A Confidence Interval for a Mean

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.

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.

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.

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.

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.

Write from a standard error of 3

• A sample of 16 objects has sample standard deviation 12.
• The lesson interval uses a standard error of 2 and a margin near 4.12.