Lower bound = sqrt((n-1)*s^2 / χ^2(α/2,n-1))
Upper bound = sqrt((n-1)*s^2 / χ^2(1-α/2,n-1))
Where:
n = sample size = 19
s = sample standard deviation = 3.0 inches
α = significance level = 0.01 (since the confidence level is 99%, the significance level is 1%)
χ^2(α/2,n-1) = chi-squared value for α/2 and n-1 degrees of freedom
χ^2(1-α/2,n-1) = chi-squared value for 1-α/2 and n-1 degrees of freedom
Using a chi-squared distribution table with 18 degrees of freedom (since n-1 = 19-1 = 18), we find that:
χ^2(0.005,18) = 38.582
χ^2(0.995,18) = 7.962
Substituting the values into the formula, we get:
Lower bound = sqrt((19-1)*3^2 / 38.582) = 1.923
Upper bound = sqrt((19-1)*3^2 / 7.962) = 4.409
Therefore, the 99% confidence interval for the population standard deviation is (1.923, 4.409) inches.