A large box of corn flakes claims to contain 510 grams of cereal. Since cereal boxes must contain at least as much product as their packaging claims, the machine that fills the boxes is set to put 513 grams in each box. The machine has a known standard deviation of 3 grams and the distribution of fills is known to be normal. At random intervals throughout the day, workers sample 5 boxes and weigh the cereal in each box. If the average is less than 511 grams, the machine is shut down and adjusted.

How often will the workers make a Type I error with this decision rule and the hypotheses vs. ? Submit your answer rounded to four decimal places.

Using the same decision rule, what is the power of the hypothesis test if the machine is actually filling boxes with an average of 510 grams? That is, calculate the chance of observing a sample mean less than 511 if the true mean is 510.