Binomial Hypothesis Testing (DP IB Maths: AI HL)

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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 X tilde straight B left parenthesis n comma space p right parenthesis
    • 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 straight P left parenthesis X less or equal than x vertical line p equals p subscript 0 right parenthesis
    • For H1 : p > p0 the p-value is straight P left parenthesis X greater or equal than x vertical line p equals p subscript 0 right parenthesis

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 X less or equal than c where c is the critical value
    • c is the largest integer such that straight P left parenthesis X less or equal than c vertical line p equals p subscript 0 right parenthesis less or equal than alpha percent sign
      • Check that straight P left parenthesis X less or equal than c plus 1 vertical line p equals p subscript 0 right parenthesis greater than alpha percent sign
  • For H1 : p > p0 the critical region is X greater or equal than c where c is the critical value
    • c is the smallest integer such that straight P left parenthesis X greater or equal than c vertical line p equals p subscript 0 right parenthesis less or equal than alpha percent sign
      • Check that straight P left parenthesis X greater or equal than c minus 1 vertical line p equals p subscript 0 right parenthesis greater than alpha percent sign

Worked example

The existing treatment for a disease is known to be effective in 85% of cases.  Dr Sabir develops a new treatment which she claims is more effective than the existing one.  To test her claim she uses the new treatment on a random sample of 60 patients with the disease and finds that the treatment was effective for 57 of them.

a)
State null and alternative hypotheses to test Dr Sabir’s claim.

4-12-4-ib-ai-hl-binomial-hyp-test-a-we-solution

b)
Perform the test using a 1% significance level. Clearly state the conclusion in context.

4-12-4-ib-ai-hl-binomial-hyp-test-b-we-solution

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Dan

Author: Dan

Expertise: Maths

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.