{10} = 1.0$     This score is positioned exactly $1$ standard deviation above the mean.

  • A student scoring a raw $x = 60\%$ yields:     $z = \frac{60 - 80}{10} = -2.0$     This score is positioned exactly $2$ standard deviations below the mean.

• Case Study: Michigan State University (MSU) Admissions Selection:

  • Two applicants, Tanya and Germaine, are competing for a single admission opening at MSU using scores from two different standardized admissions tests:

  • Tanya completed the SAT and earned a raw score of $x = 1320$, resulting in a standardized score of $z = 1.3$ (or $1.3334$).

  • Germaine completed the ACT and earned a raw score of $x = 28$, resulting in a standardized score of $z = 1.4$.

  • Conclusion: Germaine holds the superior relative standing because Germaine's standardized score ($z = 1.4$) is greater than Tanya's standardized score ($z = 1.3$). Consequently, Germaine is selected for admission.

Normal Distribution Properties and Percentiles

• Total Area of the Normal Curve:

  • The total area under the probability density curve of any normal distribution equals exactly $1$ (or $100\%$ of all observations).

• Symmetry and Central Tendency:

  • The normal distribution curve is unimodal and symmetric around the central mean.

  • The central peak represents the mean, mode, and median.

  • The median corresponds to the $50^{\text{th}}$ percentile, dividing the total distribution area into two equal halves of $0.50$ ($50\%$) each.

• Percentile Definitions:

  • The $p^{\text{th}}$ percentile is the value such that $p\%$ of the distribution falls strictly at or below that value.

  • Scoring at the $96^{\text{th}}$ percentile on an SAT indicates that the test taker performed better than $96\%$ of examinees.

  • Scoring at the $97^{\text{th}}$ percentile indicates that $97\%$ of examinees earned scores lower than or equal to that individual's score.

Finding Areas Under the Normal Curve Using Calculator Functions

• TI Calculator Navigation:

  • Access distribution commands by pressing [2nd] followed by [VARS] (the DISTR menu).

  • Select normalcdf (Normal Cumulative Distribution Function).

  • Note: Never select normalpdf; normalpdf evaluates point density rather than cumulative area.

• Function Syntax and Arguments:

  • Horizontal command syntax: normalcdf(lower, upper, μ, σ)

  • Vertical prompt fields:

  • lower: the lower bound of the shaded region.

  • upper: the upper bound of the shaded region.

  • μ: the distribution mean.

  • σ: the distribution standard deviation.

• Setting Tail Boundaries:

  • Left Tail Calculations (Area Below a Cutoff):

  • Set lower to an extreme negative value: $-10^{10}$.

  • Set upper to the cutoff score $x$.

  • Numerical formatting caution: Input the negative sign without parentheses as -10^10. Placing parentheses around the negative base ((-10)^10) evaluates to a positive number ($+10^{10}$), producing incorrect results.

  • Right Tail Calculations (Area Above a Cutoff):

  • Set lower to the cutoff score $x$.

  • Set upper to an extreme positive value: $10^{10}$.

Finding Cutoff Values Using the Inverse Normal Function

• Inverse Problem Workflow:

  • In reverse problems, the cumulative probability or percentile area is known, and the objective is to determine the unknown cutoff score $x$.

  • Access invNorm by selecting option 3 within the [2nd] [VARS] distribution menu.

• Syntax and Area Direction:

  • Mathematical syntax: invNorm(area, μ, σ)

  • Standard calculator functions strictly require the cumulative area to the left of the desired cutoff score.

  • If a problem specifies a right-tail area ($A_{\text{right}}$), the input area must be converted to the left-tail complement:     $A_{\text{left}} = 1 - A_{\text{right}}$

• Visualizing and Shading the Curve:

  • Always sketch the normal curve before operating the calculator.

  • Mark the mean at the central vertical axis ($50^{\text{th}}$ percentile).

  • Ensure tail shading reflects accurate proportions. When shading a region representing the $40^{\text{th}}$ percentile ($0.40$), the boundary line must sit entirely to the left of the central median; shading across the center peak represents an area exceeding $50\%$ and is mathematically invalid.

• Worked Example: College Program Admission Threshold:

  • A college program admits applicants who score above the $40^{\text{th}}$ percentile on the SAT.

  • Given SAT parameters: mean $\mu = 1060$, standard deviation $\sigma = 195$.

  • To find the minimum cutoff score for eligibility:

  • Identify left-tail area: $A_{\text{left}} = 0.40$

  • Execute command: invNorm(0.40, 1060, 195)

Empirical Rule and Spread Around the Mean

• The $68-95-99.7$ Rule (Empirical Rule):

  • For any bell-shaped, normal distribution:

  • Approximately $68\%$ of observations lie within $1$ standard deviation of the mean ($\mu \pm 1\sigma$).

  • Approximately $95\%$ of observations lie within $2$ standard deviations of the mean ($\mu \pm 2\sigma$).

  • Approximately $99.7\%$ of observations lie within $3$ standard deviations of the mean ($\mu \pm 3\sigma$).

• ACT Exam Spread Calculations:

  • Given ACT parameters: mean $\mu = 21$, standard deviation $\sigma = 5$ (or $5.1$).

  • One Standard Deviation Spread ($\mu \pm 1\sigma$):

  • Lower limit: $21 - 5 = 16$

  • Upper limit: $21 + 5 = 26$

  • Calculator execution: normalcdf(16, 26, 21, 5)

  • Result: Approximately $68\%$ ($0.68$) of examinees score between $16$ and $26$.

  • Two Standard Deviations Spread ($\mu \pm 2\sigma$):

  • Lower limit: $16 - 5 = 11$

  • Upper limit: $26 + 5 = 31$

  • Calculator execution: normalcdf(11, 31, 21, 5)

  • Result: Approximately $95\%$ ($0.95$) of examinees score between $11$ and $31$.

Applications: Household Food Cost Distribution Models

• USDA Monthly Food Cost Data (January 2021 Report):

  • According to United States Department of Agriculture (USDA) reports for January 2021, the monthly cost of food for a family of four following a moderate-cost food plan follows a normal distribution with:

  • Mean: $\mu = \$1109$

  • Standard Deviation: $\sigma = \$150$

• Problem Type 1: Proportion Spending Below a Lower Threshold ($x = \$950$ or $x = \$960$):

  • Objective: Find the proportion of four-person families spending less than a threshold such as $\$950$.

  • Setup:

  • lower: $-10^{10}$

  • upper: $950$

  • μ: $1109$

  • σ: $150$

  • Calculator execution: normalcdf(-10^10, 950, 1109, 150)

• Problem Type 2: Proportion Spending Within an Interval ($\$1000$ to $\$1200$):

  • Objective: Find the proportion of four-person families spending between $\$1000$ and $\$1200$ monthly.

  • Setup:

  • The mean ($\$1109$) lies inside the interval.

  • The lower boundary ($\$1000$) lies to the left of the mean.

  • The upper boundary ($\$1200$) lies to the right of the mean.

  • lower: $1000$

  • upper: $1200$

  • μ: $1109$

  • σ: $150$

  • Calculator execution: normalcdf(1000, 1200, 1109, 150)

• Problem Type 3: Expenditure Cutoff for the Lowest $10\%$:

  • Objective: Determine the maximum dollar cutoff defining the lowest spending $10\%$ ($0.10$) of families.

  • Graphical representation: A narrow tail slice on the extreme left containing $10\%$ of the distribution area (must not be shaded broadly like a $25\%$ quartile slice).

  • Setup:

  • area: $0.10$

  • μ: $1109$

  • σ: $150$

  • Calculator execution: invNorm(0.10, 1109, 150)

• Additional Testing Distribution (SAT Spread Interval):

  • For an SAT score distribution having standard deviation $\sigma = 197$ (or $196$), the proportion of test takers scoring between $800$ and $1500$ is found via:

  • normalcdf(800, 1500, μ, σ)

Questions & Discussion

• Issue: Calculator returns a decimal density value instead of a cumulative area or probability.

  • Clarification: The user inadvertently selected normalpdf instead of normalcdf. normalpdf evaluates the point value along the probability density curve, whereas cumulative tail areas require normalcdf.

• Issue: How to exit a stuck calculator screen or wrong menu.

  • Clarification: Press [2nd] followed by [MODE] (QUIT) to return to the default computation screen.

• Issue: Entering $-10^{10}$ produces negative probability or unexpected calculator outputs.

  • Clarification: Do not place parentheses around the negative sign when defining the lower bound. Entering (-10)^10 raises a negative base to an even power, transforming it into positive $10^{10}$. The correct input is -10^10 without parentheses.

• Issue: Contrasting collegiate calculator methods with high school statistics procedures.

  • Clarification: High school courses frequently rely on manual table lookups or extended step-by-step arithmetic. Utilizing direct calculator functions (normalcdf and invNorm) provides exact cumulative areas and cutoff thresholds directly from distribution parameters.