Final answer:
To estimate the standard deviation of body temperature with an 80% confidence interval, use the Chi-square distribution with the sample standard deviation s = 0.69°F and sample size n = 107 to find the critical values. Then calculate the confidence interval for the population standard deviation with lower and upper bounds.
Step-by-step explanation:
To construct an 80% confidence interval estimate for the standard deviation of body temperature of all healthy humans, we use the sample standard deviation s and the Chi-square distribution since the population standard deviation is unknown. The formula for the confidence interval of the standard deviation is:
σ = s * sqrt((n-1)/X²)
Where:
- σ is the population standard deviation
- s = 0.69°F (sample standard deviation)
- n = 107 (sample size)
- X² is the Chi-square statistic
First, we find the critical values of the Chi-square distribution for the degrees of freedom df = n - 1 = 106 at the 80% confidence level. We look these values up in the Chi-square distribution table and typically, we would find one value for the lower end and one for the upper end of the distribution. Calculate the lower and upper bounds by using the formula with each Chi-square value.
The confidence interval estimate of the standard deviation is then given by the range between the lower and upper bounds. It's important to note that because a table is needed to find the Chi-square values, here we would illustrate the method without the actual values.
In this case, after finding the appropriate Chi-square values, the calculation would yield the confidence interval for the population standard deviation with lower and upper bounds (denoted as L and U, respectively).
σ = L °F < U °F