Analyzing Categorical Variable I

If the distribution represents an entire population, it’s called a __________ ____________. This is often theoretical because we usually can’t measure everyone.

probability distribution

If the distribution represents a sample from a population, it’s called a ___________ __________, which shows how often each value occurs in the sample.

frequency distribution

What is a Binomial Distribution?

A probability distribution used when there are two possible outcomes (success or failure) for each trial.

What is a requirement for using Binomial Distribution regarding the number of trials?

There must be a fixed number of trials (n).

What must be constant for each trial in a Binomial Distribution?

The probability of success (p) must be constant.

Give an example of a situation that can be modeled by a Binomial Distribution.

Counting how many skinks have their original tail in a sample of 8 skinks.

What is the Poisson Distribution used for?

Counting the number of events that happen in a fixed space, time, or group, especially when events are rare.

Does the Poisson Distribution have a fixed number of trials?

No, there is no fixed number of trials.

What does the Poisson Distribution focus on?

The average rate of occurrence (λ).

Give an example of a situation where the Poisson Distribution might be used.

Counting how many red-green color-blind individuals appear in a sample of 100 people.

What is the Binomial Formula?

P(X=x) = inom{n}{x} p^x q^{n-x}

What does X represent in the Binomial Formula?

X represents the number of successes.

What does x represent in the Binomial Formula?

x represents a specific number of successes you're calculating the probability for.

What does n represent in the Binomial Formula?

n represents the total number of trials.

What does p represent in the Binomial Formula?

p represents the probability of success on a single trial.

What does q represent in the Binomial Formula?

q represents the probability of failure on a single trial, where q = 1 − p.

What does the binomial coefficient represent?

The binomial coefficient represents the number of ways to choose x successes from n trials.

What does X represent in the Poisson Formula?

The number of occurrences of the event.

What does x represent in the Poisson Formula?

The specific number of events you're calculating the probability for.

What does λ represent in the Poisson Formula?

The mean (average) number of occurrences in the interval.

What is e in the context of the Poisson Formula?

The base of natural logarithms (approximately 2.71828...).

What does x! represent in the Poisson Formula?

The factorial of x, representing the number of ways the events can occur.

Mean (Expected # of successes in a Binomial Distribution)

E[X] = n⋅p → trials × chance of success

Variance (Spread in a Binomial Distribution)

Var[X] = n⋅p ⋅ (1−p) → trials × chance of success × chance of failure

What is the relationship between mean and variance in a Poisson Distribution?

Mean = Variance = λ (The mean and variance will be the same)

What does λ represent in a Poisson Distribution?

λ = average number of events

When do you use the binomial test?

When you want to know if the number of "successes" you observed is different from what you'd expect by chance.

How do we interpret results from a binomial test?

If the p-value < 0.05, the result is significantly different from the expected value.

If the p-value > 0.05, the result is not significantly different — it could be due to chance.

One-Sided vs. Two-Sided Test

One-Sided Test: Tests if the result is greater than OR less than the expected value (only one direction matters).

Two-Sided Test: Tests if the result is different in either direction (higher or lower).

One sided test example

Are more skinks keeping their tails than expected?

Two sided test example

Is the number of skinks with tails different from expected (could be more or less)?

What is a P-value?

The probability that your results happened by chance if the null hypothesis is true.

What is a Type I Error (α)?

The cutoff you choose (often 0.05) for deciding if a result is significant.

What does it mean if P < α?

You reject the null hypothesis, and the result is significant.

What does it mean if P > α?

You fail to reject the null hypothesis, and the result is not significant.