4.12.3 Binomial Hypothesis Testing

What is a hypothesis test using a binomial distribution?

• You can use a binomial distribution to test whether the proportion of a population with a specified characteristic has increased or decreased
• These tests will always be one-tailed
• You will not be expected to perform a two-tailed hypothesis test with the binomial distribution
• A sample will be taken and the test statistic x will be the number of members with the characteristic which will be tested using the distribution 
• This can be thought of as the number of successes

What are the steps for a hypothesis test of a binomial proportion?

• STEP 1: Write the hypotheses
  • H₀ : p = p₀
    • Clearly state that p represents the population proportion
    • p₀ is the assumed population proportion
  • H₁ : p < p₀ or H₁ : p > p₀

• STEP 2: Calculate the p-value or find the critical region
• See below
• STEP 3: Decide whether there is evidence to reject the null hypothesis
• If the p-value < significance level then reject H₀
• If the test statistic is in the critical region then reject H₀
• STEP 4: Write your conclusion
• If you reject H₀­ then there is evidence to suggest that...
• The population proportion has decreased (for H₁ : p < p₀)
• The population proportion has increased (for H₁ :  p > p₀)
• If you accept H­₀ then there is insufficient evidence to reject the null hypothesis which suggests that...
• The population proportion has not decreased (for H₁ : p < p₀)
• The population proportion has not increased (for H₁ : p > p₀)

How do I calculate the p-value?

• The p-value is determined by the test statistic x
• The p-value is the probability that ‘a value being at least as extreme as the test statistic’ would occur if null hypothesis were true
• For H₁ : p < p₀ the p-value is 
• For H₁ : p > p₀ the p-value is 

How do I find the critical value and critical region?

• The critical value and critical region are determined by the significance level α%
• Your calculator might have an inverse binomial function that works just like the inverse normal function
• You need to use this value to find the critical value
• The value given by the inverse binomial function is normally one away from the actual critical value
• For H₁ : p < p₀ the critical region is  where c is the critical value
• c is the largest integer such that 
• Check that 
• For H₁ : p > p₀ the critical region is  where c is the critical value
• c is the smallest integer such that 
• Check that