An environmental scientist wants to estimate the mean mercury content in ppm of fish in a lake.

An environmental scientist wants to estimate the mean mercury content in ppm of fish in a lake. They collect a random sample of 50 fish from the lake, measure the mercury content of each, calculate the average mercury content for these 50 fish and a 99% confidence interval for the mean. The confidence interval is (0.82, 1.13). Which of the following are valid interpretations of the confidence interval?

I) We are 99% certain that each fish has approximately 0.82 to 1.13 ppm of mercury.

II) We expect 99% of the fish to have between 0.82 and 1.13 ppm of mercury.

III) We would expect about 99% of all possible sample means from this population to be in between 0.82 and 1.13 ppm of mercury.

IV) We are 99% certain that the confidence interval of (0.82, 1.13) includes the true mean of the mercury content of fish in the lake.

Please explain your reasoning.