9231 · 4.2
Inference using normal and t-distributions flashcards
Revision flashcards for Cambridge 9231 Inference using normal and t-distributions (syllabus 4.2). Flip, recall, then mark a real past-paper question.
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What is statistical inference?
The process of using data from a sample to deduce properties of an underlying population.
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When do you use the normal distribution for inference on the mean?
1. When the population variance (σ²) is known, and the population is normal (or n is large, by CLT). 2. When σ² is unknown, but the sample size (n) is large (typically n > 30), using the sample variance as an estimate.
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When do you use the t-distribution for inference on the mean?
When the population variance (σ²) is unknown, the sample size (n) is small, and the parent population is assumed to be normally distributed.
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What is a confidence interval for a mean?
A range of values, calculated from sample data, that is likely to contain the true population mean with a certain level of confidence.
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What is the formula for the unbiased estimate of population variance, $s^2$?
$s^2 = \frac{1}{n-1} \sum (x_i - \bar{x})^2$. This is the value you must use when the population variance is unknown.
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What is the test statistic for a hypothesis test using the normal distribution?
$Z = \frac{\bar{X} - \mu_0}{\sigma / \sqrt{n}}$ if $\sigma$ is known, or $Z = \frac{\bar{X} - \mu_0}{s / \sqrt{n}}$ if $n$ is large and $\sigma$ is unknown.
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What is the test statistic for a hypothesis test using the t-distribution?
$T = \frac{\bar{X} - \mu_0}{s / \sqrt{n}}$, where $s$ is the unbiased estimate of the population standard deviation. This statistic follows a t-distribution with $n-1$ degrees of freedom.
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What are the degrees of freedom for a single-sample t-test?
The degrees of freedom (df) are $v = n - 1$, where $n$ is the sample size.
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What is the general formula for a confidence interval?
Point Estimate ± (Critical Value) × (Standard Error). For a mean, this is $\bar{x} \pm \text{critical value} \times \frac{\text{std dev}}{\sqrt{n}}$.
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What is a common trap when calculating a t-statistic?
Using a biased sample variance (dividing by $n$) instead of the unbiased estimate (dividing by $n-1$). Exam questions may provide $\sum(x-\bar{x})^2$ or a biased variance, requiring you to calculate the correct $s^2$ yourself.
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How does the width of a confidence interval change with the confidence level?
A higher confidence level (e.g., 99% vs 95%) requires a wider interval, as you need a larger range to be more certain it contains the true mean.
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What is a p-value?
The probability of observing a test statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true. If p-value < significance level (α), you reject H₀.