Detailed Notes on Normal Distribution and Inverse Norm

Normal Distribution Problems

  • Two amusement park coasters: Killer Bee (minimum 48 inches), Honeybee (minimum 40 inches).
  • Student heights are normally distributed with a mean of 42 inches and a standard deviation of 3 inches.
  • Objective: Determine the proportion of students who can ride both or only one of the coasters.

Proportion Riding Both Coasters

  • Requirement: At least 48 inches tall to ride both.
  • Using normal Cumulative Distribution Function (CDF) from 48 to infinity.
  • Calculator input: normalcdf(48, 1e99, 42, 3)
  • P(X≥48)P(X \geq 48)
  • Result: 0.0228, or 2.28% can ride both coasters.

Proportion Riding Only Honeybee

  • Requirement: Between 40 and 48 inches.
  • Using normal CDF from 40 to 48.
  • Calculator input: normalcdf(40, 48, 42, 3)
  • P(40 \leq X < 48)
  • Result: 0.7248, or 72.48% can ride only the Honeybee coaster.

Proportion Unable to Ride Either Coaster

  • Requirement: Shorter than 40 inches.
  • Using normal CDF from negative infinity to 40.
  • Calculator input: normalcdf(-1e99, 40, 42, 3)
  • P(X < 40)
  • Result: 0.2525, or 25.25% cannot ride either coaster.

Types of Normal CDF Problems

  • Area to the left.
  • Area in the middle.
  • Area to the right.
  • All solved using normal CDF with appropriate lower, upper bounds, mean, and standard deviation.

Inverse Norm Function

  • Doing it backwards.
  • Given: proportion, percentage, percentile (area under the curve).
  • Find: value, reference values, endpoint.
  • Normal CDF: finds shaded area given a value.
  • Inverse Norm: finds the endpoint given the shaded area (probability).

Example 1: Finding Z-Score

  • Standard normal distribution (mean = 0, standard deviation = 1).
  • Find the Z-score such that the area to the left of the Z-score is 0.25.
  • Area = 0.25, mean = 0, standard deviation = 1.
  • Calculator input: invNorm(0.25, 0, 1)
  • Result: Z = -0.6745 (negative because it's to the left of zero).

Example 2: Finding GPA (Percentile)

  • Undergraduate GPA normally distributed with mean 3.2 and standard deviation 0.3.
  • Find the GPA such that 60% of students score at or below that GPA (60th percentile).
  • Area = 0.60, mean = 3.2, standard deviation = 0.3.
  • Calculator input: invNorm(0.6, 3.2, 0.3)
  • Result: GPA = 3.276

Example 3: Finding Minimum Test Score (Right Tail)

  • Standardized test scores normally distributed with mean 1012 and standard deviation 263.
  • Top 25% get to go on a field trip.
  • Find the minimum score to go on the trip.
  • Area to the right = 0.25, mean = 1012, standard deviation = 263.
  • Classic Calculator approach: invNorm(0.75, 1012, 263) because classic calculators only understand area to the left.
  • Newer Calculator approach : invNorm(0.25, 1012, 263, right)
  • Result: Minimum score = 1189

Homework Problems (6.1)

  • 6. 1: Standard Normal (mean = 0, standard deviation = 1).
  • 6. 2: Any normal distribution.
  • Solving problems the same way using normal CDF and inverse norm.
Homework Problem 15
  • Find Z given the area.
  • Area to the left of Z = 0.5 (symmetric).
  • Remainder of the area is 0.0825.
  • Z score: invNorm(0.0825,0,1) = -1.3885
Homework Problem - Middle 35%
  • Standard normal distribution.
  • Find Z scores that bound the middle 35%.
  • Area on each tail: (1 - 0.35) / 2 = 0.325.
  • Negative Z score: invNorm(0.325, 0, 1) = -0.4538.
  • Positive Z score: 0.4538 (symmetric).
  • Newer Calculators: Specify center for direct calculation.