STATS 200 MSU EXAM 1

Explanatory variable

The variable that is manipulated by researchers.

Response variable

The variable that is observed by researchers.

Treatment group

The group of subjects in a study that receives a treatment.

Control group

A separate group of subjects in a study that does not receive the treatment.

Controlled experiment

An experiment where researchers impose a treatment on one group of subjects (treatment group) and do not impose it on a separate group (control group).

Placebo effect

The psychological and physical response to treatments that have no physiological effect.

Double-blind study

A study where neither the child, the parents, the person administering the shot, nor the health professionals making a diagnosis are aware of whether the child received the vaccine or the placebo.

Confounding factors

Any underlying factors besides the treatment which affect both the treatment and the response variable.

Randomization

The process that avoids any sort of bias in the choice of treatment and control group subjects.

Observational studies

Studies where researchers are limited to simply observing subjects without interfering in their behavior.

Causal conclusions

Conclusions that can be reasonably made based on experiments but are treacherous when based on observational data.

Associations

What observational studies are generally only sufficient to show.

Parental consent

A confounding factor in the polio vaccine experiment.

Data collection in observational studies

Generally collected only by monitoring what occurs.

Primary explanatory variable in experiments

Must be assigned for each subject by the researchers.

Population of Interest

The collection of all people/cases/items we want to learn about.

Sample

A subset of the population that researchers gather data from instead of measuring the entire population of interest.

Representative Sample

A sample that accurately reflects the characteristics of the population of interest.

Random Selection

The method of selecting observations in a way that every member of the population has the exact same chance of being selected into the sample.

Convenience Sampling

Samples obtained by measuring whatever or whoever is available to be measured, which are rarely representative of a larger population.

Selection Bias

Occurs if the method for selecting participants produces a sample that does not represent the population of interest.

Nonresponse Bias

Occurs when a representative sample is chosen for a survey, but a subset cannot be contacted or does not respond.

Response Bias

Occurs when participants respond differently from how they truly feel, influenced by question wording or interviewer behavior.

Numerical Variables

Variables that measure or record a numerical quantity for each case, allowing for operations like adding and averaging.

Categorical Variables

Variables that classify cases into groups or categories, placing each case into exactly one of two or more categories.

Dot Plot Drawbacks

Dot plots are not helpful for large data sets, or when data sparsely covers a large range.

Histograms

Graphs that help answer questions about how measurements vary, the range of values, and the distribution of numerical data.

Data Variation

Refers to how measurements are concentrated in one interval, sparse, or uniform.

Range of Values

The minimum and maximum values in a data set.

Anecdotal Evidence

Conclusions based on unusual cases or recalled striking characteristics rather than systematic data.

Generalization

The process of applying results from a sample to the larger population.

Survey

A method of collecting data from a sample to understand the population of interest.

Statistical Analysis

The process of interpreting data collected from samples or populations to draw conclusions.

Symmetric and bell-shaped

Data is clustered in the middle with an approximately equal number of smaller and larger values.

Right skewed

Most data have smaller values, with just a few larger values.

Left skewed

Most data have larger values, with just a few smaller values.

Symmetric but not bell-shaped

When you imagine cutting the distribution down the middle, the data values on one side mirror the values on the other side.

Parameters

Summary values calculated from populations.

Statistics

Summary values calculated from samples.

Notation

Capital letters like 𝑋 and 𝑌 refer to the name of a numerical variable and lower-case letters with subscripts refer to individual data points.

Data points

For a variable 𝑋, the values of the variable are denoted 𝑥1, 𝑥2, 𝑥3, … , 𝑥𝑛 where 𝑛 = # items in the data set or sample size.

Central Tendency

We want to describe the 'typical' values for the data.

Mean

The numerical average value, calculated as 𝑥̄ = (𝑥1 + 𝑥2 + 𝑥3 + ⋯ + 𝑥𝑛) / 𝑛.

Population Mean

The mean of a population is denoted 'μ'.

Median

The middle value when data is arranged from smallest to largest.

Sum of squares in statistics

The total of squared deviations from the mean, which will always be positive.

Population variance measure

The average of the squared deviations from the mean for a population.

Sample variance calculation

Sample variance is calculated by dividing the sum of squares by n - 1, where n is the number of observations.

Population standard deviation formula

σ = √(Σ(xi - μ)² / n).

Sample standard deviation formula

s = √(Σ(xi - x̄)² / (n - 1)).

Boxplots and histograms in data visualization

Boxplots provide a quick look at the range and center of data, while histograms show details of the shape and variation.

Purpose of a bar plot

A bar plot visualizes the number of observations in each category of a categorical variable.

Sample proportion denotation

The sample proportion is denoted as p-hat (𝑝̂).

Contingency table use

A contingency table is used to investigate the relationship between two categorical variables.

Nominal vs ordinal variables

Nominal variables have categories without a natural order, while ordinal variables have categories with a logical ordering.

Numerical variables as categorical

Yes, numerical variables can sometimes be grouped into ranges and treated as ordinal or nominal variables.

Base rate in probability

The proportion of the population that has a particular disease or condition.

Test sensitivity measure

The rate of true positives, indicating how often it correctly identifies patients with the disease.

Specificity in medical testing

The rate at which a test correctly identifies patients without the disease, indicating true negatives.

Null hypothesis (H0)

States that there is no change, difference, or relationship between the variables being studied.

Alternative hypothesis (HA)

States that there is a change, difference, or relationship between the variables.

Null distribution

Generated to evaluate the quality of a null model, simulating data assuming the null hypothesis is true.

p-value definition

Quantifies the likelihood of observed results under the null hypothesis.

Large p-value indication

Suggests that observed results can occur often under the null hypothesis.

Small p-value indication

Indicates strong evidence against the null hypothesis.

p-value less than 0.001 significance

Indicates extremely strong evidence against the null hypothesis.

First step of the scientific method

Make an observation and formulate a research question.

Purpose of statistical inference

Uses observed data from a sample to make inferences about a larger population.

Null hypothesis role in hypothesis testing

Serves as a default position that there is no effect or relationship.

Observing data 'at least as favorable' to HA

Determining the proportion of simulated differences under the null model that are greater than the observed value.

p-value calculation for independence testing

The fraction of simulations that are more extreme than the observed difference.

p-value when testing for an increase

Calculated as the fraction of simulations that are greater than the observed difference.

p-value when testing for a decrease

Calculated as the fraction of simulations that are less than the observed difference.

Null hypothesis (Ho) for independence testing

States that the variables are independent.

Alternative hypothesis (Ha) for independence testing

States that the variables are not independent.

p-value if first proportion is greater than second

The fraction of simulations greater than the observed difference.

p-value if first proportion is smaller than second

The fraction of simulations less than the observed difference.

Bootstrap Test for Single Proportion purpose

Used to draw inferences about a single categorical variable.

Two outcomes in Bootstrap Test

Each unit is counted as either a 'success' or a 'failure.'

Example of single categorical variable in Bootstrap Test

Proportion of people voting for a candidate.

Null hypothesis (Ho) in Bootstrap Test

States that the population proportion is equal to a hypothesized value.

Alternative hypothesis (Ha) in Bootstrap Test

States that the population proportion is greater, smaller, or different from a hypothesized value.

Ƹ𝑝 definition

The proportion of successes found in a sample.

𝑝 definition

Represents the true population proportion of successes.

𝑝0 definition

Represents the hypothesized (null) value of the parameter, 𝑝.

p-value definition for hypothesis tests

The probability of observing data at least as favorable to the alternative hypothesis as the current data set.

p-value when testing if 𝑝 is different from 𝑝0

The fraction of simulations further from 𝑝0 as Ƹ𝑝.

p-value when testing if 𝑝 is greater than 𝑝0

The fraction of simulations greater than Ƹ𝑝.

p-value when testing if 𝑝 is less than 𝑝0

The fraction of simulations less than Ƹ𝑝.