Justifying a Claim about a Slope

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 $0$.

If 0 is outside the interval for β there is convincing evidence of a linear relationship

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.

Write from zero outside a slope interval

• A confidence interval for a slope is 0.2 to 1.1.
• The question is whether a linear relationship is plausible.