The concentration of a specific enzyme in a synthetic culture is assumed to follow a normal distribution. A researcher takes a random sample of five independent cultures and measures their concentrations in milligrams per litre (mg/L) as follows:
12.42,12.58,12.35,12.70,12.45 12.42, \quad 12.58, \quad 12.35, \quad 12.70, \quad 12.45 12.42,12.58,12.35,12.70,12.45Calculate unbiased estimates for the population mean and the population variance of the enzyme concentration in these cultures.
The true population standard deviation of the concentration is known to be 0.15 mg/L0.15 \text{ mg/L}0.15 mg/L. The research team requires a higher degree of precision for their upcoming study. They want to ensure that there is a probability of at least 0.980.980.98 that the estimate of the population mean, based on a random sample of size nnn, lies within 0.04 mg/L0.04 \text{ mg/L}0.04 mg/L of the actual population mean.
Determine the minimum sample size nnn required to satisfy this condition.