| English | 의미 |
|---|---|
| paired/peəd/ | 쌍 |
| independent samples/ˌɪndɪˈpendənt ˈsæmplz/ | 독립 표본 |
Pair before calculating the uncertainty
- The same three runners have before/after times (12,10), (15,12) and (11,10) minutes. Take before minus after.
- The paired differences are 2, 3 and 1 minutes, giving a mean difference of 2. The observational units for this analysis are three pairs, not six independent measurements.
Independent or paired?
- Comparing two means starts with one crucial question about the design. Independent samples 独立样本: two separate groups (e.g. treatment vs. control).
- Paired (matched-pairs) 配对: two measurements on the same or matched subjects (before/after). The design decides which procedure you use — get it right first.
The two-sample interval
- For independent samples, use a two-sample $t$-interval for $\mu_1 - \mu_2$: $$(\bar{x}_1 - \bar{x}_2) \pm t^{*}\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$$
- The variances add under the root (each group's $\frac{s^2}{n}$). Check random design, independence between groups, the 10% condition when sampling without replacement, and the data shape in each group.
The paired interval
- For paired data, first compute each pair's difference $d = x_1 - x_2$. Then run a one-sample $t$-interval on the differences:
- $$\bar{d} \pm t^{*}\frac{s_d}{\sqrt{n}}$$$n$ is the number of pairs; $df = n - 1$. Check random design and independence across pairs, and inspect the differences for skew/outliers.
훈련 전후 동일한 달리기 선수들을 측정하는 것은 어떤 설계인가?
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
동일한 주체를 두 번 측정함 → 짝지음.
짝지음 데이터에 올바른 구간은 ...
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페어를 차이로 압축한 후 일표본 t를 적용한다.
이표본 t-구간의 경우 두 분산은 루트 아래에 더해진다.
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SE = √(s1²/n1 + s2²/n2).
짝지음 데이터를 독립적인 두 표본으로 분석하면 올바른 표준오차를 얻는다.
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It ignores dependence within pairs. The resulting standard error need not always be larger; analyse the paired differences.
절차를 결정하는 핵심 질문은 ...
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연결됨 → 짝지음; 연결되지 않음 → 이표본.
설계를 도형에 대조하여 t 분석을 위한 조건을 확인하십시오.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
쌍별(t) 분석은 차이값을 관측치로 사용합니다. 가장자리 분포가 정態적(정규분포를 닮은)인 집단은 자동으로 적절한 차이값 분포를 보장하지는 않습니다.
쌍별(t) 분석에는 어떤 조건들이 필요한가요?
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
쌍 내의 종속관계는 차이값을 사용하는 이유입니다. 독립성은 서로 다른 쌍들 사이에서 요구됩니다.
Match the method to the design
- Paired data analyzed as two independent samples → wrong standard error, wrong answer. Independent data forced into a paired test → also wrong.
- The tell: is each value in group $1$ naturally linked to one in group $2$? If yes → paired; if no → two-sample.
The design dictates the procedure. If subjects are matched or measured twice (before/after), it's paired — collapse to differences and use a one-sample $t$-interval. If the two groups are separate and unlinked, it's a two-sample $t$-interval. Using the two-sample formula on paired data ignores the within-pair dependence and generally gives the wrong standard error. The direction of the error depends on the relationship within pairs.
Measure each runner's time before and after training.
- Same runners, two measurements → paired.
- Compute differences $d = \text{before} - \text{after}$ for each runner.
- Build a one-sample $t$-interval on the $d$'s (not a two-sample interval).
Carry the reasoning to a new case
- The paired standard error uses the variability of the differences.
- Ignoring pairing does not always inflate the standard error; its effect depends on the within-pair relationship.
For the three stated pairs, compute the mean before-minus-after difference in minutes.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
The differences 2, 3 and 1 sum to 6, so the mean across three pairs is 2.
Comparing two means: independent samples use a two-sample $t$-interval $(\bar{x}_1 - \bar{x}_2) \pm t^{*}\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$; paired data collapse to differences and use a one-sample $t$-interval $\bar{d} \pm t^{*}\frac{s_d}{\sqrt{n}}$. Match the method to the design, checking conditions for both groups.