Statistics and Bayesian Analysis Quiz

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Questions and Answers

Which correlation method is appropriate when the assumptions of Pearson’s Correlation are not satisfied?

Simple linear regression

Spearman's rho (correct)

Multiple correlation

Polynomial regression

What is the minimum recommended sample size for conducting Pearson’s correlation effectively?

100

30 (correct)

Which non-parametric correlation method is suggested for small sample sizes?

Pearson's correlation

Spearman's rho

Kendall's tau (correct)

Point-biserial correlation

Which of the following is NOT an assumption of Pearson’s Correlation?

The data should be on an ordinal scale Signup and view all the answers

What technique can be applied to convert non-normally distributed data for Pearson’s correlation?

Ranking the data Signup and view all the answers

What does the Bayesian view define probability as?

The degree of belief assigned to the truth of an event Signup and view all the answers

Which of the following is NOT a requirement for Bayesian analysis?

Frequentist analysis Signup and view all the answers

Which statement best captures a disadvantage of the Bayesian view?

It requires the specification of a degree of belief Signup and view all the answers

How can subjective probability be operationalized according to the Bayesian view?

By deciding on bets one is willing to take Signup and view all the answers

In the example provided regarding rain probability, what would indicate a favorable bet?

Believing it is a 70% chance of rain Signup and view all the answers

What is a characteristic of elementary events in probability?

They represent single, exclusive outcomes Signup and view all the answers

Which option best describes a common misconception about the Bayesian view of probability?

Probabilities are purely objective Signup and view all the answers

What is a potential advantage of the Bayesian approach to probability?

It allows flexible assignment of probabilities to events Signup and view all the answers

What defines an elementary event when throwing a die?

An event with a single favorable outcome Signup and view all the answers

Which event is classified as non-elementary when throwing a die?

Getting an even number Signup and view all the answers

In the context of binomial distribution, what does 'N' represent?

The number of observations Signup and view all the answers

In a binomial distribution setup, what does the variable 'X' represent?

The outcomes from the trials Signup and view all the answers

Given θ = 0.167 and N = 20, what is the type of probability being calculated?

Probability of getting exactly 4 successes Signup and view all the answers

In the equation 'Data = Model + Error', what does the term 'Model' represent?

The theoretical framework explaining data Signup and view all the answers

What key difference exists between comparison and prediction in data analysis?

Comparison focuses on past data only Signup and view all the answers

What does the term 'random variable' signify in the context of a binomial experiment?

A variable that can take any value from a set Signup and view all the answers

What is the primary focus of frequentists in probability?

Long-run frequency of events Signup and view all the answers

Which of the following is NOT a requirement for frequentist methods?

Subjective interpretation Signup and view all the answers

What disadvantage is associated with the frequentist view of probability?

It has a narrow scope meaning it can't cover all probability situations. Signup and view all the answers

How do frequentists and Bayesians primarily differ in their approach to probability?

Frequentists use long-run frequency while Bayesians incorporate prior knowledge. Signup and view all the answers

What could be deemed an advantage of the frequentist approach to probability?

It yields consistent results among different observers. Signup and view all the answers

Which of the following statements describes a major limitation of frequentist probability?

It relies too heavily on past data. Signup and view all the answers

Which analogy illustrates the difference between probability and statistics?

Observing an animal's footprint vs. predicting the animal based on the footprint. Signup and view all the answers

What does the frequentist perspective on probability NOT account for?

The influence of prior knowledge on outcomes. Signup and view all the answers

What does the 'd' form in probability distributions signify?

It specifies a particular outcome and its probability. Signup and view all the answers

In the context of the binomial distribution, what does 'size' represent?

The total number of trials in the experiment. Signup and view all the answers

Which statement is true regarding the characteristics of the normal distribution?

The mean and median are always equal. Signup and view all the answers

What does the standard deviation control in a normal distribution?

The spread of the data around the mean. Signup and view all the answers

How is the cumulative probability calculated in probability distributions?

By specifying a quantile q. Signup and view all the answers

In the notation for a normally distributed variable, which symbol represents the mean?

μ Signup and view all the answers

What is indicated by the cumulative probability being equal to 0.5 in a normal distribution?

Exactly half of the values lie below the mean. Signup and view all the answers

What is the purpose of the 'r' form in probability distributions?

To generate a specified number of random outcomes. Signup and view all the answers

What does a smaller standard deviation indicate about a data set?

The data points are tightly clustered around the mean. Signup and view all the answers

How is a binomial distribution characterized?

It consists of histogram-like bars. Signup and view all the answers

What does the output of the cor.test() function provide?

It determines whether the correlation in the population is different from zero. Signup and view all the answers

What is indicated by a p-value greater than 0.05 in a correlation test?

The correlation is not significantly different from zero. Signup and view all the answers

What happens to the normal distribution when the standard deviation increases?

The distribution becomes flatter and wider. Signup and view all the answers

What is implied when the confidence interval for a correlation coefficient includes zero?

The correlation could be zero in the population. Signup and view all the answers

If the t-statistic is far from the mean, what does that suggest?

We need to include all values greater or smaller than the t-value. Signup and view all the answers

When analyzing a dataset with a normal distribution, what effect does an increase in standard deviation have on data interpretation?

It leads to more extreme values occurring. Signup and view all the answers

Study Notes

Statistics II - Exam Study Guide

Basics of Probability: Probabilities form the basis for statistical inference. Inferential statistics are used to determine how representative data are of a population. Probability involves predicting outcomes, while statistics involves interpreting data to make inferences about the population.

Probability & Statistics

Frequentists vs Bayesians: Frequentists define probability as long-run frequency. For example, a fair coin (50% probability of heads) is expected to land heads in half of the trials. Frequentists require data, models, and design for their analysis. Bayesians, on the other hand, view probabilities as degrees of belief held by a rational agent.

Advantages & Disadvantages

Frequentist view: Objective, unambiguous, and grounded in the physical world. However, infinite sequences don't exist, and it has a limited scope regarding the analysis of events.
Bayesian View: Assigns probabilities to events based on beliefs and assumptions of an intelligent agent. Can handle events that aren't easily quantified in the physical world. However, it's subjective, requiring careful specification of belief. The Bayesian view is often considered too broad.

Probability Distributions

Elementary events: For a given observation, the outcome will be one and only one of these events.
Example: In tossing a coin, "heads" and "tails" are elementary events.

Statistics

Binomial Distribution: This distribution applies when an event happens or doesn't happen (e.g., 0 or 1). Success probability (e.g., the probability of a 'heads' outcome in a coin toss) and the number of observations (trials) are important parameters defining this distribution.
Example: Calculating probability of getting a specific number of successes (like getting heads 4 times in 20 coin tosses).

Relationship Between Models & Data

Data = Model + Error
Statistical Inference compares models to data

Using Different Distributions in R

Binomial Distribution (dbinom, pbinom, rbinom, qbinom): Used to calculate outcomes and probabilities in experiments of finite sizes (e.g., number of heads in a series of coin flips)
Example: calculating the probability of getting 4 heads in 10 coin flips.
Normal Distribution (dnorm, pnorm, rnorm, qnorm): Used to calculate outcomes and probabilities when dealing with continuous data or distributions approximated by it.

Characteristics of Normal Distribution

The area under the normal distribution curve is equal to 1.
The mean, mode, and median are all equal in a normal distribution.
The curve is symmetric around the mean (μ).
Standard deviation (σ) controls the spread, which tells us if the data is closely clustered around the mean, or spread out.

Binomial vs Normal

Binomial: Discrete (countable) plot appearance.
Normal: Continuous (uncountable) smooth curve distribution

Functions in R for Correlation

cor(), cor.test(), and rcorr() for calculating Pearson and Spearman correlations.

Sample Statistics and Population Parameters

Population parameter: Describes the characteristic of the whole population.
Sample statistic: Describes the characteristic of a smaller group (subset) taken from the population.

Running & Interpreting R Output

Linear Regression: A method to find the relationship between variables. Results (output) often include estimates, standard errors, p-values, and R-squared values.

Hypotheses & Research Questions

Hypothesis: A statement about the relationship between variables. Example: There is a relationship between the amount of exercise and overall health. Hypotheses testing attempts to rule out chance as a plausible explanation for results.

Effect Sizes

Cohen's d: Effect size measure focusing on mean differences in terms of the standard deviation, mostly tested with student's t-tests or z-tests.

Sampling Theory

Population: A comprehensive set of units to which findings are generalized.
Sample: A subset of the population from which inferences about the population are drawn.
Sampling distribution: A probability distribution of a statistic calculated to determine the distribution of outcomes for a given statistic in a population.

Correlation & Covariance

Correlation: Measures the extent to which two variables are related.
Covariance: Indicates how much two variables change together. Positive covariance signifies that they generally change in the same direction, while negative covariance means they change in opposite directions.

Partial and Semi-Partial Correlation

Partial correlation: Measures the relationship between two variables while controlling the effect of other variables.
Semi-partial correlation: Controls for a variable's effect on only one outcome variable (either X or Y).

Regression/Test Statistics

Regression line: Straight line depicting the mathematical relationship between variables.

Ordinary Least Square

Method of calculating regression line minimizing the difference between observed and predicted data.

Testing the Model (ANOVA)

ANOVA (Analysis of variance): Test showing whether the variation of the model explains the variability in data better than the variation from the mean.

Mean Squared Error (MSE)

MSE: Represents the variability in a distribution, often estimated using a model.
MSE can be divided into Sum of squares of the model (SSM) and Sum of squares of the residuals (SSR).
The proportion of variance explained by a model is often quantified as R-squared (R²).

Standard Error

Standard error: A measure of the variability of a statistic.

Null and Alternative hypotheses

Null hypothesis(H₀): A claim of "no difference" in a population.
Alternative hypothesis(Ha): Contends that the null hypothesis is false.

Types of Sampling Methods

Random Sampling: Every member has an equal chance of being selected.
Stratified Sampling: The population is divided into subgroups (strata), and random samples are taken from each subgroup.
Volunteer Sampling: Participants self-select to participate in a study.
Opportunity Sampling: Choosing participants who are readily available.
Convenience Sampling: Choosing participants that are convenient for the researcher.
Snowball Sampling: Participants recruit other potential participants (useful for hard-to-reach populations).

Confidence Intervals

Confidence intervals: Provide a range of plausible values for a population parameter (e.g., mean, proportion), based on a random sample. Example: There's a 95% chance that the average IQ is between 89 and 111.

Central Limit Theorem

Central Limit Theorem: Shows that as sample sizes grow, a distribution of sample means gets closer and closer to a normal distribution.

Type I and Type II Errors

Type I error: Rejecting a true null hypothesis (false positive)
Type II error: Failing to reject a false null hypothesis (false negative)

Interaction Terms in Multiple Regression

Interaction: When the relationship between two predictors differs depending on the level of a third variable. Example: The effect of one ingredient in a cake differs depending on the amount of another ingredient added to it.

Categorical Variable Coding

Dummy coding: Used to represent categorical variables in an analysis.
Unweighted effect coding: Assign values based in the set of groups and means.
Weighted effect coding: An approach to coding categorical variables where the values are assigned based on a weight.
Contrast coding: Useful in situations where the researcher has pre-existing hypotheses about the interactions of the variables.

Growth Models with Polynomials

Example: Determining functional relationship between weight and time. This is done using interaction terms when examining more than one predictor.

Running & Interpreting R output

Polynomial Regression: A technique for modelling curvilinear relationships.

Multiple Linear Regression

Method to model the relationship between a dependent variable and two or more independent variables.

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Description

Test your knowledge on correlation methods, particularly Pearson's correlation and Bayesian probability concepts. This quiz covers various assumptions, sample sizes, and techniques relevant to both statistical and Bayesian analyses. Perfect for students and professionals looking to refine their understanding of these key statistical principles.