Statistics Study Guide: Probability, Distributions, and Statistical Inference

Probability and Unusual Events

Defining Unusual Events

In statistics, an event is considered unusual if its probability is very low, typically less than 0.05 (5%). This threshold helps identify outcomes that are rare or unexpected under normal circumstances.

• Unusual Event: Probability < 0.05

• Example: An event with probability $\frac{11}{20} = 0.55$ is not considered unusual.

Discrete Probability Distributions

Probability Distribution Table

A probability distribution lists the probabilities of all possible outcomes for a discrete random variable. The sum of all probabilities must equal 1.

y (Number of Patients)	P(Y = y)
0	0.160
1	0.330
2	0.275
3	0.145
4	0.065
5	0.020
6	0.005

• Finding Probability: To find the probability that at least one patient has the condition, use $P(Y \geq 1) = 1 - P(Y = 0)$.

• Formula: $P(\text{at least 1}) = 1 - 0.160 = 0.840$

Conditional Probability and Diagnostic Testing

False Positives and Negatives

Diagnostic tests can yield false positives (test positive, but do not have the condition) and false negatives (test negative, but do have the condition).

• Conditional Probability: Probability of an event given another event has occurred.

• Example: Given test results, calculate the probability that a randomly selected participant did not have the flu or tested positive.

Joint and Marginal Probability

Contingency Tables

Contingency tables summarize data for two categorical variables. Probabilities can be calculated for joint or marginal events.

	Reported Poor Sleep: Yes	Reported Poor Sleep: No
Used Electronic Devices Before Bed	800	2940
Did Not Use Electronic Devices Before Bed	210	4030

• Joint Probability: Probability that a student did not use electronic devices before bed and did not report poor sleep: $P = \frac{4030}{800+2940+210+4030}$

Probability in Categorical Data

Graduation Data Table

	Science	Non-Science
Male	75,000	105,000
Female	95,000	125,000

• Marginal Probability: Probability that a randomly selected science graduate is male: $P = \frac{75,000}{75,000+95,000}$

Probability Without Replacement

Dependent Events

When selecting items without replacement, probabilities change after each selection.

• Example: Probability all three selected students are left-handed from a group of 8 (4 left-handed): $P = \frac{4}{8} \times \frac{3}{7} \times \frac{2}{6}$

Bayes' Theorem

Conditional Probability Formula

Bayes' Theorem allows calculation of the probability of an event based on prior knowledge of conditions related to the event.

• Formula: $P(G|H) = \frac{P(H|G)P(G)}{P(H|G)P(G) + P(H|\overline{G})P(\overline{G})}$

Simple Events

Counting Outcomes

A simple event is an event with only one outcome.

• Example: Drawing marble number 7 from 10 marbles is a simple event with 1 outcome.

Probability of Random Guessing

Counting Principle

When guessing digits, the probability of a correct guess is $\frac{1}{10^n}$, where n is the number of digits.

• Example: Guessing 4 digits: $\frac{1}{10^4} = 0.0001$

Linear Transformations of Mean

Adjusting Mean Values

When a mean is increased by a percentage and a fixed amount, the new mean is calculated as:

• Formula: $\text{New Mean} = \text{Old Mean} \times (1 + \text{Rate}) + \text{Fixed Charge}$

• Example: $320 \times 1.07 + 25 = 342.40 + 25 = 367.40$

Binomial and Unusual Values

Histogram Shape and Unusual Probabilities

For a binomial random variable, the histogram's shape depends on the probability of success.

• Symmetrical: Probability near 0.5

• Skewed Right: Probability < 0.5

• Skewed Left: Probability > 0.5

• Unusual Value: Probability < 0.05

Poisson Distribution

Probability of Events in Fixed Interval

The Poisson distribution models the number of events in a fixed interval of time or space.

• Formula: $P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}$

• Example: For $\lambda = 3.2$, $k = 3$:

• $P(X = 3) = \frac{3.2^3 e^{-3.2}}{3!}$

Hypergeometric Distribution

Sampling Without Replacement

Used when sampling without replacement from a finite population.

• Formula: $P(X = k) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}}$

• Example: Probability none of 6 selected bolts are substandard from 7,500 bolts, 18% substandard.

Normal Distribution

Calculating Probabilities

The normal distribution is a continuous probability distribution characterized by its mean ($\mu$) and standard deviation ($\sigma$).

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

• Example: $P(W > 100)$ for $\mu = 120$, $\sigma = 20$

Probability Between Two Values

Normal Distribution Applications

• Formula: $P(a < X < b) = P(X < b) - P(X < a)$

• Example: Nitrogen dioxide levels between 12.0 and 25.0 parts per billion

Sampling Distribution of the Mean

Central Limit Theorem

For large samples, the sampling distribution of the mean is approximately normal.

• Standard Error: $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

• Example: Probability sample mean rainfall > 22.8 mm

Sample Size Estimation

Confidence in Estimating Standard Deviation

To estimate the population standard deviation with a specified confidence and margin of error, use sample size formulas based on the desired confidence level.

• Example: Minimum sample size for 99% confidence that sample standard deviation is within 3% of population standard deviation.

Confidence Intervals

Estimating Population Mean

A confidence interval provides a range of values within which the population parameter is likely to fall.

• Formula: $\bar{x} \pm z^* \frac{s}{\sqrt{n}}$

• Example: 95% confidence interval for mean weight of dimes minted after 2000.

Interpretation: If the confidence interval includes the specification value, the mean meets the specification.

Additional info: Some formulas and context have been expanded for clarity and completeness.