In what situations would a sample not accurately represent the population?
- When the population size is too large
- When the sample is not randomly selected
- When the sample size is too small
- When the sampling method is biased
A sample might not accurately represent the population when the sampling method is biased. In this case, the sample may not be diverse enough or inclusive of all relevant aspects of the population. This can lead to skewed results and inaccurate inferences about the population. Hence, it's essential to choose an unbiased sampling method.
In a _______ distribution, all outcomes are equally likely.
- Bimodal
- Normal
- Skewed
- Uniform
In a uniform distribution, all outcomes are equally likely. This distribution is characterized by two parameters, a and b, which are the minimum and maximum values, respectively. The probability of any outcome is constant and equal across the entire range of the distribution.
How is a confidence interval calculated in statistics?
- By calculating the median and the mode
- By multiplying the sample size by the standard deviation
- By squaring the sample mean
- By using the sample mean plus and minus the standard error
A confidence interval is calculated using the sample mean plus and minus the standard error. Specifically, it is calculated by taking the point estimate and adding/subtracting the margin of error (which is the standard error multiplied by the relevant Z-value or T-value).
What are the consequences of using too large or too small a sample size in hypothesis testing?
- The sample size does not influence hypothesis testing
- Too large a sample size can dilute the effect size, and too small can exaggerate it
- Too large a sample size can lead to overfitting, and too small can lead to underfitting
- Too large a sample size can overstate evidence against the null hypothesis, and too small can lack the power to detect an effect
With a large sample size, small differences may become statistically significant, which can lead to overstating the evidence against the null hypothesis. In contrast, with a small sample size, we might not have enough power to detect an effect, even if one exists.
What does the shape of the probability density function of a normal distribution look like?
- It is skewed to the left
- It is skewed to the right
- It is symmetric and bell-shaped
- It is uniform
The probability density function of a normal distribution is symmetric and bell-shaped. It is characterized by its mean and standard deviation, with the mean indicating the center of the distribution and the standard deviation indicating the width or spread.
What does ANOVA stand for in statistics?
- Analysis of Variance
- Analytic Normality of Variables
- Automatic Number Variance Analysis
- Average Numerical Outcome Variance
ANOVA stands for Analysis of Variance. It's a statistical method used to test the differences between two or more means.
A negative value of skewness indicates that the distribution is skewed to the ________.
- Left
- Middle
- Right
- nan
A negative value of skewness indicates that the distribution is skewed to the left, meaning that the left tail of the distribution is longer or fatter than the right tail.
Explain the concept of conditional independence in probability theory.
- It is another term for mutual exclusivity
- It means that the independence of two events does not depend on the occurrence of any other events
- It means that two events are independent only when a third event does not occur
- It means that two events are independent only when a third event occurs
Conditional independence in probability theory refers to a situation where two events are independent given the occurrence of a third event. Mathematically, two events A and B are conditionally independent given a third event C if the probability of the intersection of A and B given C is the product of the probabilities of A given C and B given C.
What does the likelihood in Bayes' theorem represent?
- The posterior probability of the event
- The prior probability of the event
- The probability of the event given the evidence
- The probability of the evidence given the event
The likelihood in Bayes' theorem represents the probability of the evidence given the event. It quantifies the extent to which the evidence supports the event.
In polynomial regression, overfitting can occur when the degree of the polynomial is excessively ________.
- High
- Low
- Middle
- Zero
Overfitting can occur when the degree of the polynomial is excessively high. Overfitting refers to a situation where a model is too complex and captures not just the underlying pattern but also the noise in the data. A high-degree polynomial may fit the training data very well, but it may perform poorly on new, unseen data.