To test for the equality of the mean of a population, we can use the central limit theorem, which states that the mean of a sample of size n from any population is approximately normally distributed, with the accuracy of the approximation improving with increasing n. When the variance of the population,is not known and the sample is large, we assume that the variance of the sample,– the unbiased estimate of the population variance,– is equal to
Calculations can then be done using the normal distribution.
Example: A machine is calibrated to produce nails with a mean length of 5.4 cm. 80 nails are to be tested to ensure the machine is still calibrated correctly. The sample has mean 5.31 cm and variance soConduct a hypothesis test at the 5% level of significance, to test whether the machine is calibrated correctly.
The null and alternative hypotheses are respectively,
Using the central limit theorem, we assume
The test statistice is
The hypothesis test is two tailed, so the area of each tail is 0.025 or 2.5 %.
The critical values of z, corresponding to probability of 2.5%, are
so the null hypothesis is rejected. There is evidence that the mean length of the nail;s is not 5.4 cm. The machine needs to be recalibrated.