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Probability and Statistics

Quick questions on Sampling and the Central Limit Theorem explained: H2 Mathematics Probability and Statistics

7short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is the sampling distribution of the mean?
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If samples of size nn are drawn from a population with mean μ\mu and variance σ2\sigma^2, the sample mean Xˉ\bar{X} has
What is the Central Limit Theorem?
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The Central Limit Theorem (CLT) states that for a sufficiently large sample size nn, the sample mean is approximately normally distributed,
What are unbiased estimators?
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From a sample, the unbiased estimate of the population mean is the sample mean xˉ=∑xn\bar{x} = \dfrac{\sum x}{n}. The unbiased estimate of the population variance uses the n−1n - 1 divisor:
What is regardless of the population's distribution?
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This is what lets us use normal-based methods even when the population is not normal, provided nn is large (commonly n≥30n \geq 30).
What is q1?
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A population has σ=10\sigma = 10. Find the standard error of the mean for a sample of size 2525. [2 marks]
What is q2?
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State what the Central Limit Theorem guarantees about the sample mean. [2 marks]
What is q3?
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Why is the population variance estimated with an n−1n - 1 divisor? [1 mark]

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