Justifying a Claim about a Difference
Interpreting the difference interval
- Read it as: "We're $95\%$ confident the true difference $p_1 - p_2$ lies between the endpoints."
- Name both groups and what the proportion measures, in context.
- A positive interval suggests group $1$ is higher; a negative one, group $2$.
- The interval estimates how much the groups differ, with uncertainty.
Is a real difference plausible?
- The key question: do the two groups genuinely differ?
- Look at where the interval sits relative to zero.
- The interval is a set of plausible values for $p_1 - p_2$ — including, possibly, $0$.
- "No difference" corresponds exactly to $p_1 - p_2 = 0$.
The role of zero
- If the interval contains $0$: "no difference" is plausible → not convincing evidence of a difference.
- If the interval is entirely above or below $0$: $0$ is implausible → convincing evidence the groups differ.
- The position of zero is the whole decision.
- An all-positive interval also tells you the direction of the difference.
Justify the conclusion
- State a clear conclusion tied to the interval and to zero.
- "Since the interval $(0.02, 0.15)$ is entirely positive, we have convincing evidence group $1$'s proportion is higher."
- Or: "Since $(−0.04, 0.24)$ contains $0$, we lack convincing evidence of a difference."
- Always phrase it about the two population proportions, in context.
Zero is the pivot. For a difference interval, a captured $0$ means "no difference is plausible" — so you can not conclude the groups differ. This mirrors a two-sided test at the matching level. Don't misread a wide interval that includes $0$ as proof the groups are equal — it just means the data can't rule equality out.
Two intervals for $p_1 - p_2$:
- $(0.02,\ 0.15)$: entirely above $0$ → convincing evidence group $1$ is higher.
- $(-0.04,\ 0.24)$: contains $0$ → no convincing evidence of any difference.
- Same estimate can give different conclusions depending on width and position.
A difference-of-proportions interval estimates $p_1 - p_2$. If it contains $0$, "no difference" is plausible — no convincing evidence the groups differ; if it lies entirely above or below $0$, there is convincing evidence (and its sign gives the direction). Justify the conclusion about the two proportions in context.
Where does zero fall?
If the interval misses 0, the groups convincingly differ.
A 95% interval for p1 − p2 is (0.02, 0.15). What can you conclude?
The interval is entirely positive (above 0).
A 95% interval for p1 − p2 is (−0.04, 0.24). What can you conclude?
The interval contains 0, so 'no difference' is plausible.
If a difference interval contains 0, no difference between the groups is plausible.
0 inside → 'no difference' can't be ruled out.
An interval containing 0 proves the two proportions are exactly equal.
It only means equality can't be ruled out — not proof.
For a difference interval, the value that represents 'no difference' is...
p1 − p2 = 0 means the two proportions are equal.