How is the concept of independence used in probability theory?
- To calculate the probability of an event without any prior information
- To describe events that always occur together
- To describe events that are mutually exclusive
- To describe events that have no influence on each other
Independence in probability theory refers to situations where the occurrence of one event does not affect the occurrence of another event. In other words, Events A and B are independent if the fact that A occurs does not affect the probability of B occurring.
How many groups or variables does a one-way ANOVA test involve?
- 1
- 2
- 3 or more
- Not restricted
A one-way ANOVA involves three or more groups or categories of a single independent variable.
How does the concept of orthogonality play into PCA?
- It ensures that the principal components are uncorrelated
- It guarantees the uniqueness of the solution
- It helps in the calculation of eigenvalues
- It is essential for dimensionality reduction
Orthogonality ensures that the principal components are uncorrelated. PCA aims to find orthogonal directions (principal components) in the feature space along which the original data varies the most. These orthogonal components represent independent linear effects present in the data.
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.
What is the difference between a one-way and a two-way ANOVA?
- One-way ANOVA is for dependent variables, two-way ANOVA is for independent variables
- One-way ANOVA is for small samples, two-way ANOVA is for large samples
- One-way ANOVA tests one independent variable, while two-way ANOVA tests two
- One-way ANOVA uses an F statistic, two-way ANOVA does not
One-way ANOVA tests the effect of one independent variable on a dependent variable, while two-way ANOVA tests the effect of two independent variables on a dependent variable. Additionally, two-way ANOVA allows for the examination of interactions between the independent variables.
A _______ is a range of values, derived from a sample, that is used to estimate an unknown population parameter.
- Confidence interval
- Point estimate
- Probability
- Variance
A confidence interval is a range of values, derived from the statistical analysis of the sample data, that is likely to contain an unknown population parameter.
How does the sample size impact the accuracy of the Central Limit Theorem?
- As the sample size increases, the approximation of the sample mean to a normal distribution becomes more accurate.
- Sample size has no impact on the Central Limit Theorem.
- The Central Limit Theorem becomes less accurate as the sample size increases.
- The Central Limit Theorem is only accurate when the sample size is exactly 30.
According to the Central Limit Theorem, as the sample size increases, the distribution of the sample mean approaches a normal distribution more closely. This means the larger the sample size, the more accurately the sample mean will represent a normal distribution.
What does it mean when we say that a distribution is skewed?
- All data points are identical
- It has outliers
- It is not symmetric about its mean
- Its mean and median are not equal
When we say that a distribution is skewed, we mean that the distribution is not symmetric about its mean. In a skewed distribution, the data points are not evenly distributed around the mean, with more data on one side of the mean than the other.
What does it mean if the p-value in a Chi-square test is smaller than the significance level?
- The alternative hypothesis is true
- The null hypothesis is true
- The test result is insignificant
- There is not enough evidence to reject the null hypothesis
If the p-value in a Chi-square test is smaller than the significance level, we reject the null hypothesis in favor of the alternative hypothesis. This suggests that there is a significant association between the variables.
How does multicollinearity affect the coefficients in multiple linear regression?
- It doesn't affect the coefficients
- It makes the coefficients less interpretable
- It makes the coefficients more precise
- It makes the coefficients negative
Multicollinearity refers to a situation where two or more predictor variables in a multiple regression model are highly correlated. This high correlation can result in unstable coefficient estimates, making them less reliable and harder to interpret.