How does the choice of significance level (α) affect the conclusion of a Chi-square test for goodness of fit?
- A higher α makes it easier to reject the null hypothesis
- A higher α makes it harder to reject the null hypothesis
- α has no impact on the conclusion of the test
- α only affects the power of the test, not the conclusion
A higher significance level (α) increases the likelihood of rejecting the null hypothesis. This is because you're setting a higher bar for the amount of evidence needed to retain the null hypothesis.
How does the sample size affect the standard error of a sample mean?
- Larger sample sizes decrease the standard error
- Larger sample sizes increase the standard error
- Smaller sample sizes decrease the standard error
- The sample size has no effect on the standard error
The sample size has an inverse relationship with the standard error of a sample mean. As the sample size increases, the standard error decreases. This is because larger samples provide a better approximation of the population, reducing the variability of the sample mean around the population mean.
What does a larger sample size do to the sampling distribution of the mean?
- It decreases the spread of the distribution
- It does not affect the distribution
- It increases the spread of the distribution
- It skews the distribution
A larger sample size decreases the spread of the sampling distribution of the mean. This is because as the sample size increases, the standard error (a measure of the spread of the distribution of sample means) decreases, which means that the sampling distribution becomes more concentrated around the true population mean.
What is the relationship between the eigenvalue of a component and the variance of that component in PCA?
- It depends on the dataset
- There is no relationship
- They are directly proportional
- They are inversely proportional
The eigenvalue of a component in PCA is directly proportional to the variance of that component. In other words, a larger eigenvalue corresponds to a larger amount of variance explained by that principal component.
What conditions must be met for the Central Limit Theorem to hold true?
- The data must be collected without any bias.
- The data must be normally distributed.
- The sample must be a simple random sample, and the sample size must be sufficiently large (typically n > 30).
- The sample size must be less than 30.
The Central Limit Theorem generally applies when the following conditions are met: 1) The data should be sampled randomly, 2) The sample values must be independent of each other, and 3) The sample size should be sufficiently large (typically, n > 30 is considered sufficient).
In hypothesis testing, a Type II error is committed when the null hypothesis is ______ but we ______ to reject it.
- False, fail to reject
- False, reject
- True, fail to reject
- True, reject
A Type II error, also known as a false negative, occurs when we fail to reject a false null hypothesis. This means we've missed evidence of an effect or difference that truly exists.
What kind of relationship does Pearson's Correlation Coefficient measure?
- Exponential
- Linear
- Monotonic
- Non-linear
Pearson's correlation coefficient measures linear relationships between variables. It measures the degree to which pairs of data for these two variables lie on a line.
What is the main difference between the Wilcoxon Signed Rank Test and the paired t-test?
- All of the above
- The Wilcoxon test is non-parametric while the t-test is parametric
- The Wilcoxon test is used for ordinal data while the t-test is used for continuous data
- The Wilcoxon test uses ranks while the t-test uses actual values
The Wilcoxon Signed Rank Test is a non-parametric test that uses ranks and is used for ordinal data, while the paired t-test is a parametric test that uses actual values and is typically used for continuous data.
The ________ in a two-way ANOVA can reveal whether the effect of one independent variable depends on the level of the other independent variable.
- Effect size
- Interaction effect
- Main effect
- Post-hoc test
The interaction effect in a two-way ANOVA reveals whether the effect of one independent variable depends on the level of the other independent variable. This allows us to understand how the independent variables relate to each other.
How is Bayes' theorem related to conditional probability?
- Bayes' theorem and conditional probability are not related
- Bayes' theorem cannot be used with conditional probability
- Bayes' theorem is a specific type of conditional probability
- Bayes' theorem is used to calculate the complement of the conditional probability
Bayes' theorem is a way of finding a probability when we know certain other probabilities. The probabilities that we know are usually conditional probabilities, and Bayes' theorem is used to 'reverse' these probabilities.