## Why are tests of significance needed?

- Tests of significance are statistical tools that help us make decisions about changes to responses (process outputs).
- Without these tools, we might look at a change in a process output and think that it is important, but the change was just part of the common cause variation from the process.
- Tests of significance give us a statistical basis for determining if a change in factor levels leads to a statistically significant effect on the process response.
- While tests of significance can be standalone statistical tools, they serve as the backbone of ANOVA (analysis of variance) and of the analysis of the results from designed experiments.

### α and β Risks

- Whenever we make statistics-based decisions, we have to accept some risk in our assessments.
- There are two types of risks we face.
- We can make a mistake in saying results are different when they are actually the same. This is an α (alpha) risk.
- A β (beta) risk occurs when we say that results are the same when they are actually different.

- With tests of significance, a 5% α risk is typically used.
- We can place all of the risk on one-tail when testing for a change in one direction, or we can divide the risk over two-tails when testing for any type of difference.

### Degrees of Freedom

- Besides the α risk, there is second term that we need to use with tests of significance, the Degrees of Freedom.
- The degrees of freedom, or df, are the number of independent values we have in a calculation. Typically, this is the number of values associated with the calculation minus 1.

### Hypothesis Testing

- Hypothesis testing is an important concept needed for both tests of significance and design of experiments. A hypothesis is an assumption about the outcome of the test or experiment.
- If a hypothesis is rejected, it means that the data available are sufficient to conclude that the hypothesis is false.
- However, if the hypothesis is accepted, we can say that the data are sufficient to conclude that the hypothesis is not false but not necessarily that the hypothesis has been proven true.

### Types of Tests of Significance

- There are four major types of significance tests. The Z-test and t-test look at differences in the mean values and the chi-squared and F-tests look at differences in variances.
- With experimental designs, we use the tests of significance for samples, the t-test and the F-test, not the tests for populations.

### t-Tests

- The t-test can give a statistical basis for whether a sample is from a population or whether multiple samples indicate their populations are equal.

### F-Tests

- The F-test investigates whether two populations are equal based on the variances of two samples from those populations.