Boxes of cereal produced in a given factory are regularly inspected and must contain an average weight of 500 grams. The individual box weights are assumed to follow a normal distribution with a mean of 500 and a standard deviation of 20 grams. In routine inspections, the inspector selects a random sample of 14 cereal boxes and if the average sample weight is below 495 or above 505, the manufacturing process is stopped and the box filling machine is adjusted. a. Suppose a change occurs in the machine changing the average weight from 500 to 493 grams, without changing the standard deviation. What would be the proportion of sample mean weights outside the limits [495, 505]; remembering that proportion is a number between 0 and 1?

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...