Ch 14 Binomial Distributions Notes

Chapter 14: Binomial Distributions Notes

Overview of Binomial Distributions

• Key Topics Covered:
  • Binomial setting and distributions
  • Statistical sampling with binomial distributions
  • Calculating binomial probabilities
  • Utilizing technology for binomial calculations
  • Mean and standard deviation of binomial distributions
  • Normal approximation for binomial distributions

The Binomial Setting

• Requirements for a Binomial Setting:
  • Fixed number of observations (trials) denoted as n.
  • Observations are independent; knowing one outcome does not influence others.
  • Each observation results in one of two outcomes: "success" or "failure" (e.g., heads/tails in a coin toss).
  • Probability of success p remains constant for each trial.

Binomial Distribution

• The number of successes in n independent trials has a binomial distribution characterized by:
  • n: number of trials
  • p: probability of success
• Random Variable Definition:
  • Denoted as X, representing the count of successes (values from 0 to n).
• Note: Not all counts are binomial; verify the conditions of independence and fixed trials.

Binomial Distributions in Statistical Sampling

• Important for making inferences about population proportions (p).
• Example Case:
  • If 11% of tomatoes from a producer are unmarketable, and 10 tomatoes are inspected from a larger shipment.
  • Let X be the count of unmarketable tomatoes.
  • When sampling is less than 5% of the population, it approximates a binomial distribution.

Binomial Probability

Part I: Finding the Probability

• Formula Development:
  • The probability of getting exactly k successes in n observations involves adding probabilities from all arrangements yielding k successes.

Part II: Binomial Coefficient

• Binomial Coefficient (0): Counts the distinct arrangements of k successes in n trials.

  • Formula: [ C(n, k) = \frac{n!}{k!(n-k)!} ]

• Binomial Probability Formula:
  [ P(X = k) = C(n, k) * p^k * (1-p)^{(n-k)} ]

Binomial Probability Example

• Case Study: Count of unmarketable tomatoes.
  • Parameters: n = 10, p = 0.11.
  • a) Probability of X ≤ 1 (no more than 1 unmarketable tomato).
  • b) Probability of X ≥ 1 (at least 1 unmarketable tomato).

Utilizing Technology for Binomial Distributions

• Tools:
  • Texas Instruments Calculator, Microsoft Excel, Minitab.
• Excel Function for Probability:
  [ P(X \leq 1) = \text{BINOM.DIST}(1, 10, 0.11, 1) = 0.697209 ]
• Minitab Usage:
  • Input cumulative functions for binomial distributions to evaluate probabilities efficiently.

Binomial Mean and Standard Deviation

• Mean Formula:
  [ \mu = n * p ]
• Standard Deviation Formula:
  [ \sigma = \sqrt{n * p * (1-p)} ]
• Note: These formulas strictly apply to binomial distributions.

Normal Approximation to Binomial Distributions

• As n increases, the binomial distribution approaches a normal distribution.
• Conditions for Normal Approximation:
  • Generally applicable when both ( np ) and ( n(1-p) ) are greater than 5.
• This provides utility in estimating probabilities when n is large, and success probabilities are reasonable.