Hypothesis Testing

1. Inferential Statistics can be used for decision making. Examples include:
    
   1. Does taking an aspirin a day decrease the risk of heart attack?
   2. Does Lovastatin decrease cholesterol levels?
   3. Does the death penalty reduce homicide rates?
   4. If children watch less television will they loose weight? (JAMA article)
       
2. How do we use statistics to make decisions?
    
   1. First we set up two hypotheses, the null hypothesis and the alternative hypothesis.
       
      1. The null hypothesis is the status quo hypothesis. This is the hypothesis we will have to accept as true unless we can prove it's false. For instance, a null hypothesis might be that Lovastatin doesn't lower cholesterol levels.
          
      2. The alternative hypothesis is also called the research hypothesis. This is the hypothesis that we want to prove is true. For instance, an alternative hypothesis might be that Lovastatin lowers cholesterol levels.
          
      3. The null and alternative hypotheses are set up so that if the null hypothesis is not true then the alternative hypothesis must be true.
          
      4. These alternatives must be set up in terms of population parameters. So, if is the mean change in cholesterol levels of all teenage boys in American if they took Lovastatin we would write the hypotheses as:
          
         H_o:   0
         H_a:  < 0
          
   2. Next we determine the probability that the null hypothesis is true. If the null hypothesis is very unlikely we decide it must be untrue and therefore the alternative hypothesis is true.
       
      1. The exact probability we find is determined by the hypotheses being tested. However, the probability we find is always called the p-value.
          
      2. If the p-value is less than a pre-determined value then we reject the null hypothesis and conclude the alternative hypothesis is true. Otherwise we say we "fail to reject the null hypothesis".
          
   3. This structure is much like the judicial system where an accused is considered innocent until proven guilty. The null hypothesis if that the person is innocent and the alternative hypothesis is that the person is guilty. If the evidence is such that it makes it obvious that the probability that the accused is innocent is very small we are forced to reject the null hypothesis and conclude he's guilty. Otherwise we assume he was innocent.
       
3. Error rates. There are two errors that can be made when conducting hypothesis tests:
    
   1. You might reject the null hypothesis when it is in fact true. This is called a Type I error and the Type I error rate is denoted by the symbol (pronounced alpha).
       
   2. You might fail to reject the null hypothesis when it is in fact false. This is called a Type II error and the Type II error rate is denoted by the symbol (pronounced beta).
       
   3. We set , meaning that we decide what the Type I error rate will be. This is just like the fact that we decide what the confidence level for a confidence interval will be.
       
   4. We don't know what is. However, as the sample size increases decreases.
       
   5. As increases decreases
       
   6. The power of a hypothesis test is 1 - . This is the probability of rejected the null hypothesis when it  is false.
       
4. When do we reject H_o? We reject H_o whenever our p-value is less than . If we follow this rule then our Type I error rate will be . We call the significance level of the test.

E-mail Mr. Callahan at stat110@edcallahan.com with questions or comments about this web site or about the class itself.