Introduction to Statistical Inference and Sampling Techniques

Statistics

the science of collecting, presenting, analyzing, and interpreting data

Data

facts and information of interest

Analytics

the process of using data to make decisions; descriptive, predictive, and prescriptive

Statistical Inference

uses sample data and probability to draw general conclusions about a population

Data set

a collection of information that can be grouped by variables

Elements

the units on which data is collected

Variables

general characteristics of interest

Numerical (or Quantitative) variables

variables that are measurable and can be expressed as numbers

Categorical (or Qualitative) variables

variables that represent categories or groups

Nominal

label or name categorical variable - Favourite ice cream

Ordinal

Label or name AND can be ordered categorical variable - Preference for mint ice cream: dislike, somewhat like, love

Interval

has a fixed unit of measure; can find the difference between (or sum of) two data values; zero can be a meaningful measure numerical variable - the temperature of ice cream from different freezers

Ratio

characteristics of interval data AND also the product or quotient of two data values make sense; zero means non-existent numerical variable - the number of ounces of ice cream in sized containers

Available Data

Data that were produced in the past may be a cost-effective way to help answer a present question

Internal data

Personnel files, cash-flow reports, and inventory records are sources of available data

External data

sources may be available for free (library, internet, researchers) or for a fee (Bloomberg, Neilson Co.)

Newly Produced Data

Some questions may require data to be produced. Newly produced data can be obtained through the design of observational or experimental studies.

Observational Study

Elements are observed and variables of interest are measured. An observation is the set of measurements for a single element

Experimental Study

A treatment is deliberately imposed on the elements and their responses are measured

Cross-Sectional

Observational Studies and Experiments - if the data is collected at a single point in time

Time Series

Observational Studies and Experiments - if the data is collected over several time periods

Population

the collection of ALL elements of interest; the group in which we want to draw a conclusion about

Sample

a subset/part of the population used to gather information; gives insight about the population

Census

an attempt to gather information from every element in the population; usually impossible or too expensive - US Gov

Parameter

a number that describes a population; a fixed amount that is usually an unknown (unless you have a census); - The mean height of Purdue students

Statistic

Is a number that describes a sample; an amount that can vary depending on which sample is collected; estimates a parameter; - The mean height of a sample of 350 Purdue students

Conclusion

We can use the sample proportion to estimate the population proportion

Ordered List

Shows the least & greatest data value

Frequency Table

Shows the total counts for particular intervals

Relative Frequency Table

Shows the total count in a particular interval relative to the entire set of data; usually represented as a percent

Histogram

A display of a frequency or relative frequency table

Mean

the sum of all data values divided by the number of data values

Median

the midpoint of an ordered data set; ½ of the observations are below it and ½ are above it

Mode

the data value (or values) that occur the most

Range

Max-min: where max is maximum (largest) data value and min is the minimum (smallest) data value; range is a single number

Sample Variance

a measure of how different, on average, data values are from the mean

Sample Standard Deviation

Square root of sample variance

Interquartile Range

Q3-Q1 (Q = quartile)

Categorical Variable

Qualitative data represented by categories.

Numerical Variable

Quantitative data represented by numbers.

Bar Graph

Visual representation of categorical data.

Pie Chart

Circular chart showing proportions of categories.

Stemplot

Graph displaying quantitative data in stems and leaves.

Shape of Distribution

Describes the form of data distribution (bell/skewed).

Unusual Data

Identifies gaps or extreme values in data.

Mean (x̄)

Average value of a data set.

Median (M)

Middle value when data is ordered.

Spread

Variability or consistency of data values.

Standard Deviation (s)

Measures data variability around the mean.

Population Variance

Variance calculated for an entire population.

Interquartile Range (IQR)

Difference between Q3 and Q1.

Quartiles

Values dividing data into four equal parts.

Percentile

Value below which a percentage of data falls.

5-Number Summary

Minimum, Q1, Median, Q3, Maximum values.

Boxplot

Graphical summary of the 5-number summary.

Modified Boxplot

Boxplot that highlights outliers.

Outlier

Data point significantly different from others.

Z-score

Standardized score indicating distance from mean.

Empirical Rule

Describes data distribution in bell-shaped curves.

Skewness

Measure of asymmetry in data distribution.

Kurtosis

Measure of data distribution's peakedness.

Descriptive Statistics

Summarizes and describes characteristics of data.

z-scores

A measure of the number of standard deviations a data value is from the mean.

Random Experiment

A random process that generates well-defined outcomes.

Sample space, S

The set of all possible outcomes (or sample points).

Event

A collection of one or more outcomes (or sample points).

Probability

A measure of the likelihood that an event will occur.

Classical (Theoretical) Method

Assumes all outcomes are equally likely.

Relative Frequency Method

Conduct many trials of a random experiment to estimate probabilities.

Counting Rule

If a random experiment consists of a sequence of k steps with n1 outcomes for the 1st step, n2 outcomes for the 2nd step, ..., nk outcomes for the kth step, then there are (n1)(n2)...(nk) total outcomes for the experiment.

Factorial

A notation for showing a special product, n! = n × (n - 1) × (n - 2) × ... × 1.

Permutations

The arrangements of n things taken r at a time, where order matters.

Combinations

The selection of n things taken r at a time, where order does not matter.

Complement of An Event

The set of all outcomes in S that are not outcomes of event A.

Union of A and B

The set of all outcomes that belong to A or B or both, denoted A ∪ B.

Intersection of A and B

The set of all outcomes that belong to both A and B, denoted A ∩ B.

Venn Diagrams

Visual displays of the relationship of the outcomes of combined events and the sample space.

Addition Law for 2 events

P(A ∪ B) = P(A) + P(B) - P(A ∩ B).

Mutually Exclusive Events

Two events that do not share any outcomes and cannot occur at the same time.

Complementary events

Events that cannot both occur at the same time and share no outcomes.

Probability Statement

P(A) = the number of favorable outcomes / TOTAL number of outcomes.

P(A)

The probability of event A occurring.

P(A and B)

The probability that both events A and B occur.

P(A or B)

The probability that either event A or event B occurs.

P(not A)

The probability that event A does not occur.

P(G)

Probability that a home has a garage.

P(P)

Probability that a home has a swimming pool.

P(G and P)

Probability that a home has both a garage and a swimming pool.

P(G or P)

Probability that a home has either a garage or a swimming pool.

N

The total number of employees or items in a sample.

P(L)

Probability that an employee completed their work late.

P(D)

Probability that an employee's work was defective.

P(L and D)

Probability that an employee's work was both late and defective.

P(L or D)

Probability that an employee's work was either late or defective.

Joint Distribution

Probability distribution of two events occurring together.

Marginal Distribution

Distribution of a single event from joint distribution.

Joint Probability

Probability of two events happening simultaneously.

Marginal Probability

Probability of a single event occurring.

Conditional Probability

Probability of an event given another event has occurred.

Multiplication Law

P(A and B) = P(A) * P(B|A).