45-733 PROBABILITY AND STATISTICS I

1. H_o: m = m_o
   H₁: m = m₁ > m_o
   

   Test: If > m_o + c then reject H_o
   

   1. To find c when s² is known
      
      

      

      To solve for c (a) fix a (c) and use Z table.
      
      
   2. To find c when s² is not known, large sample

      

      To solve for c (a) fix a (c) and use Z table.
      
      
   3. To find c when s² is not known, small sample from N(m, s²)
      
      

      

      Where T has a t distribution with n - 1 degrees of freedom. To solve for c (a) fix a (c) and use t table.
      
      
2. H_o: m = m_o
   H₁: m ¹ m_o
   
   Test: If > m_o + c or - < m_o - c then reject H_o.
   1. To find c when s² is known
      
      
      
      To solve for c (a) fix a (c) and use Z table.
      
      
   2. To find c when s ² is not known, large sample
      
      
      
      To solve for c (a) fix a (c) and use Z table.
      
      
   3. To find c when s² is not known, small sample from N(m , s²)
      
      
      
      Where T has a t distribution with n - 1 degrees of freedom. To solve for c (a) fix a (c) and use t table.
      
      
3. H_o: m ₁ - m₂ = 0
   H₁: m₁ - m₂ ¹ 0
   
   Test: If then reject H_o.
   
   
   1. To find c when s₁² and s₂² are known
      
      
      
      To solve for c (a) fix a (c) and use Z table.
      
      
   2. To find c when are not known, large sample
      
      
      
      To solve for c (a) fix a (c) and use Z table.
      
      
   3. To find c when are not known, small samples from N(m , s²)
      
      
      
      Where T has a t distribution with n + m - 2 degrees of freedom and s² is given by:
      
      
      
      To solve for c (a) fix a (c) and use t table.
      
      
4. H_o: s² =
   H₁: s² =
   
   Test: If then reject H_o.
   
   To find c₂ set
   
   
   
   and use the chi-square table with n - 1 degrees of freedom.
   
   
5. H_o: s² =
   H₁: s² =
   
   Test: If then reject H_o.
   
   To find c₁ and c₂ set
   
   

   

   

   and use the chi-square table with n - 1 degrees of freedom.
   
   
6. H_o:
   H₁:
   
   Test: If or
   
   then reject H_o. To find k₁ and k₂ set
   
   
   
   
   
   Use the F table with n - 1 and m - 1 degrees of freedom to find k₂ and then use the reciprocal of the entry for m - 1 and n - 1 degrees of freedom for k₁.