The ________ of a random variable is the sum of the probabilities of all possible outcomes.
- Distribution
- Expected value
- Mean
- Variance
The "expected value" of a random variable is the sum of all possible values it can take, each multiplied by the probability of that outcome. It gives us the mean or average value of the random variable and is a fundamental concept in probability theory and statistics.
What assumptions are made when conducting an ANOVA test?
- Independent observations, no outliers, equal sample sizes
- Independent observations, normal distribution of variables, no outliers
- Independent observations, normally distributed residuals, homoscedasticity
- No missing data, normally distributed residuals, no outliers
ANOVA makes three key assumptions: 1) Observations are independent. 2) Residuals (the differences between the observed and predicted values) are normally distributed. 3) The variance of the residuals is the same for all groups (homoscedasticity).
What does a scatter plot with points clustered tightly around a line indicate?
- A strong correlation
- A weak correlation
- An undefined correlation
- No correlation
When points in a scatter plot are clustered tightly around a line, it indicates a strong correlation between the two variables. The line is typically a line of best fit or regression line.
The _________ states that the sampling distribution of the sample means approaches a normal distribution as the sample size gets larger—no matter what the shape of the population distribution.
- Central Limit Theorem
- Law of Large Numbers
- Probability Rule
- Sampling Distribution
The Central Limit Theorem states that the sampling distribution of the sample means approaches a normal distribution as the sample size gets larger—no matter what the shape of the population distribution. This allows us to apply normal probability calculations to situations that might not initially seem appropriate for them.
________ is a measure of asymmetry of a probability distribution.
- Mean
- Median
- Mode
- Skewness
Skewness is a measure of the asymmetry of a probability distribution about its mean. It quantifies the direction and extent of skew (departure from horizontal symmetry) in the data.
In _________ sampling, the population is divided into subgroups, and a simple random sample is drawn from each subgroup.
- Cluster
- Simple Random
- Stratified
- Systematic
In stratified sampling, the population is divided into non-overlapping groups, or strata, such as age groups, income levels, or gender. Then, a simple random sample is taken from each stratum. Stratified random sampling can provide more precise estimates if the strata are relevant to the characteristic of interest.
A low p-value (less than 0.05) in a t-test suggests that you can reject the _______ hypothesis.
- alternative
- both a and b
- nan
- nan
A low p-value in a t-test suggests that you can reject the null hypothesis. The p-value represents the probability that the results are due to random chance, so a lower p-value means the results are less likely to be due to chance.
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.