Hypothesis Testing for the Proportion of a Binomial Distribution (Edexcel IAL Maths)

Binomial Hypothesis Testing

Spec point — 4.6

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 H_**0 : 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 and comparing this with the significance level

    • You may have to use a normal approximation to calculate the probability

  • 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 $\mathrm{P} ( X \le c ) \le \alpha \%$ and $\mathrm{P} ( X \le c + 1 ) > \alpha \%$ then c is the critical value
    • For H₁: p > ... * if $\mathrm{P} ( X \ge c ) \le \alpha \%$ and $\mathrm{P} ( X \ge c - 1 ) > \alpha \%$ 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

  • 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 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 $X \sim B ( n , p )$ 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 necessary probability for the test
• Step 5. Compare the observed value of the test statistic with the critical value(s) or the probability 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 Jacques, a breadmaker, claims that fewer than 40% of people that shop in a particular supermarket buy his brand of bread. Jacques takes a random sample of 12 customers that have purchased bread and asks them which brand of bread they have purchased. He records that 2 of them had purchased his brand of bread. Test, at the 10% level of significance, whether Jacques’ claim is justified.

Exam Hint

• Most of the time the values of n and p will be in the tables
• if not, you will have to use the formula to calculate the probabilities or use a normal approximation