What is the assumption made when computing the Pearson correlation coefficient?
- The correlation is zero
- The variables are independent
- The variables are normally distributed
- There is a linear relationship between variables
When computing the Pearson correlation coefficient, it is assumed that there is a linear relationship between the variables. Furthermore, it's also assumed that the variables are continuous and that the data is homoscedastic (i.e., the variance of the errors is the same across all levels of the variables).
How is the variance related to the standard deviation in a data set?
- The variance is the average of the standard deviation
- The variance is the square of the standard deviation
- The variance is the square root of the standard deviation
- The variance is twice the standard deviation
The variance is the square of the standard deviation. Standard deviation is a measure of dispersion in a dataset and variance is a square of it, meaning that they both represent the same concept of dispersion, but in different units.
What does kurtosis measure in a dataset?
- Central tendency
- Dispersion
- Skewness
- The "tailedness" of the distribution
Kurtosis is a statistical measure that defines how heavily the tails of a distribution differ from the tails of a normal distribution. In other words, kurtosis identifies whether the tails of a given distribution contain extreme values.
A statistical test has more power to detect an effect if the effect size is ______.
- Equal to the sample size
- Large
- Small
- Unchanged
The power of a test is influenced by the effect size - the magnitude of the difference or relationship you're testing for. Larger effect sizes increase the power of a test because they create a larger signal relative to the noise, making it easier to detect an effect if one exists.
How does the Central Limit Theorem relate to the use of Z-tests?
- It allows for the assumption that the sample mean distribution is normally distributed
- It enables the calculation of the sample standard deviation
- It increases the power of the test
- It reduces the impact of outliers in the sample
The Central Limit Theorem states that, with a large enough sample size, the distribution of the sample mean will be approximately normally distributed. This allows us to use Z-tests even when the population is not normally distributed.
In what kind of scenario is the Central Limit Theorem used?
- It's used only when dealing with a uniform distribution.
- It's used to determine whether an event will occur.
- It's used to predict the future.
- It's used when we want to make inferences about a population based on a sample.
The Central Limit Theorem (CLT) is often used in scenarios where we are interested in the average outcome of a large number of independent or nearly independent events. This is commonly the case when we are making inferences about a population based on a sample.
What does a residual plot tell us about the fit of the model?
- It indicates how well the model's predictions match the actual data
- It indicates the variance of the residuals
- It shows the correlation between the dependent and independent variables
- It shows the relationship between the dependent and independent variables
A residual plot shows the residuals on the y-axis and the independent variable on the x-axis. If the points in a residual plot are randomly dispersed around the horizontal axis, a linear regression model is appropriate for the data; otherwise, a non-linear model is more appropriate.
Can PCA be used for both supervised and unsupervised learning?
- No
- Only for supervised learning
- Only for unsupervised learning
- Yes
No, PCA is a technique for unsupervised learning. It does not use any class label information in its algorithm, making it unsupervised. However, the transformed dataset from PCA can be used for subsequent supervised learning tasks.
What is the effect of outliers on PCA?
- It depends on the distribution of the data
- They can distort the principal components
- They enhance the performance of PCA
- They have no effect on PCA
Outliers can significantly distort the principal components identified by PCA, as they can artificially inflate the variance along their direction. It's generally a good practice to address outliers before applying PCA.
What is the concept of "Type I" error in the context of hypothesis testing?
- Failing to reject a false null hypothesis
- Failing to reject a true alternative hypothesis
- Rejecting a false alternative hypothesis
- Rejecting a true null hypothesis
A Type I error in hypothesis testing is the incorrect rejection of a true null hypothesis, often signified by the Greek letter alpha (α). In other words, a Type I error happens when the researcher incorrectly concludes that the null hypothesis is false when, in fact, it is true.