Stat 211: Elementary Inferential Statistics - Study Notes

Stat 211: Elementary Inferential Statistics

Unit 1: Data and Collection

Identifying Data

• Date: Introduction to Statistics

Definitions: Statistics and Statistic

• Statistics (plural): The science of learning from data. It involves collecting, organizing, analyzing, and interpreting information to understand patterns, make decisions, and draw conclusions in the presence of uncertainty.

• Statistic (singular): A numerical summary that describes a characteristic of a SAMPLE.

Population vs Sample

• Population: The entire group of individuals or cases that we are interested in studying or drawing conclusions about.

• Sample: A subset of the population that is actually observed or collected for analysis. It is used to estimate information about the population.

• Parameters & Statistics:

  • Populations have parameters: numerical measurements describing characteristics of a population (usually unknown, denoted by Greek letters).

  • Samples have statistics: numerical measurements describing characteristics of a sample (calculated to estimate parameters).

Types of Statistics

• A statistic is a number that summarizes data from a sample and serves various roles:

  • Descriptive: Describes what was observed.

  • Summary: Condenses raw data into a meaningful value.

  • Explanatory: Helps explore relationships between variables.

  • Predictive: Used to forecast outcomes.

Statistical Analysis

• Descriptive Statistics: Summarizing and organizing data.

• Inferential Statistics: Making predictions or decisions about a population based on a sample.

The Process of a Statistical Study

1. Identify Goals

2. Draw from Population

3. Sample Population

4. Draw Conclusions

5. Summarize Sample Statistics

Applications of Statistics

Data Collection Examples

• Application: Facebook

  • Data Collection: Gathers user behavior (likes, shares, clicks, time on posts).

  • Descriptive Statistics: Summarizes activity (average time spent, most liked posts, user demographics).

  • Inferential Statistics: Makes predictions (What posts will you like next? Who might you know?).

• Application: Texting while Driving

  • Study Details: 40 drivers divided into three groups: sober, drunk, and texting.

  • Measured: Reaction times in simulated emergencies.

  • Findings: Reaction times of texting drivers were significantly slower than both sober and drunk drivers.

• Application: Driving While Black

  • Study Details: 2,533 traffic stops in Cincinnati, OH.

  • Focus: Investigated disproportionate stops and searches by race.

  • Discussion Point: Raises the question: Can these outcomes be explained by random variation alone?

Data Structures

• Organization: Tables, matrices, and data frames keep data tidy and accessible.

• Efficiency: Enables faster searching, sorting, grouping, and computation.

• Interpretation: Makes it easier to summarize, graph, and derive insights.

• Analysis-Ready: Most tools require data in specific formats (e.g., rows = observations, columns = variables).

Datasets, Variables, and Cases

• Cases: The units or subjects being studied.

• Variables: The characteristics or attributes measured on each case.

Variable Types

• Categorical (Qualitative): Places values into groups or categories.

• Quantitative: Numerical values where magnitude matters.

• Identifier: Unique labels for each case (not used in analysis).

Categorical Variables (Qualitative)

• Nominal: No natural order (e.g., types of fruit).

• Ordinal: Categories with a meaningful order (e.g., education level).

Quantitative Variables

• Discrete: Countable values (e.g., number of students).

• Continuous: Any value in an interval (e.g., weight).

Quantitative Measurement Types

• Interval: Equal spacing, but no true zero (e.g., temperature in Celsius).

• Ratio: Equal spacing with a true zero (e.g., height).

Practice: Classify These Variables

1. Age: Quantitative (Continuous)

2. Musical Genres: Categorical (Nominal)

3. Price of Computer: Quantitative (Continuous)

4. Marital Status: Categorical (Nominal)

5. Number of Pixels Displayed: Quantitative (Discrete)

6. Time Needed to Complete Exam: Quantitative (Continuous)

Sampling Techniques

Sampling Overview

• Goal: Learn about the entire group of individuals called the population.

• Problem: It is usually impossible to collect data on the entire population (called a census).

• Compromise: Collect data on a smaller group of individuals (called a sample) selected from the population.

• Challenge: Obtain a sample that is perfectly representative of the population while avoiding Bias.

• Bias: The over- or under-emphasizing of some characteristic that is pertinent to the study.

Census

• A census is when data is collected for the entire population.

• Reasons for Not Conducting a Census:

  • Difficult to complete.

  • Hard/expensive to locate everyone.

  • Impractical in manufacturing.

  • Populations change over time.

  • Opinions may change.

  • The U.S. Census: Conducted every 10 years as required by the U.S. Constitution. Used for congressional representation, federal funding, research, and planning.

Randomization

• Definition: Randomizing protects us from the influences we know and don't know are in the data.

• Key Idea: On average, a randomized sample will look like the population.

Sample Size

• Sample Size Definition: The number of individuals in the sample.

• Guideline: A few hundred may be enough for a proportion; in general, the bigger the better.

• Incomplete Sampling Frame: Not all individuals in the population are included in the list from which the sample is taken.

Sampling Methods

1. Simple Random Sampling (SRS): - Choosing a subset of a population where each member has an equal chance of being selected.

   • Example: Dining Hall - Use Excel and the function = rand() to assign random values, then sort.

2. Stratified Random Sampling: - Sampling by dividing the population into homogenous groups called strata and selecting proportionate amounts from each group.

   • Benefits: More precise estimates, less variability, detect differences among groups.

   • Simpson's Paradox: A trend appears in separate groups but disappears or reverses when groups are combined.

3. Cluster Sampling: - Divide the population into clusters that are mutually homogenous yet internally heterogeneous, then sample whole clusters at random.

   • Benefits: Less time and cost, natural groupings (e.g., dorms, classes, majors), easier administration (e.g., professors can mandate responses).

   • Analogy: Stratified Sampling = takes a bite from each layer of a pie; Cluster Sampling = takes a vertical slice through the whole cake.

4. Systematic Sampling: - Choose every nth person after selecting a random starting point (the order must not be related to the outcome).

   • Example: Select students in a dorm systematically.

5. Multistage Sampling: - Sampling schemes that combine several methods are called multistage samples.

Visualization of Sampling Methods

• SRS: Randomly selected sample.

• Stratified: Grouped by strata attributes.

• Systematic: Selected at intervals.

• Cluster: Whole groups sampled.

• Multistage: Combination of methods.

Practice Quiz: Name that Sampling Method

1. Pick every 10th passenger on a flight → Systematic

2. Randomly choose 5 from first class, 25 from coach → Stratified

3. Randomly generate 30 seat numbers → Simple Random Sampling

4. Survey everyone sitting in window seats → Cluster Sampling

Bad Sampling Techniques

• Voluntary Response Sampling: Individuals choose to participate, prone to bias.

  • Examples: "Tell Us What You Think" website.

• Convenience Sampling: Drawn from those easiest to reach, prone to bias.

  • Example: Stopping people outside a dining hall.

Bias in Statistical Studies

What is Bias?

• Definition: Bias is the degree to which a procedure systematically over- or under-estimates a population value.

• A procedure is unbiased if it produces the true population parameter on average.

Major Categories of Bias

1. Selection Bias: Affects the sample selection process.

2. Response Bias: Affects the responses provided by respondents.

Types of Selection Bias

• Voluntary Response Bias:

  • Occurs when people choose to participate in a survey, often reflecting only the opinions of those with strong opinions.

  • Example: A survey asking people to call in their support for a new issue.

• Non-Response Bias:

  • Occurs when people are unwilling to participate and those who do not respond may differ in opinions from those who do.

• Sampling Bias:

  • Some individuals are more likely to be selected than others.

  • Example: Favoring a specific group unintentionally.

Types of Response Bias

• Social Acceptability Bias:

  • People may give answers they think are more socially acceptable.

  • Example: Over-reporting favorable behaviors like recycling.

• Leading Question Bias:

  • The wording of the question suggests a preferred response.

  • Example: "Don’t you agree that our policy is beneficial?"

• Acquiescence Bias:

  • The tendency to agree with statements, regardless of true beliefs.

  • Example: Likely in surveys that use scales (e.g., Strongly Agree to Strongly Disagree).

• Self-Interest Bias:

  • Arises when individuals or organizations have a self-interest in the outcome, which can influence both the study and how the results are analyzed.

Key Takeaway for Types of Bias

Practice: Identify the Type of Bias

1. Local business owner asks residents to call a hotline to show support for a new stadium: Type of Bias: Voluntary Response Bias

2. Police chief sends uniformed officers to ask if residents think the police are doing a bad job: Type of Bias: Response Bias (Social Acceptability Bias)

3. Candidate's campaign website claims only 11% support a rival's policy: Type of Bias: Self-Interest Bias

4. Bank mails 8,000 surveys; only 500 are returned: Type of Bias: Non-Response Bias

5. Fitness center asks, “Why do YOU love our new 24-hour access policy?”: Type of Bias: Leading Question Bias