To avoid difficulties with the Federal Trade Commission or state and local consumer protection agencies , a beverage bottler must make reasonably certain that 12 - ounce bottles actually contain 12 ounces of beverage . To determine whether a bottling ma chine is working satisfactorily , one bottler randomly samples 12 bottles per hour and measures the amount of beverage in each bottle . The mean 7 of the 12 fill measure ments is used to decide whether to readjust the amount of beverage delivered per bot tle by the filling machine . If records show that the amount of fill per bottle is normally distributed , with a standard deviation of .2 ounce , and if the bottling machine is set to produce a mean fill per bottle of 12.1 ounces , what is the approximate probability that the sample mean x of the 12 test bottles is less than 12 ounces ?

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 + ...