Justifying a Claim about a Mean · 用区间支持关于均值的主张
A mean interval does not describe most individual objects
- A valid 95% interval for mean apple mass is (145.9, 154.1) grams. A claim that the mean is 150 g is compatible.
- The interval does not say that 95% of individual apples weigh in this range. Individual masses can be much more spread out than the uncertainty in their 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."
A valid 95% interval estimates population mean apple mass. Which interpretations are justified?
Keep the fixed population mean, individual observations and repeated-sampling method separate.
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?"
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.
Holding sample standard deviation and confidence fixed, increasing n makes a t 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.
Holding n and confidence fixed, a larger sample standard deviation makes a t interval...
Larger s → larger SE → wider interval.
An interval is (145.9, 154.1) g. What is its margin of error in grams?
Half the width is (154.1 - 145.9)/2 = 4.1 g.
What changes the width
- Larger sample size, holding sample SD and confidence fixed → narrower interval (smaller SE, and $t^{*}$ shrinks toward $z^{*}$). Higher confidence level → wider interval (larger $t^{*}$).
- More variable data (larger $s_x$) → wider interval. Increasing $n$ can offset the widening from a higher confidence level; the amount must be checked.
"$95\%$ 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.
Carry the reasoning to a new case
- For fixed sample standard deviation and confidence, increasing n reduces standard error.
- Across different samples the standard deviation can change, so narrower is not an unconditional guarantee.
Match each quantity to the relevant interpretation.
Keep the population mean, method coverage and individual values separate.
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.