9709 · 6.4
Sampling and estimation flashcards
Revision flashcards for Cambridge 9709 Sampling and estimation (syllabus 6.4). Flip, recall, then mark a real past-paper question.
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What is a population?
The entire set of items or individuals from which a sample is drawn. For example, all the light bulbs produced by a factory.
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What is a sample?
A subset of the population selected for observation and analysis. It should ideally be random to be representative.
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What is an unbiased estimator?
A statistic whose expected value (the average of its values from all possible samples) is equal to the population parameter it is estimating. E.g., $E(\bar{X}) = \mu$.
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What is the formula for an unbiased estimate of the population mean, $\mu$?
The sample mean, $\bar{x} = \frac{\sum x}{n}$.
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What is the formula for an unbiased estimate of the population variance, $\sigma^2$?
Denoted $s^2$, it is calculated as $s^2 = \frac{1}{n-1} \left( \sum x^2 - \frac{(\sum x)^2}{n} \right)$. Note the $\frac{1}{n-1}$ factor.
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Why do we divide by $n-1$ for the unbiased estimate of variance?
Dividing by $n$ tends to underestimate the true population variance. Dividing by $n-1$ (Bessel's correction) corrects for this bias, making the estimate more accurate on average.
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State the Central Limit Theorem (CLT).
If a random sample of size $n$ (where $n$ is large, typically $n > 30$) is taken from any population with mean $\mu$ and variance $\sigma^2$, then the sample mean $\bar{X}$ is approximately normally distributed with mean $\mu$ and variance $\frac{\sigma^2}{n}$. So, $\bar{X} \sim N(\mu, \frac{\sigma^2}{n})$ approximately.
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What is a confidence interval?
A range of values, calculated from sample data, that is likely to contain an unknown population parameter. The confidence level gives the probability that the method produces an interval containing the parameter.
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What is the formula for a confidence interval for the mean $\mu$ when $\sigma^2$ is known?
$\bar{x} \pm z \times \frac{\sigma}{\sqrt{n}}$, where $z$ is the critical value from the standard normal distribution corresponding to the desired confidence level.
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What are the z-values for 90%, 95%, and 99% confidence intervals?
90% CI: $z = 1.645$ 95% CI: $z = 1.960$ 99% CI: $z = 2.576$
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True or False: A 95% confidence interval means there is a 95% probability that the true population mean $\mu$ lies within my calculated interval.
False. This is a common misconception. The correct interpretation is that if we were to take many samples and construct an interval for each, 95% of those intervals would contain the true mean $\mu$. The mean $\mu$ is a fixed value, not a random variable.
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How does increasing the sample size, $n$, affect the width of a confidence interval?
Increasing the sample size $n$ decreases the width of the confidence interval, as $\frac{\sigma}{\sqrt{n}}$ gets smaller. This means our estimate becomes more precise.