Justifying a Claim about a Slope
Introduced| English |
|---|
| linear relationship/ˈlɪnɪə rɪˈleɪʃənʃɪp/ |
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 $\beta$.
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 $0$.
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.
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.
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.
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.
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.
Does the slope differ from zero?
A slope of 0 (flat line) means no linear relationship.
A 95% interval for a slope is (2.1, 6.3). You can conclude...
The interval is entirely positive (above 0).
A 95% interval for a slope is (−1.0, 3.4). You can conclude...
The interval contains 0, so a flat slope is plausible.
For a slope, the value that represents 'no linear relationship' is...
β = 0 means x doesn't help predict y.
A slope interval that contains 0 proves there is no relationship at all.
It only means a flat slope can't be ruled out.
A slope of β = 0 corresponds to no ___ relationship (one word).
β = 0 means no linear relationship.