Probability Distributions and Statistical Inference: Binomial and Normal Applications

Probability Distributions in Statistics

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. It is commonly used when outcomes are binary (e.g., success/failure).

• Key Formula: The probability of observing exactly k successes in n trials with probability p is given by: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

• Mean and Standard Deviation: $\mu = np$ $\sigma = \sqrt{np(1-p)}$

• Applications: Used to model scenarios such as the number of people not sanitizing hands, number of murders by knife, or number of patients responding to a drug.

Example: Hand Sanitizing Habits

• Scenario: Probability that exactly 7 out of 30 people do not sanitize hands, given $p = 0.32$.

• Calculation: $P(X = 7) = \binom{30}{7} (0.32)^7 (0.68)^{23}$ Result: 0.0983

• Unusual Events: If the probability of an event is less than 0.05, it is considered unusual.

Example: Drug Effectiveness

• Scenario: In a sample of 250 patients, 60% respond to a drug. Probability that at least 155 respond:

• Calculation: $P(X \geq 155) = 1 - \text{binomcdf}(250, 0.6, 154)$ Result: 0.041

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by mean ($\mu$) and standard deviation ($\sigma$). It is used to model many natural phenomena.

• Standardization (Z-score): Converts a value to the number of standard deviations from the mean. $z = \frac{x - \mu}{\sigma}$

• Applications: Used to model pregnancy lengths, sample means, and other continuous data.

Example: Human Pregnancy Lengths

• Scenario: Mean = 266 days, $\sigma$ = 16 days. Proportion lasting more than 270 days:

• Calculation: $z = \frac{270 - 266}{16} = 0.25$ $P(z > 0.25) = 1 - P(z < 0.25) = 1 - 0.5987 = 0.4013$

• Interpretation: About 40.13% of pregnancies last more than 270 days.

Example: Preterm Babies

• Scenario: Gestation less than 224 days.

• Calculation: $z = \frac{224 - 266}{16} = -2.63$ $P(z < -2.63) = 0.0043$

• Interpretation: Since 0.0043 < 0.05, this is considered unusual.

Statistical Inference: Unusual Events

In statistics, an event is considered unusual if its probability is less than 0.05. This threshold helps in identifying rare or significant outcomes in data analysis.

• Example: Observing 150 murders by knife in a sample of 300, when expected is 105, is unusual if 150 > $\mu + 2\sigma$.

Using Cumulative Distribution Functions (CDF)

The binomcdf function calculates the cumulative probability up to a certain number of successes in a binomial distribution. It is useful for finding probabilities of 'at most' or 'at least' scenarios.

• Example: Probability that at most 4 out of 30 do not sanitize hands: $P(X \leq 4) = \text{binomcdf}(30, 0.32, 4) = 0.0171$

Summary Table: Binomial vs. Normal Distribution

Feature	Binomial Distribution	Normal Distribution
Type	Discrete	Continuous
Parameters	n (trials), p (success probability)	μ (mean), σ (standard deviation)
Shape	Symmetric (for p ≈ 0.5)	Bell-shaped, symmetric
Common Uses	Counting successes/failures	Modeling measurements, averages

Additional info:

• When using binomial probabilities for large n and p not near 0 or 1, the normal approximation may be used.

• Sample mean and standard deviation formulas for binomial: $\mu = np$, $\sigma = \sqrt{np(1-p)}$.

• For normal distribution, probabilities are found using z-tables or statistical software.