All Flashcards
Explain the concept of the Central Limit Theorem (CLT) in the context of t-tests.
If the sample size (n) is at least 30, the sampling distribution of the sample mean is approximately normal, allowing us to use the t-distribution.
Explain the importance of random sampling in a t-test.
Random sampling ensures that the sample is representative of the population, allowing for valid inferences about the population mean.
Explain the independence condition in the context of t-tests.
The population size must be at least 10 times the sample size (10n) to ensure that the observations are independent.
Explain the relationship between the p-value and the significance level (α).
If the p-value is less than or equal to α, we reject the null hypothesis. If the p-value is greater than α, we fail to reject the null hypothesis.
Explain Type I error.
Rejecting the null hypothesis when it is true. The probability of committing a Type I error is equal to the significance level (α).
Explain the purpose of checking conditions before performing a t-test.
Checking conditions (randomness, independence, normality) ensures that the assumptions underlying the t-test are met, making the results of the test valid and reliable.
What are the differences between a one-tailed and a two-tailed t-test?
One-tailed: Tests for a directional difference (μ < μ₀ or μ > μ₀). Rejection region is on one side of the distribution. | Two-tailed: Tests for any difference (μ ≠ μ₀). Rejection region is on both sides of the distribution.
What are the differences between using a t-test and a z-test?
T-test: Used when the population standard deviation (σ) is unknown and estimated from the sample. | Z-test: Used when the population standard deviation (σ) is known.
What are the differences between the null and alternative hypotheses?
Null Hypothesis: A statement of no effect or no difference (μ = μ₀). It is what we are trying to disprove. | Alternative Hypothesis: The opposite of the null hypothesis, what we are trying to find evidence for (μ ≠ μ₀, μ < μ₀, or μ > μ₀).
What are the differences between Type I and Type II errors?
Type I Error: Rejecting the null hypothesis when it is true (false positive). Probability is α. | Type II Error: Failing to reject the null hypothesis when it is false (false negative). Probability is β.
What is the significance level (α)?
The probability of rejecting the null hypothesis when it is actually true (Type I error).
Define the null hypothesis (H₀).
A statement of no effect or no difference, which we are trying to disprove. Expressed as H₀: μ = μ₀.
Define the alternative hypothesis (Hₐ).
The opposite of the null hypothesis. It is what we are trying to find evidence for (μ ≠ μ₀, μ < μ₀, or μ > μ₀).
What is a one-sample t-test?
A test used to compare a sample mean to a known or hypothesized population mean when the population standard deviation (σ) is unknown.
What is the rejection region?
The area under the t-distribution curve where, if our sample statistic falls, we reject the null hypothesis.