Inference using normal and t-distributions
Introduced| English | Meaning |
|---|---|
| sample/ˈsæmpl/ | A subset of a population selected for observation, measurement or analysis. |
| distribution/ˌdɪstrɪˈbjuːʃn/ | The pattern of values or outcomes and their frequencies or probabilities. |
| variance/ˈveərɪəns/ | A measure of spread based on the squared deviations of values from their mean. |
| tail/teɪl/ | tail |
| confidence interval/ˈkɒnfɪdəns ˈɪntəvl/ | An interval calculated from sample data using a method with a stated long-run probability of containing the population parameter. |
| degrees of freedom/dɪˈɡriːz ɒv ˈfriːdəm/ | The number of independent values free to vary after relevant constraints have been imposed. |
| standard error/ˈstændəd ˈerə/ | The standard deviation of a statistic's sampling distribution, often estimated from a sample. |
| paired/peəd/ | Describing observations linked in meaningful pairs, such as repeated measurements on the same individual. |
| pooled estimate/puːld ˈestɪmət/ | In the common-variance model, an estimate combining independent sample variances with weights based on their degrees of freedom. |
Small samples 样本, honest uncertainty
- With a huge sample you can use the normal distribution 分布. But with only, say, 8 measurements and an unknown spread, the normal is too confident.
- William Gosset, brewing at Guinness, invented the $t$-distribution to handle exactly this — small samples with unknown variance 方差.
When to use the t-distribution
- Use the $t$-distribution when the sample is small and the population variance is unknown (you estimate it with the sample's $s$).
- It is bell-shaped like the normal but with heavier tails 尾部 — wider, to admit the extra uncertainty.
The t-distribution is bell-shaped like the normal but with heavier tails; as the sample grows it approaches the normal.
Inference & the normal curve
P(Z < z)
Tests and intervals read areas under the normal curve — slide z to find them.
You use the t-distribution (rather than the normal) when:
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Small n with unknown σ → estimate with s and use t.
The t-distribution has lighter tails than the normal distribution.
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False. The t-distribution has heavier tails, reflecting the extra uncertainty from estimating the variance; it approaches the normal as the degrees of freedom grow.
Confidence interval 置信区间 for the mean
- A confidence interval for the population mean:
- $t$ comes from the $t$-tables with $n - 1$ degrees of freedom 自由度; $\dfrac{s}{\sqrt n}$ is the standard error 标准误.
For a single sample of size n = 10, how many degrees of freedom does the t-value use?
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Degrees of freedom = n − 1 = 9.
The quantity s/√n is called the standard ______.
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The standard error of the mean is s/√n.
Worked example
- A sample of $n = 10$ has $s = 4$; for 95% with 9 degrees of freedom, $t = 2.262$.
- Margin $= t\dfrac{s}{\sqrt n} = 2.262 \times \dfrac{4}{\sqrt{10}} = 2.86$ (2 dp).
The t-distribution has a lower peak and heavier tails than the normal
A sample n = 10 has s = 4. With t = 2.262, what is the margin t × s/√n? (2 dp)
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2.262 × 4/√10 = 2.262 × 1.265 = 2.86.
Comparing two populations
- A 2-sample or paired 配对 $t$-test compares two populations.
- When the two are assumed to share a variance, combine them into a pooled estimate 合并估计 of $s^2$.
Degrees of freedom, not $n$. The $t$-value uses $n - 1$ degrees of freedom, not $n$. Reading the wrong row of the table is the most common error.
- For a small sample, base a hypothesis test on the $t$-distribution: a 2-sample t-test or a paired sample t-test.
You've got it
- small sample + unknown variance → use the $t$-distribution ($n-1$ degrees of freedom)
- CI for the mean: $\bar{x} \pm t\dfrac{s}{\sqrt{n}}$, with standard error $\dfrac{s}{\sqrt n}$
- compare two populations with a 2-sample or paired $t$-test (pool the variance when shared)
Write from 15 degrees of freedom
- A single sample has size 16.
- Find the degrees of freedom for the t-value. The table uses n minus 1, not n.
A single sample has size 16. How many degrees of freedom does the t-value use, and why is that not 16?
One degree of freedom is used in estimating the sample standard deviation, leaving n minus 1.
How many degrees of freedom are used?
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16 minus 1 is 15. Other options — 16 (the sample size); 9 (the lesson value for a sample of size 10).