A process that cuts lengths of metal with high precision has historically produced metal  rods with length that is normally distributed and with a mean of 800 millimetres (mm).  Management at the factory is concerned that after an overhaul of the machine the mean  length of the rods being produced could differ from the required 800 mm. If it is found that  the rods are not being produced to specifications then the machine will have to be stopped  for readjustment, with a consequent loss of production. The production line manager tasked  with investigating the issue finds that a (random) sample of 25 rods yielded a mean length  of 806 mm and a sample standard deviation of 10 mm.  i. The mean of the sample is clearly not the required 800 mm. Say why this is not enough  information in itself to say if the machine is operating correctly or not.  ii. State the null and alternative hypotheses for this situation.

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 + ...