Math · Introductory statistics · Worked example

Build a 95% t interval for a mean

A random sample of 16 packages has mean mass 52.3 g and standard deviation 6.4 g. Build a 95% confidence interval for the mean mass of all packages.

Check the conditions

The packages were sampled at random and weighed independently. With only 16 of them, the interval relies on masses that are roughly normal, without strong skew or outliers.

Find the standard error

\frac{6.4}{\sqrt{16}} = 1.6

Find t*

There are 15 degrees of freedom, and 95% confidence leaves 2.5% in each tail, so t* ≈ 2.131.

Find the margin and the interval

\begin{array}{l} 2.131 \cdot 1.6 \approx 3.41 \\ 52.3 - 3.41 = 48.89 \\ 52.3 + 3.41 = 55.71 \end{array}

Result

From about 48.89 g to 55.71 g.

Your turn

Find the 95% margin of error for n = 25 and s = 10.

Show the answer and explanation

About 4.13.

SE = 10/5 = 2, and t* with 24 degrees of freedom is about 2.064, so the margin is 2.064 × 2 ≈ 4.13.

\begin{array}{l} \frac{10}{\sqrt{25}} = 2 \\ 2.064 \cdot 2 \approx 4.13 \end{array}

Keep exploring

In Statistics, the t table with α = 0.025 and 15 degrees of freedom gives critical t = 2.13145. Change α to 0.005 for 99% confidence: t* = 2.94671, and the interval widens to about 47.59 to 57.01 g.

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