Detailed Notes on Combinations and Probability

Basic Understanding of Combinations

• Problem Summary: Selecting a committee of 6 members from a pool of 8 parents and 4 teachers.
  • Requirement: 3 parents and 3 teachers.
• Computation Using Combinations:
  • Total committee size: 6
  • Use combination formula:
  • Selecting 3 parents from 8: C(8, 3)
  • Selecting 3 teachers from 4: C(4, 3)
  • Total selection: C(12, 6)
• Important Note: When using a calculator, ensure it follows the correct order of inputs: larger number first.

Understanding Percentiles and Normal Distributions

• Percentile Problem: Given mean and standard deviation to find the 67th percentile of a data set (e.g., chips in a bag).
  • Steps:
  • Use the Inverse Norm function on calculator.
  • Inputs required: area (0.67), mean (24.5), standard deviation (0.2).
  • Result: Approximately 24.588.
• Normal Distribution: Avoid using the Z chart directly for accuracy; calculators are preferred.

Additional Calculations in Normal Distribution

• Example Problem: Length of time finding parking follows a normal distribution with a mean of 4.5 and a standard deviation of 1.
  • Finding Exceeded Time: Area exceeds a certain value based on computations.
  • For cutoff of 70: Area calculation is 1 - 0.78758, yielding area of 0.242.

Binomial Approximations with Normal Distribution

• Newborn Cats Gender Probability: Given stats to approximate female births.
  • Use N = n imes p to find the mean and ext{SD} = ext{sqrt}(n imes p imes (1-p)) for the standard deviation when approximating binomial distribution.
  • For 100 births, mean = 50, standard deviation = 5.
• Use of Normal CDF on Calculator
  • Setup lower bound (55) and upper bound (very high number, e.g., 1000). Adjust boundaries by 0.5 for approximation.

Working with Confidence Intervals

• Finding Standard Error: Standard deviation divided by the square root of the sample size.
• Z and T Intervals: Differentiate based on sample size:
  • Z interval for large samples; T interval for smaller samples (n < 30).
• Example of T Interval Calculation: Given mean, standard deviation, and confidence level; determine the interval.

Key Statistics Formulas

• Sample Size Calculation:
  • N = \frac{p(1-p) z^2}{E^2}
  • Where E is the margin of error, z is the critical value based on confidence level.
  • E.g., for 0.04 and 95% confidence, z = 1.96.

Finalizing Calculations

• Practice on Calculators: Familiarize with confidence interval, normal CDF, and other statistical functions to ease exam pressure.
• Preparation Strategies: Review previous works, practice similar problems without referencing notes, and plan for additional support if needed during exam preparation.