Statistics Midterm

Population

A well-defined collection of objects of interest to a study

Sample

A subset of individuals from a population of interest

Inferential Statistics

The process and techniques associated with generalizing the behavior of the population from the behavior of the sample

Categorical/Qualitative Data

Type of data that can be classified or can define an attribute of the observed individual

Ordinal Data

Type of categorical data that can be ordered and ranked such as year of student or letter grade recieved

Nominal Data

Type of categorical data that does not have a specific order, such as handedness or political affiliation

Numeric/Quantitative Data

Data that is the result of counting the number of units that have a specific characteristic of interest or a measurement associated with an individual

Discrete Data

Type of numeric data that takes on specific whole number values over a range, such as number of beds at a hospital or number of individuals who support decreasing taxes

Continuous Data

Type of numeric data that can take on any possible values in a range (can be decimals) such as the weight of something or a length of time

Relative Frequency

Calculated for qualitative data by taking the number of times something occurred and dividing by the total number of observations in the dataset

Measurement of Central Tendency

Summary number that identifies the center, typical value, or overall trend of a dataset

Mode

Measure of central tendency, the most repeated numeric value in the sample

Median

Measure of central tendency, the middle value of the sample that divides the number of observations in the sample in half

Mean

Measure of central tendency, the center of mass of the sample found by adding every value and dividing by the number of observations

Measurements of Variability/Spread

Summary number that describes how scattered values are within a dataset

Range

Difference between minimum and maximum value of a sample

Interquartile Range

Difference between the upper and lower quartile, representing the middle 50% of the data

Variance

Measure of how much the datapoints vary in units squared

Standard Deviation

Describes the average difference from an observation in the sample to the mean of the sample in units

Pie Chart

Visualizes the relative frequency of the observations from a sample in a circular graph with slices

Bar Chart

Uses rectangular bars to represent the frequency/percentage of the sample in each category

Histogram

Contains bins that touch each other and count the number of datapoints within each bin/bar

Boxplot

Visual representation of the 5 number summary (Min, 1st Q, Median, 3rd Q, Max)

Probability Plot

Determines the likely distribution that the observations from a sample were taken from, commonly normal. If dots fall along straight line, they match.

Central Location

Where the main peak of the distribution falls

Shape

The number of peaks and where they are centered/if they are symmetrical

Spread

The amount of variability a distribution has

Chance Experiment

Experiment run where the outcome of the experiment is completely random or left to chance

Sample Space (S)

The collection of all possible outcomes from a chance experiment (order matters)

Proper Listing of Sample Space (2 Coin Example)

INCORRECT: {HH, TT, HT}

CORRECT: {HH, TT, HT, TH}

Calculating the Sample Space

Size of S = kⁿ
k = number of individual outcomes per item
n = number of identical objects with said outcomes

OR

Size of S = (kⁿ)(p^m)
p and m are the same as k and n but may have a different number of outcomes or number of items

Event

A subset of the sample space that contains outcomes of interest to the researcher

Simple Event

An event from the sample space that has exactly one outcome

Null Event

An event that contains no outcomes

Classical Probability Calculation

P(X) = Outcomes of event X/Size of sample space
P(S) = 1
P(Null) = 0

Impossible Event

An event in the sample space with P(A) = 0; will never happen in the experiment

Certain Event

An event in the sample space with P(A) = 1; will happen every time the experiment is run guaranteed

Law of Complements

All events that are not part of the event in question (A) are the complement of that event (A’)

Complement Equations

A + A^c = S
P(A) + P(A)^c = P(S) = 1
P(A) = 1 - P(A)^c
P(A)^c = 1 - P(A)

Intersection of Events (∩)

All elements contained in two events at the same time; “AND”

Union of Events (∪)

All elements contained in one event, the other, or both; inclusive “OR”

Venn Diagram

A way to visually represent the events and probabilities of them within a sample space

De Morgan’s Law

The complement of the union between events is equivalent to the intersection between the complements of each event
P(𝐴∩𝐵)^c = 𝑃(𝐴^c∪𝐵^c)

Conditional Probability

When the probability that an event may happen depends on information about another event that has previously happened

Conditional Probability Equations

P(A|B) = Knowing B happened, what is the probability of A happening?
P(A|B) = P(A∩B)/P(B)

Disjoint Events

Mutually exclusive events that share no outcomes in common, P(A∩B) = 0

Independence of Events

When the knowledge of an event happening does not give information about the knowledge of another event happening

Total Probability

The overall probability of each of the possible paths that can be followed
P(A) = Sum{P(A|B_i)P(B_i)}

Bayes’ Theorem

Conditional probability where the time dependency of the condition has been flipped; “(Blank) happened, what is the probability (blank) was the path chosen that led to that outcome?”

Experimental/Empirical Probability

Method to estimate the probability of an event based on the results gathered during a sample experiment
P(X = x) = number of successful trials/total number of trials

Law of Large Numbers

As the total number of trials of an experiment approaches infinity, the probability calculated experimentally will approach the true probability of the even of interest

Random Variable

Numeric variable whose value represents the outcome of a chance experiment

Expected Value

The long term average value of a random variable that may or may not be actually achievable

Expected Value Equations

DRV: E(X) = u_x = Sum{xP(X=x)}
CRV: E(X) = u_x = Integral{xf(x)dx}

Variance of RVs

Measures a variable’s tendency to deviate from its expected value

Variance of RV Equations

DRV: V(X) = sigma_x squared = Sum{x²P(X=x)} - u_x²
CRV: V(X) = sigma_x squared = Integral{(x-u_x)²f(x)dx}

Standard Deviation

The square root of variance, describes variance in the same units as the original units

Binomial Distribution

Discrete
- Independent success vs. failure trials
- Number of trials fixed at the start of the experiment
- Number of successes is between 0 and n
- P(S) is constant for all individual trials
E(X) = nP(S), V(X) = nP(S)[1-P(S)]

Hypergeometric Distribution

Discrete
- X = number of successes present in collected sample
- fixed number of non-independent trials
- population size (N) is finite so P(S) is non-constant
- a = number of successes in population, ^a _n = initial P(S)
E(X) = n(^a _n), V(X) = n(a/N)(1-a/N)(N-n/N-1)

Poisson Distribution

Discrete
- X = number over successes over change in time/position (ex. 5 successes in one hour)
- 𝜆 = rate of successes (constant)
- x >= 0 over delta t
E(X) = V(X) = 𝜆

Geometric Distribution

Discrete
- X = number of trials performed before 1st successful outcome
- P(S) = probability of success of individual trials (constant)
- Independent trials
- 1st success must happen on last trial of experiment
E(X # of trials) = 1/P(S), E(X # of failures) = 1-P(S)/P(S), V(X) = 1-P(S)/P(S)²

Negative Binomial Distribution

Discrete
- X = number of trials needed for r number of successes
- P(S) = probability of success of individual trials (constant)
- Independent trials
- rth success must happen on final trial of experiment, x >= r
E(X # of trials) = r/P(S), E(X # of failures) = r(1-P(S))/P(S), V(X) = r(1-P(S))/P(S)²

Multinomial Distribution

Discrete
- Like the binomial but with more than 2 disjointed outcomes
- Trials of the experiment are independent of each other
- P(A), P(B), and P(C) are constant over the experiment
- a + b + c = n

Uniform Distribution

Continuous
- Density function remains constant over range of x defined (a to b)
u_x = ½(b+a), sigma_x² = (1/12)(b-a)²

Exponential Distribution

Continuous
- Variable represents number of successes to happen over an interval of fixed length
u_x = 1/lambda = beta, sigma_x² = 1/lambda² = beta²

Normal Distribution

Continuous
- Symmetric distribution where data clusters around a central peak
u_x = u, sigma_x² = sigma²
- Special case: standard/z
* u_x = 0, sigma_x² = 1

Lognormal Distribution

Continuous
- If we have a variable W known to have a normal distribution, the relationship X = e^W would produce (some long ass function lol)

Weibull Distribution

Continuous
- For any random variable X which follows a density function with a shape alpha and a scale beta
u_x = Beta(Gamma)(1+1/a), sigma_x² = Beta²(Gamma(1+2/a) - (Gamma(1+1/a))²)

Sampling Distribution

The probability distribution associated with a statistic

Standard Error

The standard deviation of the sampling distribution of a statistic

Central Limit Theorem Part 1

If we sample from a known normal population (mean = u, stdev = sigma), the distribution of the sample mean will also be normal (mean = u, sterr = sigma/radical n) for ANY sample size n > 1

Central Limit Theorem Part 2

If we sample from a random population (normality not known), the distribution of the sample mean will be APPROXIMATELY normal (mean = u, sterr = sigma/radical n) if the sample size n >= 30 (“sufficiently large”)

Central Limit Theorem Part 3

If we sample from a known Binomial population with a percentage of success p, the distribution of the sample proportion p-hat will be approximately normal (mean = p, sterr = radical(p(1-p)/n)) for a sample size np(1-p) >= 10 (“sufficiently large”)

Point Estimate

An estimate for a parameter that is a single calculated value from a sample. Good estimates are unbiased and have little variability

Parameters and Point Estimators

Parameter Estimator
u x-bar
sigma s
pi p-hat

Interval Estimate

An estimate for a population parameter that uses a range of values we expect or are confident that the parameter’s value will fall between, AKA a confidence interval

General Interval Estimate Layout

Point Estimate +- Margin of Error

Margin of Error = Critical Value x Standard Error of Estimate

Confidence Interval Validity Interpretation

Due to the sample size (being/not being) large enough (__ > __), under CLT #_, we (may/may not) assume our sample mean is approximately normal and the process selected is (valid/invalid).

Confidence Interval Bounds Interpretation

We have a ____(probably 95)% confidence in the method selected that has constructed bounds that we expect will contain the true mean ___ between the bounds __ and _ (on average).