Justifying a Claim about a Difference
IntroducedA positive estimate can still be uncertain
- For library 1 minus library 2, an interval $(-0.04,0.24)$ includes zero. It permits differences in either direction as well as equality.
- A positive point estimate alone is not enough to conclude that the first population proportion is higher.
Read zero as the equal-proportion claim
- In a difference interval, zero means $p_1-p_2=0$. It does not mean that nobody in either group supports the proposal.
- An interval containing zero leaves equality plausible under the interval procedure. It does not prove the populations are exactly equal.
For a difference interval, the value that represents 'no difference' is...
p1 − p2 = 0 means the two proportions are equal.
Inspect the direction of the whole interval
- An interval $(0.02,0.15)$ is entirely positive: it supports a higher proportion in population 1 by this interval procedure.
- An interval $(-0.15,-0.02)$ supports a lower proportion in population 1. Name the group order before interpreting the signs.
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.
Match each interval for p1 - p2 to its interval-based conclusion.
Check the sign of the entire interval and whether it contains zero.
Express differences in percentage points
- A difference of 0.02 is two percentage points. The interval $(0.02,0.15)$ spans two to fifteen percentage points.
- A relative percentage change also needs a reference proportion. Do not label the same endpoints as a 2% to 15% relative increase without that calculation.
An interval entirely below zero supports p₂ > p₁. Read the subtraction order before naming the higher group.
An interval containing 0 proves the two proportions are exactly equal.
It only means equality can't be ruled out — not proof.
Express the lower endpoint 0.02 as a difference in percentage points; type the number only.
A proportion difference of 0.02 is two percentage points, not a two-percent relative increase.
Proportions are 0.60 and 0.50. What is their difference in percentage points?
(0.60 - 0.50) × 100 = 10 percentage points. The relative increase from 0.50 is 20%, a different measure.
Do not swap in a different test automatically
- The usual interval has an unpooled standard error; the equality test uses a pooled standard error. They need not give identical borderline decisions.
- Use the interval to describe plausible differences. If the question asks for a hypothesis test, perform that specified test and state its alternative.
Containing zero leaves equality plausible under the interval method; it does not prove no difference.
Conclude only what the interval supports
- For $(-0.04,0.24)$, say the interval does not establish a population difference. Avoid claiming equal support or zero practical effect.
- The wide interval still allows important differences. Discuss its endpoints and study assumptions instead of treating non-exclusion of zero as proof of no effect.
For $(-0.04,0.24)$, say the interval does not establish a population difference. Avoid claiming equal support or zero practical effect.