Final Answer:
a. The estimated average weight of the total fish in the farm is approximately 1.62 kg.
b. The range for the percentage of marked fish in the farm is approximately 11.1% to 18.9%.
Step-by-step explanation:
a. To estimate the average weight of all fish in the farm, we use the sample mean (1.6 kg) as an estimate for the population mean since the sample is considered representative. With a sample size of 120, a standard deviation of 0.21 kg, and a 95% confidence level, we compute the confidence interval for the population mean using the formula \(\bar{x} \pm Z \cdot \frac{\sigma}{\sqrt{n}}\). Substituting the given values yields an average weight estimate of approximately 1.62 kg.
b. The percentage of marked fish in the sample is \( \frac{15}{120} \times 100\% = 12.5\% \). To estimate the range for the entire farm, we use the margin of error in the sample percentage to create the confidence interval for the population percentage. At a 95% confidence level, the range is approximately 11.1% to 18.9%.
c. To infer the number of fish within this range, we consider the total number of fish. Assuming the lower and upper bounds of the percentage range, the estimated number of marked fish would range between 111 to 189 fish in the entire farm.
d. To calculate the farm's revenue from selling all the fish, we multiply the estimated total number of fish in the farm (including marked and unmarked) by the average weight obtained earlier (1.62 kg) and then by the selling price per kilogram (VND 27,000/kg).