Hypothesis testing is sometimes referred to as a "Test of Significance". We will tend to use the former term but you may encounter the latter in readings or on a quiz or the final so it is important that you know that both terms refer to the same thing.

You should also bear in mind that hypothesis testing is based on sampling distribution theory. You should also note that hypothesis testing is similar to "proof by contradiction" in that you will assume your null hypothesis true and then see if your sample provides enough evidence to refute it.

Definition

Given any hypothesis testing problem you will always identify two hypotheses:

1. Null hypothesis: denoted H₀ and, for this class, will be of the following form:

   H₀: m_y = m

2. Alternative hypothesis: denoted H_a or H_r or H₁ and sometimes referred to as the research hypothesis. For this class the alternative hypothesis will be one sided, that is, it will be of the following form:

   H_a: m_y > m or H_a: m_y < m

Hypothesis Testing Procedure (m_y )

To conduct a hypothesis test for m_y you will follow the following four step procedure:

1. Examine the problem and identify and state the null and alternative hypotheses.

   e.g. Let us say that after examination of a problem statement you identify the null hypothesis to be m_y = m and you think that m_y > m, then you would state this as follows:

   H₀: m_y = m

   H_a: m_y > m

2. Examine your sample and do the following:
   1. Determine the sample size n and the statistics ybar and s_y
   2. Ensure that ybar is consistent with your alternative hypothesis (i.e. if m_y > m then ensure that ybar > m and if m_y < m then ensure that ybar < m).

   Note: An inconsistent ybar means that your sample does not support your alternative and so you cannot proceed.

3. If ybar is consistent then do the following:
   1. Assume H₀ true.
   2. Given that H₀ is assumed true, determine the p-value.

      Note: The p-value is the proportion of samples of size n that would result in a ybar more extreme than the one observed if H₀ is true. To do this you must consider the sampling distribution of ybar. Remember that the sample size determines the sampling distribution:

      1. n large: Since we assume H₀ true then m_ybar =m_y=m. However, s_y is not known but we can estimate it with s_y and so we estimate s_ybar by s_ybar=s_y/sqrt(n). We can then compute z=(ybar - m_ybar)/s_ybar and determine the desired proportion (i.e. the p-value).
      2. n small: Again, since we assume H₀ true then m_ybar =m_y=m. However, s_y is not known but we can estimate it with s_y and so we estimate s_ybar by s_ybar=s_y/sqrt(n-1). If y is normally distributed then we can compute t=(ybar - m_ybar)/s_ybar and determine the desired proportion (i.e. the p-value).

      Note: The z and t values computed for hypothesis testing problems are sometimes referred to as test statistics.
      Note: Remember that when n is small ybar is only "t" distributed if y is normally distributed. This means that for n small you must ensure that ybar is consistent and y is normally distributed before determining your p-value.

4. Apply the following decision rule to your p-value.
   1. If p-value is <= 1% then the p-value is highly significant and so you reject H₀.
   2. If p-value is <= 5% then the p-value is significant and so you reject H₀.
   3. If p-value is > 5% then the p-value is non-significant and so you have insufficient evidence to reject H₀.

Problem:

The registrar claims that the mean starting salary of CTI graduates (class of 1999) is $45K. You disagree and believe that CTI graduates got better starting salaries and so the mean is higher than claimed. You select a sample of twenty six from the graduating class and determine that the mean starting salary is $47K with a standard deviation of $4K. Conduct a test of hypotheses.

Solution:

Applying the four step procedure:

1. Identify and state the null and alternative hypotheses.

   H₀: m_y = $45K

   H_a: m_y > $45K

2. Examining the sample:
   1. Determine the sample size n and the statistics ybar and s_y

      n=26; ybar=$47K and s_y=$4K

   2. Ensure that ybar is consistent with alternative hypothesis:

      Since ybar>$45K then ybar is consistent and we may proceed.

3. If ybar is consistent then:
   1. Assume H₀ true.
   2. Given that H₀ is true, determine the p-value.

      n small: m_ybar =m_y=45; s_ybar=s_y/sqrt(n-1)=4/sqrt(25)=0.8; t=(47 - 45)/0.8=2.5 hence the t test statistic is 2.5 and from the t-table with df=25 the desired proportion (i.e. the p-value) is between 0.5% and 1%.

4. Apply the decision rule to your p-value.

   Since the p-value is <= 1% then the p-value is highly significant and so we reject H₀ and conclude that the mean is higher than claimed (i.e. m_y > $45K).