What is the impact of PCA on the interpretability of the original features?
- It depends on the data
- It doesn't affect interpretability
- It enhances interpretability
- It reduces interpretability
PCA typically reduces the interpretability of the original features. This is because each principal component is a linear combination of all the original features, making it difficult to understand how individual features affect the outcome.
What is the primary application of Bayes' Theorem in statistics?
- To calculate the mean of a data set
- To calculate the standard deviation
- To determine if two events are independent
- To update prior beliefs given new data
Bayes' Theorem is primarily used to update prior beliefs given new data. It's a way to go from a prior probability to a posterior probability, which is a more accurate estimate because it incorporates new evidence.
The ______ Rule of Probability is used when we want to find the probability that either of two events happens.
- Addition
- Division
- Multiplication
- Subtraction
The Addition Rule of Probability is used when we want to find the probability that either of two events happens. This rule states that the probability of either of two mutually exclusive events occurring is the sum of their individual probabilities.
What role does Bayes' theorem play in machine learning algorithms?
- It is not used in machine learning algorithms
- It is used to calculate error rates
- It is used to divide the data into training and test sets
- It is used to update prior beliefs based on new data
Bayes' theorem is used in various machine learning algorithms to update prior beliefs based on new data. For example, in Bayesian classifiers, it is used to estimate the parameters of the model and make predictions.
What is the relationship between the Kruskal-Wallis Test and the Mann-Whitney U Test?
- The Kruskal-Wallis Test is an extension of the Mann-Whitney U Test
- There is no relationship
- They are opposites
- They are the same
The Kruskal-Wallis Test is an extension of the Mann-Whitney U Test for more than two independent groups.
What is the correlation coefficient in the context of a scatter plot?
- A measure of the correlation between two variables
- A measure of the spread of data points
- The slope of the line of best fit
- The y-intercept of the line of best fit
The correlation coefficient, often denoted by r, is a numerical measure that quantifies the degree of correlation between two variables. It ranges from -1 to +1, with -1 indicating a perfect negative correlation, +1 indicating a perfect positive correlation, and 0 indicating no linear correlation.
In the Mann-Whitney U test, what does a lower U value indicate?
- A greater dissimilarity between the groups
- A greater similarity between the groups
- A higher correlation between the variables
- A lower correlation between the variables
In the Mann-Whitney U test, a lower U value indicates a greater dissimilarity between the groups. This means that it is more likely that values from one group are larger than values from the other group.
If a null hypothesis is rejected, what can we infer about the alternative hypothesis?
- It has no relation to the null hypothesis
- It is likely to be true
- It is rejected as well
- It needs to be tested separately
If a null hypothesis is rejected, it means that the alternative hypothesis is likely to be true. We can infer that there's enough evidence in our data to support the claim of the alternative hypothesis.
What is heteroscedasticity in the context of residual analysis?
- It is the assumption that residuals have constant variance
- It is the condition where residuals have varying variance
- It is the linear relationship between residuals and the dependent variable
- It refers to the independence of residuals
Heteroscedasticity refers to a situation where the variance of the errors or the residuals is not constant across all levels of the independent variables. This violates one of the assumptions of linear regression and can result in inefficient estimates of the regression coefficients.
How do Type I and Type II errors relate to the power of a statistical test?
- Both decrease the power of a test
- Both increase the power of a test
- Type I errors decrease the power, Type II errors increase it
- Type I errors increase the power, Type II errors decrease it
The power of a test is the probability that it correctly rejects a false null hypothesis (true positive). It's the complement of a Type II error. As Type I error probability increases, power also increases because we're more willing to reject the null hypothesis. However, a Type II error decreases power because it's a missed opportunity to reject a false null hypothesis.