Suppose an investigator takes a random sample of \(n=50\) birth weights from several teaching hospitals located in an inner-city neighborhood. In her random sample, the sample mean \(\bar{x}\) is 3,150 grams and the standard deviation is 250 grams.

(a) Calculate a \(95 \%\) confidence interval for the population mean birth weight in these hospitals.

(b) The typical weight of a baby at birth for the US population is 3,250 grams. The investigator suspects that the birth weights of babies in these teaching hospitals is different than \(3,250 \mathrm{grams}\), but she is not sure if it is smaller (from malnutrition) or larger (because of obesity prevalence in mothers giving birth at these hospitals). Carry out the hypothesis test that she would conduct.