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QLDSpecialist MathematicsQuick questions

Unit 4: Further calculus, and statistical inference

Quick questions on Distribution of the sample mean and the central limit theorem (QCE Specialist Mathematics Unit 4)

6short 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 sample mean is random?
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If you take a random sample of size nn from a population and compute its mean Xˉ\bar{X}, a different sample gives a different mean. So Xˉ\bar{X} is a random variable. Its distribution is called the sampling distribution of the mean, and it has its own mean and standard deviation.
What is the central limit theorem?
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The central limit theorem states that for a sufficiently large sample size nn, the distribution of the sample mean Xˉ\bar{X} is approximately normal,
What are standardising to find probabilities?
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To compute a probability for Xˉ\bar{X}, standardise using the standard error:
What is effect of sample size?
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Because the standard error is σn\dfrac{\sigma}{\sqrt{n}}, quadrupling the sample size halves the spread of Xˉ\bar{X}. This is why larger samples produce tighter estimates and narrower confidence intervals: the sampling distribution concentrates around μ\mu.
What is state the sampling distribution?
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With n=36n = 36, the standard error is
What is interpret?
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There is about a 2.3%2.3\% chance that a sample of 3636 has a mean above 5454. Note that a single observation above 5454 would only be Z=54−5012≈0.33Z = \frac{54-50}{12} \approx 0.33, far more likely; averaging over 3636 values makes a mean that high much rarer.

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