The p-value in a hypothesis test is the probability of getting a sample statistic as extreme as the test statistic, given that the _______ hypothesis is true.
- Alternative
- Null
- Original
- Random
In the context of hypothesis testing, the p-value is the probability of observing a test statistic as extreme as the one calculated, assuming that the null hypothesis is true.
What are the assumptions required for a distribution to be considered a Poisson distribution?
- The events are dependent on each other
- The events are occurring at a constant mean rate and independently of the time since the last event
- The events have more than two possible outcomes
- The number of trials is fixed
The key assumptions for a Poisson distribution are that the events are happening at a constant mean rate and independently of the time since the last event. This is often used for modeling the number of times an event occurs in a given interval of time or space.
What's the difference between a histogram and a bar plot?
- Bar plots are for continuous data, histograms for categorical data
- Both are for continuous data only
- Histograms are for continuous data, bar plots for categorical data
- There is no difference
The main difference between a histogram and a bar plot is the type of data they represent. A histogram is used for continuous data, where the bins represent ranges of data, while a bar plot is used for categorical data to compare the frequency or count of different categories.
What is the error term in a simple linear regression model?
- It is the dependent variable
- It is the difference between the observed and predicted values
- It is the independent variable
- It is the slope of the regression line
The error term in a simple linear regression model is the difference between the observed and predicted values. It captures the variability in the dependent variable that is not explained by the independent variable in the model.
What can be inferred if the residuals are not randomly distributed in the residual plot?
- The data has no outliers
- The data is perfectly linear
- The linear regression model is a perfect fit for the data
- The linear regression model is not a good fit for the data
If the residuals are not randomly distributed (e.g., if they form a pattern), it suggests that the linear regression model is not a good fit for the data. This could be because the relationship between the variables is not linear, or because the data exhibits heteroscedasticity (unequal variances of errors), among other reasons.
What type of data is used in the Chi-square test for goodness of fit?
- Categorical data
- Continuous data
- Interval data
- Ordinal data
The Chi-square test for goodness of fit is used with categorical data. It compares the observed frequencies in each category with the frequencies we would expect to see if the data followed the theoretical distribution.
What is the null hypothesis in the Mann-Whitney U test?
- The groups have different variances
- The groups have equal variances
- There is a significant difference between the groups
- There is no significant difference between the groups
In the Mann-Whitney U test, the null hypothesis is that there is no significant difference between the groups. More specifically, it states that the probability that a randomly selected value from the first group is greater than a randomly selected value from the second group is equal to 0.5.
How does sample size affect the width of a confidence interval?
- Increasing the sample size decreases the width of the confidence interval
- Increasing the sample size has no effect on the width of the confidence interval
- Increasing the sample size increases the width of the confidence interval
- The relationship between sample size and the width of the confidence interval is unpredictable
Increasing the sample size decreases the width of the confidence interval. The larger the sample size, the more information you have, and thus the less uncertainty (which translates into a smaller standard error and narrower confidence interval).
If A and B are independent events, the probability of both occurring is ________.
- P(A + B)
- P(A / B)
- P(A ∩ B)
- P(A ∪ B)
If A and B are independent events, the probability of both occurring is P(A ∩ B) which is equal to P(A) * P(B). This is the fundamental characteristic of independent events in probability.
Why is it important to consider the power of a test when designing a study?
- To ensure the study can detect an effect if it exists
- To ensure the study does not detect an effect if it does not exist
- To maximize the chance of a Type I error
- To minimize the chance of a Type I error
The power of a test is the ability of the test to detect an effect if it truly exists. It's the probability that the test correctly rejects a false null hypothesis. High power is desirable because it means the test is less likely to make a Type II error (false negative). When designing a study, it's important to choose a sample size and significance level that will provide enough power to detect an effect if one exists.