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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 is the general form of the null hypothesis?
H₀: μ = μ₀, where μ is the population mean and μ₀ is the hypothesized population mean.
How to calculate the t-statistic for a one-sample t-test?
t = (x̄ - μ₀) / (s / √n), where x̄ is the sample mean, μ₀ is the hypothesized population mean, s is the sample standard deviation, and n is the sample size.
How to determine the degrees of freedom (df) for a one-sample t-test?
df = n - 1, where n is the sample size.
What is the relationship between α and confidence interval?
Confidence Level = 1 - α. For example, α = 0.05 corresponds to a 95% confidence interval.
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.