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.

What is the sum of squares in statistics?

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

What does the population variance measure?

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

How is sample variance calculated?

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

What is the formula for population standard deviation?

The formula for population standard deviation is σ = √(Σ(xi - μ)² / n).

What is the formula for sample standard deviation?

The formula for sample standard deviation is s = √(Σ(xi - x̄)² / (n - 1)).

What do boxplots and histograms show in data visualization?

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

What is the purpose of a bar plot?

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

How is the sample proportion denoted?

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

What is a contingency table used for?

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

What distinguishes nominal variables from ordinal variables?

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

Can numerical variables be treated as categorical?

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

What is the base rate in probability?

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

What does a test's sensitivity measure?

A test's sensitivity measures the rate of true positives, indicating how often it correctly identifies patients with the disease.

What is specificity in the context of medical testing?

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

What is the null hypothesis (H0)?

The null hypothesis states that there is no change, difference, or relationship between the variables being studied.

What is the alternative hypothesis (HA)?

The alternative hypothesis states that there is a change, difference, or relationship between the variables.

What is a null distribution?

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

What is a p-value?

A p-value quantifies the likelihood of observed results under the null hypothesis, helping to assess the null model.

What does a large p-value indicate?

A large p-value suggests that observed results can occur often under the null hypothesis, indicating little evidence against it.

What does a small p-value suggest?

A small p-value indicates strong evidence against the null hypothesis, suggesting that the null model may not be a good explanation.

What is the significance of a p-value less than 0.001?

A p-value less than 0.001 indicates extremely strong evidence against the null hypothesis.

What is the first step of the scientific method?

The first step is to make an observation and formulate a research question.

What is the purpose of statistical inference?

Statistical inference uses observed data from a sample to make inferences about a larger population.

What is the role of the null hypothesis in hypothesis testing?

The null hypothesis serves as a default position that there is no effect or relationship, which is tested against the alternative hypothesis.

What does it mean to observe data 'at least as favorable' to the alternative hypothesis?

It refers to determining the proportion of simulated differences under the null model that are greater than the observed value.

How is the p-value calculated when testing if two variables are independent?

The p-value is the fraction of simulations that are more extreme than the observed difference of Ƹ𝑝1 − Ƹ𝑝2 and -(Ƹ𝑝1 − Ƹ𝑝2).

What is the p-value calculation when testing for an increase?

It is calculated as the fraction of simulations that are greater than the observed difference of Ƹ𝑝1 − Ƹ𝑝2.

What is the p-value calculation when testing for a decrease?

It is calculated as the fraction of simulations that are less than the observed difference of Ƹ𝑝1 − Ƹ𝑝2.

What is the null hypothesis (Ho) when testing if two variables are independent?

Ho states that the variables are independent.

What is the alternative hypothesis (Ha) when testing if two variables are independent?

Ha states that the variables are not independent.

What is the p-value when testing if the first proportion is greater than the second?

It is the fraction of simulations greater than the observed difference.

What is the p-value when testing if the first proportion is smaller than the second?

It is the fraction of simulations less than the observed difference.

What is the Bootstrap Test for a Single Proportion used for?

It is used to draw inferences about a single categorical variable.

What are the two outcomes for each observational unit in the Bootstrap Test?

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

What is an example of a single categorical variable in the Bootstrap Test?

Proportion of people voting for a candidate.

What is the null hypothesis (Ho) in the Bootstrap Test?

Ho states that the population proportion is equal to a hypothesized value.

What is the alternative hypothesis (Ha) in the Bootstrap Test?

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

What does Ƹ𝑝 represent?

It represents the proportion of successes found in a sample.

What does 𝑝 represent?

It represents the true population proportion of successes.

What does 𝑝0 represent?

It represents the hypothesized (null) value of the parameter, 𝑝.

What is the definition of p-value?

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

What is the p-value when testing if 𝑝 is different from 𝑝0?

It is the fraction of simulations further from 𝑝0 as Ƹ𝑝.

What is the p-value when testing if 𝑝 is greater than 𝑝0?

It is the fraction of simulations greater than Ƹ𝑝.

What is the p-value when testing if 𝑝 is less than 𝑝0?

It is the fraction of simulations less than Ƹ𝑝.