Statistics Exam 2 Review: Probability, Distributions, and Confidence Intervals

Probability Concepts and Random Variables

Key Terms and Definitions

• Mutually Exclusive Events: Events that cannot occur at the same time. If one event happens, the other cannot.

• Degrees of Freedom: The number of independent values or quantities which can be assigned to a statistical distribution.

• Random Variable: A variable whose value is subject to variations due to chance (randomness).

• Mean: The average value of a set of numbers, calculated as the sum divided by the count.

• Z-score: The number of standard deviations a data point is from the mean. Formula: $z = \frac{x - \mu}{\sigma}$

• Normal QQ-plot: A graphical tool to assess if a dataset is approximately normally distributed.

Probability Rules

• Probability of 0.9999: If an event has a probability of 0.9999, it is almost certain to occur, but not guaranteed.

• Probability of 0: If an event has a probability of 0, it is impossible for it to occur.

• Probability of Union of Mutually Exclusive Events: For mutually exclusive events A and B: $P(A \cup B) = P(A) + P(B)$

• Total Area Under Density Curve: The total area under any probability density curve equals 1.

• Sum of Probabilities: The sum of the probability of an event and its complement equals 1: $P(A) + P(A^c) = 1$

Discrete vs. Continuous Random Variables

• Discrete Random Variable: Takes on a countable number of distinct values.

• Continuous Random Variable: Can take on any value within a given range.

Example Table: Discrete Random Variable

x	P(x)
1	0.1
2	0.3
3	0.45
4	0.15

Additional info: To find mean and standard deviation, use:

• Mean: $\mu = \sum x P(x)$

• Standard deviation: $\sigma = \sqrt{\sum (x - \mu)^2 P(x)}$

Probability Applications: M&M Example

Probability Calculations with Categorical Data

Given a bowl of 150 M&Ms with different types and colors, probabilities can be calculated for selecting certain colors or types.

Type	Brown	Red	Yellow	Orange	Green	Blue
Plain	12	8	14	11	9	6
Peanut	17	12	11	8	11	11

• Probability of Yellow: $P(\text{Yellow}) = \frac{\text{Total Yellow}}{150}$

• Probability of Plain and Green: $P(\text{Plain and Green}) = \frac{9}{150}$

• Probability of Plain or Red: $P(\text{Plain or Red}) = \frac{12 + 8}{150}$

• Column Totals: Sum each color column for total counts.

Properties of Events and Distributions

Mutually Exclusive vs. Independent Events

• Mutually Exclusive: Events that cannot happen at the same time.

• Independent: The occurrence of one event does not affect the probability of the other.

• Comparison: Mutually exclusive events are not necessarily independent.

Empirical Rule for Normal Distribution

• Approximately 68% of observations fall within 1 standard deviation of the mean.

• Approximately 95% within 2 standard deviations.

• Approximately 99.7% within 3 standard deviations.

Confidence Intervals

Width of Confidence Interval

• Increasing Sample Size: Decreases the width of the confidence interval.

• Decreasing Confidence Level: Decreases the width of the confidence interval.

Constructing Confidence Intervals

• Point Estimate for Mean: $\bar{x}$

• Confidence Interval Formula (Known $\sigma$): $\bar{x} \pm z^* \frac{\sigma}{\sqrt{n}}$

• Interpretation: The interval gives a range of plausible values for the population mean.

Binomial Distribution Applications

Survey Example: Americans Favoring Dogs or Cats

• Random Variable: Number of Americans favoring dogs (binomial random variable).

• Mean: $\mu = np$

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

• Probability Calculation: Use binomial probability formula for specific outcomes.

Normal Distribution and Z-scores

Standardization and Probability Calculations

• Z-score: $z = \frac{x - \mu}{\sigma}$

• Probability for Range: $P(a < x < b) = P(z_a < z < z_b)$

Statistical Symbols and Parameters

Common Symbols Table

Symbol	Name of Symbol	Parameter or Statistic
$\mu$	Population Mean	Parameter
$\bar{x}$	Sample Mean	Statistic

Summary

• This review covers foundational concepts in probability, random variables, distributions, and confidence intervals.

• Key formulas and definitions are provided for exam preparation.

• Tables and examples illustrate practical applications of statistical methods.