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What is the Central Limit Theorem (CLT)?
The distribution of sample means will be approximately normal, regardless of the population's distribution, if sample size is large enough.
Define 'sample mean'.
The average of a set of observations taken from a larger population.
What is a 'simple random sample' (SRS)?
A sample where each member of the population has an equal chance of being chosen.
Define 'independence' in the context of sampling.
Each data point in the sample does not influence the others.
What is a 'sampling distribution'?
The probability distribution of a statistic (like the sample mean) derived from all possible samples of a given size from a population.
Explain the importance of the Central Limit Theorem.
Allows us to make inferences about population means using sample means, even when the population distribution is unknown.
Explain how sample size affects the sampling distribution.
Larger sample sizes result in a sampling distribution that is less spread out and more closely approximates a normal distribution.
Explain the concept of 'normality' in the context of the CLT.
The CLT states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the population's distribution.
Explain the role of randomness in the CLT.
Random sampling reduces bias and ensures that the sample is representative of the population, which is crucial for the CLT to hold.
Explain why a large sample is important when using the CLT.
A large sample size ensures that the sampling distribution of the sample mean is approximately normal, even if the population distribution is not. It also reduces variability.
What are the differences between population distribution and sampling distribution?
Population distribution: Distribution of all values in a population | Sampling distribution: Distribution of a statistic (e.g., sample mean) from multiple samples taken from the population.
What are the differences between a sample and a population?
Sample: A subset of the population used for analysis. | Population: The entire group of individuals or items of interest.
What are the differences between using CLT with small and large sample sizes?
Small sample size: CLT may not apply if the population is not normal; sampling distribution may not be normal. | Large sample size: CLT applies; sampling distribution is approximately normal regardless of population distribution.
What are the differences between a parameter and a statistic?
Parameter: A value that describes a population. | Statistic: A value that describes a sample.
What are the differences between a biased and unbiased sample?
Biased sample: A sample that does not accurately represent the population. | Unbiased sample: A sample that accurately represents the population.