A state fish hatchery raises trout for stocking streams and lakes. The size of the fish at release time can be controlled to a fair degree by varying the rate of feeding. The target is a mean of 14 ounces; if the fish are too small, those who catch the fish aren’t happy. A random sample of 82 fish were weighed at time of release and it was determined that the mean was 13.78 ounces with a standard deviation of 1.84 ounces. Test to determine if the fish being released have a population mean less than 14 ounces at the 0.05 significance level.

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 + ...