Null hypothesis vs. alternative hypothesis (hypotheses) 
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   We use H0 as a null hypothesis
          Ha as an alternative hypothesis. 
   
   Below we have a few examples. 
         1)  H0   : x_mean= x0
             Ha(1): x_mean < x0
             Ha(2): x_mean > x0

         2)  H0   : x_mean > x0
             Ha   : x_mean < x0

         3)  H0:    var = var0
             Ha(1): var < var0
             Ha(2): var > var0

         4)  H0:    var > var0
             Ha:    var < var0

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alpha: significant level vs. confidence interval 
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Alpha: significant level usually takes three values 1%, 5%, and 10%. 

     Confidence interval = 1- alpha

One-sided vs. two-sided tests
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   For the following hypothesis test, it is a 2-sided test. 
   
             H0   : x_mean= x0
             Ha(1): x_mean < x0
             Ha(2): x_mean > x0

   when estimating the critical value, we use alpha/2. 
   Below is an example of a one-sided test. 
            H0   : x_mean > x0
            Ha   : x_mean < x0
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Three decision rules
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   For example, we have a null hypothesis

                H0   x_mean= x0

   1) Range Rule: if x0 is inside the range   --> accept H0
                  if x0 is outside the range  --> reject 

   2) Distance Rule: if the distance > critical distance  --> reject 
                                     < critical distance  --> accept

   3) Shaded Area Rule: if  the area < alpha   --> reject
                                     > alpha   --> accept

 Note #1 How to estimate the range?
          Range      -->   mean +- margin of error
          lower bound    = mean - margin of error
          upper bound    = mean + margin of error 

         Margin of error =  t(critical value) * stdErr
                        std
           stdErr =    -------
                      sqrt(n)
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