Justifying a Claim about a Mean
ВведеноA 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?
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
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...
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
160 lies outside → evidence against it.
'95% confidence' means there is a 95% probability that μ is in this particular interval.
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
μ is fixed; confidence describes the method over many samples.
Holding sample standard deviation and confidence fixed, increasing n makes a t interval...
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Larger n → smaller SE → narrower interval.
A claimed mean inside the confidence interval is plausible and cannot be ruled out.
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Inside → consistent with the data.
Holding n and confidence fixed, a larger sample standard deviation makes a t interval...
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
Larger s → larger SE → wider interval.
An interval is (145.9, 154.1) g. What is its margin of error in grams?
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
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.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
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.
Write from an interval for the population mean
- A t interval for a mean is 148 grams to 156 grams.
- A reader says 95 percent of the objects weigh between those values. The lesson mean 4.1 is another context.
What does the interval estimate?
A mean interval is narrower than the distribution of individuals.
What does 148 to 156 estimate?
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
The endpoints are for the mean.