Hisuggart, an energy drink, bottling comany has high speed manufacturing line that drops 750 mg of sugar in each bottle of their half gallon beverage. the actual drop is a random variable with the mean of 750mg and standard deviation of 10mg. sometimes the sugar drop machine gets out of adjustment and drops 730 mg of sugar with standard deviation of 10. to adjust the machine plant manager has to shut down the line. to ensure unnecessary shot down plant manager uses the following procedure. a random sample of 30 bottles are obtained from each shift, and the sugar content of each sample is carefully measured and the measurements are averaged. if the average is less than 745mg then the line will be shot down for maintenance.
a . if x-bar denotes the sample average for the random sample of the 30 bottles , what would be the value of E (X-bar) and Var(X-bar).
b. plant manager is very concerned about the "false alarm" which is the condition when bottling machine is adjusted perfectly, yet aline shut down is requested. based on your answer to part a, what is the probability of such an error?