Math · Introductory statistics · Worked example

Build a confidence interval from raw data

Use the sleep data 4, 8, 6, 5, 3, 7, 9 and 6 hours, a random sample of eight students, to build a 95% confidence interval for the mean hours of sleep.

Summarize the sample

n = 8, x̄ = 6 and s = 2, as in the standard deviation example.

Find the standard error

\frac{2}{\sqrt{8}} \approx 0.707

Find t* and the margin

With 7 degrees of freedom, t* ≈ 2.365.

2.365 \cdot 0.707 \approx 1.67

Build the interval

\begin{array}{l} 6 - 1.67 = 4.33 \\ 6 + 1.67 = 7.67 \end{array}

Result

About 4.33 to 7.67 hours.

Your turn

Would a 90% interval from the same data be wider or narrower? Find it.

Show the answer and explanation

Narrower: about 4.66 to 7.34 hours.

With 7 degrees of freedom and 5% in each tail, t* ≈ 1.895, so the margin is 1.895 × 0.707 ≈ 1.34.

\begin{array}{l} 1.895 \cdot 0.707 \approx 1.34 \\ 6 - 1.34 = 4.66 \\ 6 + 1.34 = 7.34 \end{array}

Keep exploring

In Statistics, the Summary gives the mean 6, the sample standard deviation 2 and the standard error 0.707107 directly.

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Summarize the data in Statistics Open worked example on a board Confidence interval in Math Reference

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