Hypothesis Testing ( p)

The following four step procedure is the hypothesis testing procedure for p. It is similar to the procedure for m_y but, since it is based on the sampling distribution of p, it is different in several subtle ways. Since we have only discussed the sampling distribution of p for n large, our procedure will be for n large only.

Hypothesis Testing Procedure ( p )

To conduct a hypothesis test for p 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 p = m and you think that p > m, then you would state this as follows:

   H₀: p = m

   H_a: p > m

   Note: m may be expressed as either a percentage or a decimal.

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

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

3. If p 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 sample proportion, p, more extreme than the one observed (given H₀ true):

      Since we assume H₀ true then m_p =p=m. Notice that assuming H₀ true means that we can state s_p exactly (i.e. we do not estimate it) since s_p=sqrt(p(1-p)/n)=sqrt(m(1-m)/n). We can then compute z=(p - m_p)/s_p and determine the desired proportion (i.e. the 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 75% of CTI graduates (class of 1999) received job offers prior to graduation. You disagree and believe that more CTI graduates received offers before graduating. You select a sample of 100 graduates and discover that 86 received offers before graduating. Conduct a test of hypotheses.

Solution:

Applying the four step procedure:

1. Identify and state the null and alternative hypotheses.

   H₀: p = 0.75

   H_a: p > 0.75

2. Examining the sample:
   1. Determine the sample size n and the statistic p

      n=100; p=86/100=0.86

   2. Ensure that p is consistent with alternative hypothesis:

      Since p=0.86 then p>0.75 and so p is consistent and we may proceed.

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

      m_p=p=0.75; s_p=sqrt(p(1-p)/n)=sqrt(0.75(1-0.75)/100)=0.0433; z=(0.86 - 0.75)/0.0433=2.54 hence from the standard normal table the desired proportion (i.e. the p-value) is (100-98.92)/2=0.54%.

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 population proportion is higher than claimed (i.e. p > 0.75).