What is a residual in the context of simple linear regression?
- The difference between the observed and predicted values
- The difference between the predicted and observed values of the independent variable
- The error in the slope of the regression line
- The observed value of the dependent variable
A residual is the difference between the observed value of the dependent variable (y) and the predicted value (ŷ), given by the regression model. It represents the error of the estimate.
How do changes in the scale of measurement affect the correlation coefficient?
- They decrease the correlation coefficient
- They do not affect the correlation coefficient
- They increase the correlation coefficient
- They reverse the sign of the correlation coefficient
The correlation coefficient is not affected by changes in the center (mean) or scale (standard deviation) of the variables. This is because correlation measures the strength of a relationship between variables relative to their variability. It's a dimensionless quantity, so changes in the scale of measurements of the variables do not change it.
How is the Bartlett's Test of Sphericity used in factor analysis?
- It tests the assumption of equal variances
- It tests the assumption of linearity
- It tests the assumption of normality
- It tests the assumption that the variables are uncorrelated
Bartlett's Test of Sphericity is used in factor analysis to test the hypothesis that the variables are uncorrelated in the population. In other words, the population correlation matrix is an identity matrix. A significant test indicates that a factor analysis may be useful with your data.
In multiple linear regression, what does each coefficient represent?
- The average change in the dependent variable for one unit of change in the independent variable, holding all other independent variables constant
- The correlation between the dependent variable and the independent variable
- The error term in the regression model
- The total variation in the dependent variable explained by the independent variable
In multiple linear regression, each coefficient represents the average change in the dependent variable for one unit of change in the independent variable, while holding all other independent variables constant.
A wider confidence interval indicates a higher level of _______ about the estimate.
- Certainty
- Standard deviation
- Uncertainty
- Variance
A wider confidence interval suggests a higher level of uncertainty about the estimate because the range of values for the population parameter is larger.
How does PCA relate to the Singular Value Decomposition (SVD) technique?
- PCA can be implemented using SVD
- SVD is a prerequisite for PCA
- SVD is a type of PCA
- They are entirely different techniques
PCA can be implemented using SVD. Both techniques can be used for dimensionality reduction, and they both rely on eigenvalue decomposition, but SVD decomposes the data matrix directly, while PCA works on the covariance matrix of the data.
The Chi-square test for goodness of fit tests the hypothesis that the observed distribution follows a ________ distribution.
- expected
- normal
- skewed
- uniform
The Chi-square test for goodness of fit is used to determine whether the observed distribution of data follows an expected distribution.
In Bayes' theorem, what does the prior probability represent?
- The likelihood of the evidence
- The probability of an event before evidence is observed
- The probability of the evidence given the event
- The updated probability after evidence is observed
The prior probability in Bayes' Theorem is the initial or original probability of an event before new evidence is taken into account. It represents our initial belief about the event.
A histogram with two peaks is known as a ________ distribution.
- Bimodal
- Multimodal
- Normal
- Uniform
A histogram with two distinct peaks is referred to as a bimodal distribution. This might suggest that the data contains two different groups, each with their own mode, or most common value.
What assumption does the Chi-square test for goodness of fit make about the observations?
- The observations are correlated
- The observations are independent
- The observations are normally distributed
- The observations are paired
The Chi-square test for goodness of fit assumes that the observations are independent, which means that the outcome of one observation does not affect the outcome of another.