Hypothesis Testing


Hypothesis Testing
Whenever we have a decision to make about a population characteristic, we make a hypothesis.  Some examples are:

        m  > 3     

or 

        m  5.

Suppose that we want to test the hypothesis that m 5.  Then we can think of our opponent suggesting that m = 5.  We call the opponent's hypothesis the null hypothesis and write:

        H0:    m = 5 

and our hypothesis the alternative hypothesis and write

        H1:    m 5

For the null hypothesis we always use equality, since we are comparing m with a previously determined mean.

For the alternative hypothesis, we have the choices: < , > , or   .

 

 

Procedures in  Hypothesis Testing

When we test a hypothesis we proceed as follows:

 

  1. Formulate the null and alternative hypothesis.

  2. Choose a level of significance.

  3. Determine the sample size.  (Same as confidence intervals)

  4. Collect data.

  5. Calculate z (or t) score.

  6. Utilize the table to determine if the z score falls within the acceptance region.

  7. Decide to 

    1. Reject the null hypothesis and therefore accept the alternative hypothesis or 

    2. Fail to reject the null hypothesis and therefore state that there is not enough evidence to suggest the truth of the alternative hypothesis.

Errors in Hypothesis Tests 

We define a type I error as the event of rejecting the null hypothesis when the null hypothesis was true.  The probability of a type I error  (a) is called the significance level.

We define a type II error (with probability b) as the event of failing to reject the null hypothesis when the null hypothesis was false.

 

Example

Suppose that you are a lawyer that is trying to establish that a company has been unfair to minorities with regard to salary increases.  Suppose the mean salary increase per year is 8%. 

You set the null hypothesis to be

        H0:  m = .08 

        H1:  m < .08

 

Q.  What is a type I error?

A.  We put sanctions on the company, when they were not being discriminatory.

 

Q.  What is a type II error?

A.  We allow the company to go about its discriminatory ways.

Note:  Larger a results in a smaller b, and smaller a  results in a larger b.

 

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