A Confidence Interval for a Mean
Introduced| English |
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| degrees of freedom/dɪˈɡriːz ɒv ˈfriːdəm/ |
How precise is a 150 gram sample mean?
- A fictional random sample of 25 apples has mean mass 150 g and sample standard deviation 10 g. We want a 95% interval for the population mean mass.
- The result estimates a population mean. It does not describe the mass range containing 95% of individual apples.
Check the sample before using t
- Check the random design and independence. For sampling without replacement, 25 should be at most 10% of the population size.
- With a sample this small, inspect the distribution for strong skew and outliers. A normal population gives the exact t model; sample-size rules support approximation rather than guarantee it.
For a one-sample t-interval with n = 25, what are the degrees of freedom (n − 1)?
df = n − 1 = 25 − 1 = 24.
Match each check to its purpose.
These checks answer different questions; one cannot substitute for the others.
A sample of thirty observations guarantees that every t procedure is valid.
Thirty is a useful approximation guideline, not a repair for biased design, dependence or extreme observations.
Use the correct degrees of freedom
- The degrees of freedom 自由度 are $df=n-1=24$. The critical t value for a 95% interval is approximately $2.064$.
- This critical value exceeds the normal value 1.96. It accounts for the extra uncertainty from using the sample standard deviation.
s = 10, n = 25. Compute the standard error of the mean s/√n.
10/√25 = 10/5 = 2.
Calculate uncertainty without changing the units
- The standard error is $SE=s/\sqrt n=10/5=2$ g. The interval is $\bar x\pm t^*SE$.
- The margin of error is approximately $2.064(2)=4.128$ g. Endpoints round to $(145.9,154.1)$ g.
The margin of error is about 4.128 g on each side, making the total width about 8.256 g. Margin and width are different quantities.
With SE = 2 and t* = 2.06, find the margin of error t*·SE. Two decimals.
2.06 × 2 = 4.12.
Change one factor at a time
- Holding the observed standard deviation and confidence fixed, increasing sample size reduces standard error and changes the critical t value.
- Higher confidence requires a larger critical value and a wider interval. Across different samples, the observed standard deviation can change too.
The reference curve, degrees of freedom and tail rule must match the selected procedure.
For a confidence interval for a mean (σ unknown), the critical value comes from...
Use t* with n − 1 degrees of freedom.
Holding other observed factors fixed, which widen a t interval?
The margin is t* times s/sqrt(n). Larger variability or a higher confidence critical value widens the interval.
State the target and the limits
- We are 95% confident that the population mean apple mass is between 145.9 g and 154.1 g, using the checked t method.
- The confidence level describes long-run coverage of the method. It does not assign a 95% probability to the fixed mean after this interval has been observed.
We are 95% confident that the population mean apple mass is between 145.9 g and 154.1 g, using the checked t method.
The standard error of the mean is s divided by n.
It's s/√n, not s/n.