Statistics for One Categorical Variable and Probability

Flashcards covering key concepts from Lesson 6 on probability rules, theoretical vs. empirical probability, the Birthday Problem, and categorical data representations.

What statistic is primary for summarizing categorical data and probability?

Proportions, which measure the relationship of parts to a whole.

What is the fundamental distinction between probability and statistics?

Probability calculates the chance of a specific outcome when population parameters are known, whereas statistics draws a sample to infer unknown population parameters.

What range of values can a probability take?

Probabilities are always between $0$ and $1$, where $0$ indicates an impossible event and $1$ indicates a certain event.

In the Birthday Problem, how many people must be in a room for the probability of at least two people sharing a birthday to exceed $50\%$?

$23$ people, which results in a $50.7\%$ probability.

What is the probability that at least two people share a birthday in a group of $50$ people?

$$97.0\%$$

What is the theoretical probability of at least one pair of matching birthdays among the last $40$ Oscar-winning Best Actresses?

$$89.1\%$$

What are the two core assumptions required when using the formula for theoretical probability?

First, that you can count the total number of favorable and possible outcomes; second, that all outcomes are equally likely.

What is the formula for calculating theoretical probability?

$$\text{Theoretical probability} = \frac{\text{number of favorable outcomes}}{\text{number of possible outcomes}}$$

What is the formula for calculating empirical (experimental) probability?

$$\text{Empirical probability} = \frac{\text{number of successes}}{\text{sample size}}$$

If a coin is flipped $200$ times resulting in $94$ heads and $106$ tails, what are the empirical and theoretical probabilities of flipping heads?

The empirical probability is $\frac{94}{200} = 0.47$ ($47\%$), while the theoretical probability is $\frac{1}{2} = 0.50$ ($50\%$).

What is a frequency table?

A summary table of categorical data displaying counts of how often each category occurs.

How is relative frequency calculated in a frequency table?

By dividing the category frequency by the total sample size: $\text{Relative Frequency} = \frac{\text{Frequency}}{\text{Total}}$.

What is the definition of the mode in categorical data?

The category or categories that occur most frequently (have the largest count or frequency).

Why is an "Other" category created when constructing frequency tables for categorical data?

To combine categories that have very low individual counts (such as single occurrences), keeping the graph or table concise.

Why are relative frequency bar charts particularly useful compared to standard frequency bar charts?

They plot proportions on the vertical axis, making it easier to compare two samples that have different sample sizes.

What is a major visual limitation of pie charts compared to bar charts?

It is generally more difficult to accurately compare group sizes or slice areas in a pie chart than bar lengths in a bar chart, especially when categories have nearly identical proportions.