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, H_1?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,?H_1?will be?H₁?:?p ≠ ...?
• To carry out a hypothesis test with the binomial distribution,?the?random variable?used in the testwill be the number of?successes?in a defined number of?trials
• When defining the distribution, 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
• 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 much bigger than the other
    • If the probability in the null hypothesis is 0.5 the critical regions will have an equal size

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 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