Hints for Hypothesis Testing

 

Rejection Regions

  1. Use the t table or short z table to find the t or z critical value associated with the given level of significance, a. 
     
  2. Plot this critical value on the horizontal axis with the t or z curve located above the axis.
     
  3. Shade in either the left tail, right tail, or both tails, depending on whether H0 has a "<", ">" or a not equal.
     
  4. Calculate the test statistic using the proper formula:
     

  5. Plot this value on the z or t axis.
     
  6. If this value is in the shaded region, reject H0, accept H1 and conclude that there is sufficient evidence to conclude that the alternative hypothesis is true.  If the value is not in the shaded region, then fail to reject H0, and state that there is insufficient evidence to make a conclusion.

P-Values

  1. Calculate the test statistic using the proper formula:
     

  2. Use the z or t table to find the corresponding probability.
     
  3. If this is a left tailed test "<", then this is the P-Value.  If this is a right tailed test ">" then the P-Value is 1 minus this number.  If this is a two tailed test and the result is less than 0.5, then the double this number to get the P-Value.  If this is a two tailed test and the result is greater than 0.5 then first subtract from 1 and then double the result to get the P-Value.
     
  4. If the P-Value is less than a, reject H0, accept H1 and conclude that there is sufficient evidence to conclude that the alternative hypothesis is true.  If the P-Value is greater than a, then fail to reject H0, and state that there is insufficient evidence to make a conclusion.

z-Table vs t-Table

  • Use the z-Table if the standard deviation is know and the original distribution is normal.  This will almost never happen in the real world, but the text book has problems with it.
     
  • It is ok. to use the z-Table if you have a large sample size (greater than 30 if you only need one decimal place of accuracy and greater than 100 if you need two decimal places.
     
  • If the standard deviation is unknown, you must use the t-table for small samples and it is slightly more accurate to use the t-table for large samples.