# Other statistical tools: Point and interval estimation

The previous section has introduced some statistical terms that are not widely used by engineers but are very familiar to statisticians. This section is a review of some of the statistical terms and procedures dealing with error estimation for the average and standard deviation as well as their confidence limits.

A good number to use for statistically significant data is 30. It is a good threshold when using some of the six sigma processes such as calculating defect rates. This is based on the fact that a t distribution with v degrees of freedom = 29 approaches the normal distribution. It can be from Table 5.2 that the data for the value of tα,30 is close to the value of the standard normal distribution. The error E is calculated as the difference between the value and the 2 value from the normal distribution. For a significance of 0.025, or confidence of 97.5%, the error is less than 5%. Note that this point of z = 1.96 is close to the z = 2

or the 2 CT point. For the 3 〇• point, or 99.9%, the error approaches 10%. The defect rate can thus be calculated using the f-distribution with small samples and known errors.

The relationship between the error and the sample size can be expanded to include the general conditions in which the standard deviationα is known from the sample and the number of the sample taken is large (>30). The maximum error E produced when sample average X is used to estimate m the population average, can be calculated in the following equation. In addition, the random sample size needed to estimate the average of a population, with a confidence of (1 — a)% can also be shown as:

Where E is the error, α is the standard deviations of the population> and n is the sample size used in calculating the error.

If the sample size n is small (<30)，and the sample is drawn from a normal distribution of the population, the standard deviation of populationσ is not known, but the sample standard deviation s can be calculated from the sample. In this case, the error made when the sample average X is used to estimate population average n is as follows: