5.2.1 Binomial Hypothesis Testing

How is a hypothesis test carried out with the binomial distribution?

• The population parameter being tested will be the probability, p  in a binomial distribution B(n , p)
• A hypothesis test is used when the assumed probability is questioned
• The null hypothesis, H₀  and alternative hypothesis, H₁ will always be given in terms of p
  • Make sure you clearly define p before writing the hypotheses
  • The null hypothesis will always be H₀ : p = ...
  • The alternative hypothesis will depend on if it is a one-tailed or two-tailed test
  • A one-tailed test would test to see if the value of p  has either increased or decreased
    • The alternative hypothesis, H₁ will be H₁ : p > ...  or  H₁ : p < ...  
  • A two-tailed test would test to see if the value of p  has changed
    • The alternative hypothesis, H₁ will be H₁ : p ≠ ... 
• To carry out a hypothesis test with the binomial distribution, the test statistic will be the number of successes in a defined number of trials
• When defining the test statistic, remember that the value of p  is being tested, so this should be written as p in the original definition, followed by the null hypothesis stating the assumed value of p
• The binomial distribution will be used to calculate the probability of the test statistic taking the observed value or a more extreme value
• The hypothesis test can be carried out by
  • either calculating the probability of the test statistic taking the observed or a more extreme value (p – value) and comparing this with the significance level
  • or by finding the critical region and seeing whether the observed value of the test statistic lies within it
    • • Finding the critical region can be more useful for considering more than one observed value or for further testing

How is the critical value found in a hypothesis test with the binomial distribution?

• The critical value will be the first value to fall within the critical region
  • The binomial distribution is a discrete distribution so the probability of the observed value being within the critical region, given a true null hypothesis may be less than the significance level
  • This is the actual significance level and is the probability of incorrectly rejecting the null hypothesis
• For a one-tailed test use your calculator to find the first value for which the probability of that or a more extreme value is less than the given significance level
  
  • Check that the next value would cause this probability to be greater than the significance level
    • For H₁ : p < ... if  and  then c is the critical value
    • For H₁ : p > ...  if   and  then c is the critical value
• For a two-tailed test you will need to find both critical values, one at each end of the distribution
  • Use your calculator to find the first value for which the probability of that or a more extreme value is less than half of the given significance level in both the upper and lower tails
  • Often one of the critical regions will be much bigger than the other

What steps should I follow when carrying out a hypothesis test with the binomial distribution?

• Step 1. Define the probability, p
• Step 2. Write the null and alternative hypotheses clearly using the form

H₀ : p = ...

H₁ : p = ...

• Step 3. Define the test statistic, usually  where n  is a defined number of trials and p is the population parameter to be tested
• Step 4. Calculate either the critical value(s) or the p – value for the test
• Step 5. Compare the observed value of the test statistic with the critical value(s) or the p - value with the significance level
• Step 6. Decide whether there is enough evidence to reject H₀ or whether it has to be accepted
• Step 7.Write a conclusion in context

Worked Example

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