What is the difference between frequentist and Bayesian statistics?
- Bayesians use Bayes' theorem, frequentists do not
- Frequentists believe in probability and Bayesians do not
- Frequentists interpret probability as a long-run frequency, Bayesians as a degree of belief
- There is no difference
Frequentist statistics interprets probability as the long-run frequency of events, whereas Bayesian statistics interprets probability as a degree of belief or as subjective probability. The Bayesian approach uses Bayes' theorem to update probabilities based on new data.
What are confidence intervals used for in statistics?
- To determine the median of a sample
- To determine the spread of data in a sample
- To estimate the population parameter
- To find the mean of a sample
Confidence intervals are used to estimate the range within which the true population parameter lies with a certain degree of confidence. They do not specifically determine the mean, median, or spread of a sample.
How does skewness affect the mean and median of a dataset?
- In a positively skewed distribution, the mean is greater than the median
- In a positively skewed distribution, the median is greater than the mean
- Skewness affects only the mean
- Skewness does not affect the mean and median
In a positively skewed distribution, the mean is greater than the median as the mean gets pulled in the direction of the skew (towards the right tail). In a negatively skewed distribution, the mean is less than the median as the mean gets pulled towards the left tail.
What can cause the Chi-square test for goodness of fit to be biased?
- Having a large sample size
- Having a small sample size
- Having equal expected frequencies in all categories
- Having normally distributed data
A small sample size can lead to unreliable results in a Chi-square test for goodness of fit. This can be due to the fact that the test requires a sufficient number of observations in each category to provide a reliable estimate of the distribution.
A ________ distribution has a constant probability.
- Binomial
- Normal
- Poisson
- Uniform
A uniform distribution is a type of probability distribution in which all outcomes are equally likely. This implies a constant probability for all outcomes.
In the factor analysis, the _______ measures the amount of variance in all the variables which is accounted for by that factor.
- communality
- eigenvalue
- factor variance
- total variance
In the factor analysis, the eigenvalue measures the amount of variance in all the variables which is accounted for by that factor.
Why might you perform a paired t-test?
- All of the above
- To compare the means of the same group at two different times
- To compare the means of two different populations
- To compare two independent groups
A paired t-test is used to compare the means of the same group at two different times or under two different conditions. It is not used to compare independent groups or different populations.
The ________ of a random variable is the sum of the probabilities of all possible outcomes.
- Distribution
- Expected value
- Mean
- Variance
The "expected value" of a random variable is the sum of all possible values it can take, each multiplied by the probability of that outcome. It gives us the mean or average value of the random variable and is a fundamental concept in probability theory and statistics.
What assumptions are made when conducting an ANOVA test?
- Independent observations, no outliers, equal sample sizes
- Independent observations, normal distribution of variables, no outliers
- Independent observations, normally distributed residuals, homoscedasticity
- No missing data, normally distributed residuals, no outliers
ANOVA makes three key assumptions: 1) Observations are independent. 2) Residuals (the differences between the observed and predicted values) are normally distributed. 3) The variance of the residuals is the same for all groups (homoscedasticity).
What is the principle of inclusion and exclusion in probability theory?
- It is used to calculate the conditional probability of an event
- It is used to calculate the probability of the intersection of events
- It is used to calculate the probability of the union of events
- It is used to prove the independence of events
The principle of inclusion and exclusion is a counting principle used to calculate the probability of the union of multiple events. It's based on the idea that the union's probability should add the individual probabilities and subtract the probabilities of intersections to avoid double-counting.