To construct an 80% confidence interval estimate of the standard deviation for body temperature, we will go through the following steps:
1. Determine the Sample Size (n):
Given:
[tex]\( n = 103 \)[/tex]
2. Identify the Sample Mean ([tex]\(\bar{x}\)[/tex]) and Sample Standard Deviation (s):
Given:
[tex]\(\bar{x} = 98.70^{\circ} F\)[/tex]
[tex]\( s = 0.67^{\circ} F\)[/tex]
3. Calculate the Degrees of Freedom (df):
The degrees of freedom is [tex]\( df = n - 1 \)[/tex].
[tex]\( df = 103 - 1 = 102 \)[/tex]
4. Find the Critical Values for Chi-Square Distribution:
For an 80% confidence level, the alpha ([tex]\(\alpha\)[/tex]) is 0.20. Since we want an 80% confidence interval, we need the critical values that mark the middle 80% of the distribution. These values split the remaining 20% in the tails, with 10% in each tail (so we use [tex]\(\alpha/2 = 0.10\)[/tex] for the lower tail and [tex]\(1 - \alpha/2 = 0.90\)[/tex] for the upper tail).
From the Chi-Square distribution table:
- Lower critical value [tex]\( \chi^2_{0.10, 102} = 84.177 \)[/tex]
- Upper critical value [tex]\( \chi^2_{0.90, 102} = 120.679 \)[/tex]
5. Calculate the Confidence Interval for the Variance:
The confidence interval for the variance ([tex]\(\sigma^2\)[/tex]) is given by:
[tex]\[
\left( \frac{(n - 1) s^2}{\chi^2_{upper}}, \frac{(n - 1) s^2}{\chi^2_{lower}} \right)
\][/tex]
Plugging in the values:
[tex]\[
\left( \frac{(103 - 1) \times (0.67)^2}{120.679}, \frac{(103 - 1) \times (0.67)^2}{84.177} \right)
\][/tex]
[tex]\[
= \left( \frac{102 \times 0.4489}{120.679}, \frac{102 \times 0.4489}{84.177} \right)
\][/tex]
[tex]\[
= \left( \frac{45.7878}{120.679}, \frac{45.7878}{84.177} \right)
\][/tex]
[tex]\[
= \left( 0.3795, 0.544,
\][/tex]
6. Calculate the Confidence Interval for the Standard Deviation:
To find the confidence interval for the standard deviation ([tex]\(\sigma\)[/tex]), take the square root of the variance bounds:
[tex]\[
\left( \sqrt{0.3795}, \sqrt{0.5443} \right)
\][/tex]
[tex]\[
= \left( 0.616, 0.737 \right)
\][/tex]
Therefore, the 80% confidence interval estimate for the standard deviation of body temperature of all healthy humans is:
[tex]\[ 0.62^{\circ} F < \sigma < 0.74^{\circ} F \][/tex]
(Note: The values are rounded to two decimal places as requested).
