Why might you perform a paired t-test?
- All of the above
- To compare the means of the same group at two different times
- To compare the means of two different populations
- To compare two independent groups
A paired t-test is used to compare the means of the same group at two different times or under two different conditions. It is not used to compare independent groups or different populations.
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.
How does the sample size impact the accuracy of the Central Limit Theorem?
- As the sample size increases, the approximation of the sample mean to a normal distribution becomes more accurate.
- Sample size has no impact on the Central Limit Theorem.
- The Central Limit Theorem becomes less accurate as the sample size increases.
- The Central Limit Theorem is only accurate when the sample size is exactly 30.
According to the Central Limit Theorem, as the sample size increases, the distribution of the sample mean approaches a normal distribution more closely. This means the larger the sample size, the more accurately the sample mean will represent a normal distribution.
Non-parametric statistical methods do not require the data to follow a specific ________.
- distribution
- pattern
- sequence
- trend
Non-parametric statistical methods do not require the data to follow a specific distribution, which is why they are often used when the assumptions of parametric tests are violated.
What does the peak of a distribution represent?
- The mean of the data
- The median of the data
- The mode of the data
- The range of the data
The peak of a distribution represents the mode of the data, that is, the value(s) that appear most frequently in the data set. In a perfectly symmetrical distribution, the mode, median, and mean coincide at the peak.
What is the potential outcome if we fail to reject the null hypothesis?
- The null hypothesis is definitely true
- The sample size was too small
- The significance level was too high
- There is not enough evidence in the data to support the alternative hypothesis
If we fail to reject the null hypothesis, this means that there is not enough evidence in the data to support the alternative hypothesis. We do not say the null hypothesis is true, because it is possible that a type II error (false negative) occurred.
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.