Section 9.2: What is a Test of Significance?

Remember… H o is the Null Hypothesis ▫When you are using a mathematical statement, the null hypothesis uses ≥, ≤, and = functions. H a is the Alternative Hypothesis ▫When you are using a mathematical statement, the alternative hypothesis uses, and ≠ functions.

Before you begin an experiment… You make a decision… ▫Is it your goal to reject the null hypothesis? ▫Is it your goal to fail to reject the null hypothesis (does not necessarily mean you accept it)? Remember, the only way to be completely certain is to test the entire population in your experiment—not practical. So you need to take a sample instead and use a sample proportion as your data in which draw your conclusions.

You can’t always be right… You may reject the null hypothesis, but it turned out to be true. They refer to this as a Type 1 Error. You may fail to reject the null hypothesis, but it was false. They refer to this as a Type 2 Error.

The Four Outcomes There are four outcomes of a hypothesis test of an experiment… Decision Ho is Correct Ho is Incorrect Do not reject Ho True Conclusion Type 2 Error Reject Ho Type 1 Error True Conclusion

Which error would be more serious? Is it worse to say something is wrong, when it is right? Or is it worse to say something is right, when it is wrong? Like in a jury trial, which is worse: to put an innocent man in jail, or set a guilty man free? ▫Innocent man put in jail is a Type 1 Error. ▫Guilty man set free is a Type 2 Error.

Alpha and Beta… In a hypothesis test, the level of significance is your maximum allowable probability of making a type 1 error, denoted by α (alpha). The probability of making a type 2 error is denoted by β (beta). By setting it to a small value, you are saying you want the probability of rejecting the true null hypothesis to be small. The most commonly used levels of significance are α = 0.10, 0.05, and 0.01 (10%, 5%, and 1%).

About the level of significance… You do not want to always set the level of significance to By reducing the probability of making a Type 1 Error, you are increasing the probability of making a Type 2 Error.

The Jury’s Errors… ▫A jury is instructed that the prosecution must provide proof of guilt “beyond a reasonable doubt”. If the probability of a Type 1 error is small (an innocent man goes to jail), then the probability of a Type 2 error (a guilty man goes free) increases. ▫The relationship between α and β is an inverse one: as one goes up, the other goes down; and vice versa.

Other Thoughts… Failing to reject a Null Hypothesis does not mean you have accepted it as true. It means there was not enough evidence to reject it. (Like in a trial, if they find you “Not Guilty”, they are not saying you are “Innocent”, they are saying there was not enough evidence to prove your guilt.)