Justifying a Claim from an Interval
Interpreting the interval
- Say it right: "We are $95\%$ confident that the true proportion $p$ lies between the endpoints."
- The interval is about the population parameter $p$, in context.
- Name the population and what $p$ measures — not just bare numbers.
- The interval is a range of plausible values for $p$.
What the confidence level means
- The confidence level describes the method, over many samples — not one interval.
- "$95\%$ confident" means: if we repeated the sampling many times, about $95\%$ of the intervals would capture $p$.
- Any single interval either does or doesn't contain $p$ — we just don't know which.
- It is not "a $95\%$ probability that $p$ is in this interval."
Justifying a claim
- To test a claimed value (say $p = 0.5$): is it inside the interval?
- Inside → that value is plausible; you can't rule it out.
- Outside → the interval gives evidence against that value.
- The interval turns "is the claim believable?" into "is it in the range?"
What changes the margin of error
- Larger sample size $n$ → smaller margin of error (narrower, more precise).
- Higher confidence level → larger margin of error (wider, to be surer).
- There's a trade-off: precision vs. confidence.
- To get both narrower and more confident, you need a bigger $n$.
"$95\%$ confidence" is a statement about the method, not a probability for one interval. Never say "there is a $95\%$ chance $p$ is in $(0.5, 0.7)$" — $p$ is fixed, so it's either in or out. The correct idea: $95\%$ of intervals built this way would capture $p$ over many repeated samples.
A $95\%$ interval for the proportion who support a policy is $(0.52,\ 0.61)$.
- Interpret: we're $95\%$ confident the true support is between $52\%$ and $61\%$.
- Claim "half support it" ($p = 0.5$)? $0.5$ is outside $(0.52, 0.61)$ → evidence against it.
- Claim "$55\%$"? $0.55$ is inside → plausible, can't be ruled out.
Interpret a proportion interval as: "$95\%$ confident the true $p$ is between the endpoints," where the confidence level describes the method over repeated samples. A value outside the interval is evidence against that claim. Larger $n$ shrinks the margin of error; higher confidence widens it.
Capturing p over repeated samples
95% of intervals built this way would capture the true p.
The correct interpretation of a 95% confidence level is...
Confidence describes the method over repeated sampling.
A 95% interval of (0.52, 0.61) gives evidence against the claim that p = 0.50.
0.50 is outside the interval, so it's implausible.
If a claimed value lies inside the confidence interval, you can rule it out.
Inside the interval → plausible, cannot be ruled out.
Increasing the sample size, holding confidence fixed, makes the margin of error...
Larger n → smaller SE → smaller margin of error.
To get a narrower interval AND higher confidence, you need a larger sample size.
Bigger n overcomes the precision-vs-confidence trade-off.