Interval estimation provides a/an ________ for the parameter of interest.
- Exact value
- Mean
- Median
- Range
Interval estimation provides a range (or interval) of values for the parameter of interest. This is more informative than a point estimate, as it gives a measure of uncertainty.
How can you detect the presence of interaction effects in your data?
- By adding interaction terms in the regression model and checking their significance
- By checking the coefficients of the independent variables
- By comparing the fit of the model with and without polynomial terms
- By examining the correlation between variables
To detect the presence of interaction effects in your data, you can include interaction terms in your regression model and then check the significance of these terms. If the interaction term is statistically significant, this suggests that the effect of one variable on the dependent variable depends on the level of another variable.
The function used to describe the likelihood of a random variable that is continuous is called a ________.
- Cumulative Distribution Function
- Probability Density Function
- Probability Mass Function
- Random Function
For continuous random variables, the probability density function (PDF) is used to specify the probability of the random variable falling within a particular range of values, as opposed to taking on any one value.
What are the implications of violating the assumption of homoscedasticity in ANOVA?
- It can lead to incorrect conclusions about the differences between group means
- It has no implications
- It leads to a decrease in the F-statistic
- It leads to an increase in the F-statistic
Violating the assumption of homoscedasticity (equal variances across groups) in ANOVA can lead to incorrect conclusions about the differences between group means, i.e., the results of the ANOVA test could be misleading. This might cause Type I errors (rejecting a true null hypothesis) or Type II errors (failing to reject a false null hypothesis).
What does 'silhouette score' represent in cluster analysis?
- The average size of the clusters
- The distance between clusters
- The level of similarity within clusters and dissimilarity between clusters
- The number of clusters
The silhouette score is a measure of the similarity of an object to its own cluster (cohesion) compared to other clusters (separation). It represents how similar an object is to its own cluster compared to other clusters. The score ranges from -1 to 1, with high values indicating that the object is well matched to its own cluster and poorly matched to neighboring clusters.
When would you use a t-test instead of a Z-test?
- All of the above
- When the data is not normally distributed
- When the population standard deviation is unknown
- When the sample size is very large
T-tests are typically used when the population standard deviation is unknown. The sample size or normality of data isn't the primary deciding factor.
In what scenarios is the use of Bayes' theorem considered controversial in statistics?
- All of the above
- When the events are independent
- When the prior is subjective or not based on data
- When the sample size is very large
The use of Bayes' Theorem is controversial when the prior probability is subjective or not based on data. Critics argue that this introduces personal bias into the statistical analysis. However, Bayesians argue that all modeling involves subjective choices.
What is the difference between one-way and two-way ANOVA?
- One-way ANOVA compares one group, two-way ANOVA compares two groups
- One-way ANOVA compares two groups, two-way ANOVA compares more than two groups
- One-way ANOVA considers one independent variable, two-way ANOVA considers two independent variables
- One-way ANOVA considers two independent variables, two-way ANOVA considers one independent variable
The key difference between one-way and two-way ANOVA lies in the number of independent variables they consider. A one-way ANOVA is used when there is one independent variable, whereas a two-way ANOVA is used when there are two independent variables.
What is the Durbin-Watson statistic used for in residual analysis?
- To check for autocorrelation
- To check for heteroscedasticity
- To check for linearity of the relationship
- To check for normality of residuals
The Durbin-Watson statistic is a test statistic used to detect the presence of autocorrelation (a relationship between values separated from each other by a given time lag) in the residuals (prediction errors) from a regression analysis.
How does one interpret the coefficients in a multiple linear regression model?
- They show the average change in the dependent variable for a one unit change in the independent variable, ceteris paribus
- They show the correlation between the dependent and independent variables
- They show the error term in the regression model
- They show the total variation in the dependent variable explained by the independent variables
Each coefficient in a multiple linear regression model represents the average change in the dependent variable for a one unit change in the corresponding independent variable, while keeping all other independent variables constant. This is known as ceteris paribus, or "all else being equal."