Quantitative Methods

							Business Research Methods

An Overview of Hypothesis Testing
• Introduction
   – The purpose of hypothesis testing is to determine
      whether there is enough statistical evidence in favor
      of a certain belief about a parameter.
   – Examples
      • Is there statistical evidence in a random sample of potential
        customers, that support the hypothesis that more than
        ???% of the potential customers will purchase a new
        product?
      • Is a new drug effective in curing a certain disease? A
        sample of patients is randomly selected. Half of them are
        given the drug where half are given a placebo. The
        improvement in the patients conditions is then measured
        and compared.
• Concept of hypothesis testing
  – The critical concepts of hypothesis testing.
     • There are two hypotheses (about a population parameter(s))
         – H0 - the null hypothesis        [ for example m = 5]
         – H1 - the alternative hypothesis [m > 5] This is what you want to prove

     • Assume the null hypothesis is
       true.
          – Build a statistic related to the
            parameter hypothesized.
          – Pose the question: How
            probable is it to obtain a statistic
            value at least as extreme as the
            one observed from the sample?
                                                               m=5        x
– Continued
  • Make one of the following two decisions (based on
    the test):
     – Reject the null hypothesis in favor of the alternative
       hypothesis.
     – Do not reject the null hypothesis in favor of the alternative
       hypothesis.
  • Two types of errors are possible when making
    the decision whether to reject H0
     – Type I error - reject H0 when it is true.
     – Type II error - do not reject H0 when it is false.
– Describing the p-value
  • If the p-value is less than 1%, there is
    overwhelming evidence that support the
    alternative hypothesis.
  • If the p-value is between 1% and 5%, there is a
    strong evidence that supports the alternative
    hypothesis.
  • If the p-value is between 5% and 10% there is a
    weak evidence that supports the alternative
    hypothesis.
  • If the p-value exceeds 10%, there is no evidence
    that supports of the alternative hypothesis.
   • The p-value method (α-level)
      – The p-value can be used when making
        decisions based on rejection region
        methods as follows:
           • Define the hypotheses to test, and the required
             significance level a.
           • Perform the sampling procedure, calculate the
             a = 0.05
             test statistic and the p-value associated with it.
                       The p-value
           • Compare the p-value to a. Reject the null
m x  170 hypothesis only if p <a; otherwise, do not reject
xL  175 .34 thenull hypothesis.
               x 178
Conclusions of a test of Hypothesis
      • If we reject the null hypothesis, we conclude that
        there is enough evidence to infer that the
        alternative hypothesis is true.
      • If we do not reject the null hypothesis, we
        conclude that there is not enough statistical
        evidence to infer that the alternative hypothesis
        is true.
                      The alternative hypothesis
                      is the more important
                      one. It represents what
                      we are investigating.
• As for Power…
   Let’s Review:
  – Judging the test

    • A hypothesis test is effectively defined by the
      significance level a and by the the sample size
      n.
    • If the probability of a type II error b is judged to
      be too large, we can reduce it by
       – increasing a, and/or
       – increasing the sample size.
                                                      By increasing the sample size
                                                      the standard deviation of the
                                                      sampling distribution of the
                                          a
                                                      mean decreases. Thus, x L
                                                      decreases.

                                 xxLLLLLLL
                                   xxxxx                          xL  m
                                                        za                , thus
                                                                      n
                  b1 > b2
                                                                           
As a result b decreases                                xL  m  z a
                                                                           n
                       x xx
                     x LxLLLL
 – In example 1, suppose n increases from 400 to
   1000.                       65
               xL  m  z a       170  1.645              173.38
                               n                    1000
                           173.38  180
              b  P( Z                   )  P( Z  3.22)  0
                            65   1000
– In summary,
  • By increasing the sample size, we reduce the
    probability of type II error.
  • Hence, we shall accept the null hypothesis
    when it is false less frequently.
– Power of a test
  • The power of a test is defined as 1 - b.
  • It represents the probability to reject the null
    hypothesis when it is false.