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어휘

English	의미
linear relationship/ˈlɪnɪə rɪˈleɪʃənʃɪp/	선형 관계

English

Interpreting the slope interval

Read it as: "We're $95\%$ confident the true slope $\beta$ is between the endpoints."
Translate the slope: predicted change in $y$ per one-unit increase in $x$, in context.
Positive endpoints → each extra unit of $x$ predicts more $y$; negative → less.
The interval is a range of plausible values for $\beta$.

English

Is a linear relationship plausible?

The key question: is there really a linear relationship 线性关系 between $x$ and $y$?
"No linear relationship" corresponds exactly to slope $\beta = 0$.
A flat line ($\beta = 0$) means $x$ doesn't help predict $y$.
So we ask where the interval sits relative to $0$.

English

The role of zero

Interval contains $0$ → a slope of $0$ is plausible → not convincing evidence of a linear relationship.
Interval entirely above or below $0$ → convincing evidence of a (positive or negative) linear relationship.
The position of zero decides the claim.
An all-positive interval also tells you the direction of the association.

English

Justify the conclusion

Tie the conclusion to the interval and to zero, in context.
"Since $(2.1, 6.3)$ is entirely positive, there's convincing evidence of a positive linear relationship."
Or "Since $(-1.0, 3.4)$ contains $0$, there's no convincing evidence of a linear relationship."
Phrase it about the population slope and the real variables.

English

Watch out

For a slope interval, zero means "no linear relationship." A captured $0$ says a flat line is plausible — so you can not conclude $x$ and $y$ are linearly related. This matches a two-sided $t$-test of $H_0: \beta = 0$. Don't read an interval containing $0$ as proof there's no relationship — it just can't rule a flat slope out.

English

Example

Two $95\%$ intervals for a slope $\beta$:

$(2.1,\ 6.3)$: entirely positive → convincing evidence of a positive linear relationship.
$(-1.0,\ 3.4)$: contains $0$ → no convincing evidence of a linear relationship.
The captured $0$ flips the conclusion.

English

Key idea

A slope interval estimates $\beta$. If it contains $0$, a flat line is plausible — no convincing evidence of a linear relationship; if it's entirely above or below $0$, there is convincing evidence (with direction). Justify the conclusion about the population slope in context.

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