Modify the formula for \(E\) on page 216 so that it applies to large samples which constitute substantial portions of finite populations, and use the resulting formula for the following problems:

(a) A sample of 50 scores on the admission test for a school of engineering is drawn at random from the scores of the 420 persons who applied to the school in 2015. If the sample mean and the standard deviation are \(\bar{x}=546\) and \(s=85\), what can we assert with \(95 \%\) confidence about the maximum error if \(\bar{x}=546\) is used as an estimate of the mean score of all the applicants?

(b) A random sample of 40 drums of a chemical, drawn from among 200 such drums whose weights can be expected to have the standard deviation \(\sigma=12.2\) pounds, has a mean weight of 240.8 pounds. If we estimate the mean weight of all 200 drums as 240.8 pounds, what can we assert with \(99 \%\) confidence about the maximum error?