Questions: A simple random sample from a population with a normal distribution of 100 body temperatures has x̄=99.10°F and s=0.66°F. Construct a 95% confidence interval estimate of the standard deviation of body temperature of all healthy humans. σ < σ (Round to two decimal places as needed.)

Solution Steps

The sample standard deviation \( s \) is given as \( 0.66 \, ^\circ F \). The sample variance \( s^2 \) is calculated as follows:

\[ s^2 = (0.66)^2 = 0.4356 \]

To construct the confidence interval for the variance of a single population with an unknown population mean, we use the formula:

\[ \left( \frac{(n - 1)s^2}{\chi^2_{\alpha/2}}, \frac{(n - 1)s^2}{\chi^2_{1 - \alpha/2}} \right) \]

Substituting the values:

• Sample size \( n = 100 \)
• Sample variance \( s^2 = 0.4356 \)

The confidence interval for the variance is calculated as:

\[ CI = \left( \frac{(100 - 1) \times 0.4356}{\chi^2_{0.025}}, \frac{(100 - 1) \times 0.4356}{\chi^2_{0.975}} \right) = (0.34, 0.59) \]

The confidence interval for the standard deviation is obtained by taking the square root of the variance confidence interval:

\[ \left( \sqrt{0.34}, \sqrt{0.59} \right) \approx (0.58, 0.77) \]

Final Answer