Lower and upper confidence interval estimate ahead of two months for cod catch (no trend in cod catch but it is changing slowly) for smoothing constant á=0.225 was found 300 and 477.12 tons. If the estimate of ó2 from the monthly catch of two years is s2=900. The confidence level at which the forecast of the cod catch was made: a) 99.2% , b) 99.8% , or c) 99.6%
To determine the confidence level, we can use the formula:
Confidence Level = 1 - α
where α is the significance level.
Given that the smoothing constant (á) is 0.225, we can calculate the variance of the forecast error (ó^2) using the formula:
ó^2 = á * s^2 / (1 - á)
where s^2 is the estimate of ó^2 from the monthly catch of two years (s^2 = 900).
By substituting the given values into the formula, we can calculate the value of ó^2:
ó^2 = 0.225 * 900 / (1 - 0.225) = 0.225 * 900 / 0.775 ≈ 260.13
Next, we can calculate the standard deviation of the forecast error (ó) by taking the square root of ó^2:
ó = √260.13 ≈ 16.125
The confidence interval can be calculated using the formula:
Lower Limit = Forecast - Z * ó
Upper Limit = Forecast + Z * ó
where Z is the Z-score, which corresponds to the desired confidence level.
To determine the confidence level, we need to find the Z-score using a cumulative normal distribution table.
For a):
Z = 2.805 (corresponding to 99.2% confidence level)
Lower Limit = 300 - 2.805 * 16.125 ≈ 256.71
Upper Limit = 300 + 2.805 * 16.125 ≈ 343.29
For b):
Z = 3.095 (corresponding to 99.8% confidence level)
Lower Limit = 300 - 3.095 * 16.125 ≈ 241.26
Upper Limit = 300 + 3.095 * 16.125 ≈ 358.74
For c):
Z = 2.898 (corresponding to 99.6% confidence level)
Lower Limit = 300 - 2.898 * 16.125 ≈ 253.76
Upper Limit = 300 + 2.898 * 16.125 ≈ 346.24
Therefore, the confidence level at which the forecast of cod catch was made is c) 99.6%.