Justifying a Claim about a Mean
Interpreting the mean interval
- Say it right: "We are $95\%$ confident the true mean $\mu$ lies between the endpoints."
- Name the population and what $\mu$ measures, with units, in context.
- The interval is a range of plausible values for the population mean.
- It's about $\mu$, not about a single data value.
What the confidence level means
- The confidence level describes the method over many samples — not one interval.
- "$95\%$ confident" = about $95\%$ of intervals built this way would capture $\mu$.
- A single interval either contains $\mu$ or it doesn't; we don't know which.
- It is not "$95\%$ probability $\mu$ is in this interval."
Justifying a claim
- To test a claimed mean (say $\mu = 100$): is it inside the interval?
- Inside → that value is plausible; can't be ruled out.
- Outside → the interval is evidence against that claim.
- The interval answers "is this claimed mean believable?"
What changes the width
- Larger sample size → narrower interval (smaller SE, and $t^{*}$ shrinks toward $z^{*}$).
- Higher confidence level → wider interval (larger $t^{*}$).
- More variable data (larger $s_x$) → wider interval.
- To be both narrower and more confident, increase $n$.
"$95\%$ confidence" describes the method, not one interval. Never say "$\mu$ has a $95\%$ chance of being in $(145.9, 154.1)$" — $\mu$ is fixed, so it's in or out. The right idea: $95\%$ of intervals built this way capture $\mu$ across repeated samples. This is exactly the proportion-interval logic, now for a mean.
A $95\%$ interval for mean apple weight is $(145.9,\ 154.1)$ g.
- Interpret: we're $95\%$ confident the true mean weight is between $145.9$ and $154.1$ g.
- Claim "mean is $160$ g"? $160$ is outside → evidence against it.
- Claim "mean is $150$ g"? $150$ is inside → plausible.
Interpret a mean interval as "$95\%$ confident $\mu$ is between the endpoints," where the confidence level describes the method over repeated samples. A value outside is evidence against that claim. Larger $n$ narrows the interval; higher confidence (or more variable data) widens it.
Capturing μ over repeated samples
95% of these intervals would contain the true mean μ.
A 95% interval for a mean is (145.9, 154.1). The claim μ = 160 is...
160 lies outside → evidence against it.
'95% confidence' means there is a 95% probability that μ is in this particular interval.
μ is fixed; confidence describes the method over many samples.
Increasing the sample size makes a mean interval...
Larger n → smaller SE → narrower interval.
A claimed mean inside the confidence interval is plausible and cannot be ruled out.
Inside → consistent with the data.
More variable data (larger s) makes the interval...
Larger s → larger SE → wider interval.