Even though the actual concentration of sodium in the drinking water is within the limit, there is a0.0974 probability that the water department will erroneously advise its customers of an above-limit concentration of sodium. Suppose that the water department wants to reduce its risk of erroneously notifying its customers that the sodium concentration is above the limit. Assuming the water department can’t change the mean or the standard deviation of the sodium concentration in the drinking water, is there anything the department can do to reduce the risk of notifying its customers that the sodium concentration is above the limit when it actually is not? No, there is nothing it can do. It can collect more specimens over the course of the year. It can collect fewer specimens over the course of the year.

The concentration of sodium in water is estimated within certain limits, this means, it was estimated using an interval. The water department has to make this estimation more accurate to prevent informing the wrong concentration. If neither the mean and the variance can change, assuming this is an interval for the population mean of sodium concentration, to get a better estimation you have to reduce the amplitude and thus the margin of error of the interval. Working with the margin of error you get that:

d= Z₁₋α * δ

√n

You have two ways to reduce the margin of error of the interval without modifying the standard deviation.

1) Changing the confidence level.

The confidence level and the margin of error have a direct relationship if you increase or decrease the confidence level, the margin of error will increase or decrease accordingly:

↑d= ↑Z₁₋α * δ

√n

2) Changing the sample size.

The sample size and the margin of error have an indirect relationship, this means that when you increase the sample size, the margin of error will decrease, and when you decrease the sample size, the margin of error will increase:

↓d= Z₁₋α * δ

↑√n

If the objective is to reduce the chances of informing the wrong limits, the best way of action is to reduce the margin of error of the estimation. To do so the water department should take more specimens over the year.

I hope it helps!