Question

3. The following data from [1] are carapace lengths (mm) of lobsters caught in some region: 78 66 65 63 60 60 58 56 52 50

Assume that the lengths of lobster carapaces in the region have a Normal distribution.

(a) Give a 95% confidence interval for the mean carapace length of lobsters in the region.

(b) Give a 95% confidence interval for the variance of carapace lengths of lobsters in the region.

Answer 1

sample mean, xbar = 60.8
sample standard deviation, s = 7.9694
sample size, n = 10
degrees of freedom, df = n - 1 = 9

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, tc = t(α/2, df) = 2.262

CI = (xbar - tc * s/sqrt(n) , xbar + tc * s/sqrt(n))
CI = (60.8 - 2.262 * 7.9694/sqrt(10) , 60.8 + 2.262 * 7.9694/sqrt(10))
CI = (55.10 , 66.50)

b)

Here s = 7.9694 and n = 10
df = 10 - 1 = 9

α = 1 - 0.95 = 0.05
The critical values for α = 0.05 and df = 9 are Χ^2(1-α/2,n-1) = 2.7 and Χ^2(α/2,n-1) = 19.023

CI = (9*7.9694^2/19.023 , 9*7.9694^2/2.7)
CI = (30.0479 , 211.7045)