Geometric Hypothesis Testing (Edexcel A Level Further Maths): Revision Note

Geometric Hypothesis Testing

How do I test for the parameter p of a Geometric distribution?

• If , test for the probability of success, , using the following hypotheses

  • 

  • or or

  • with significance level 

    • For example,  for 5%

• You will be given an observed value, , in the question

  • This is the number of trials it takes to see the first success

    • For example, "They thought the coin was fair (), but last week it took 5 flips to get the first tail ()"

  • It can help to compare  with the expected number of trials to see the first success,

    • For example, "they expected a fair coin () to take attempts to see the first tail"

• Assuming 

  • Find the probability that  is the observed value , or more extreme than that

  • For the extreme values are

    • Note the "change in inequality direction"

    • A lower probability of success means a higher number of attempts to first reach that success

  • For  the extreme values are

    • A higher probability of success means a lower number of attempts to first reach that success

  • For  compare  with 

    • If is less than , then extreme values are

    • If is more than , then the extreme values are

  • If the total probability of these values is (or  for two-tailed tests)

    • Write that "there is sufficient evidence to reject "

  • If not, write that "there is insufficient evidence to reject " 

• Write a conclusion in context

  • For example

    • "the probability of success is less than "

    • or "the probability of success has not changed from "

How do I find the critical region for a Geometric hypothesis test?

• If  

  • Assume that

  • Then test different integer values, , to get  as close to as possible, without exceeding it

    • Use the formula to help

    • The integer that's the nearest is called the critical value

    • Checking one integer lower should show that is

  • The critical region is

    • Note that the inequality is the opposite way round to 

  • Instead of testing integers, you can also use logarithms to solve the critical region inequalities

    • Beware when dividing both sides by

    • so the inequality must be "flipped"

• If

  • It's the same process, but with  as close to as possible, without exceeding it

    • Use the formula to help

  • The critical region is 

• If

  • The critical region is  or

    • is as close to  as possible, without exceeding it

    • is as close to as possible, without exceeding it

• Your calculator may have an 'Inverse Geometric Distribution' function that can help with finding critical values

  • But always check those values against the requirements of the question

  • The calculator may not always give the exact answer you are looking for

What is the actual significance level?

• As the geometric model is discrete, it's not possible to get a critical region whose probability sums to  exactly

  • That's because  can only take integer values

• Whatever it does sum to is called the actual significance level

  • The actual amount of probability in the tail (or tails)

• For example, if  has the critical region 

  • Then will be just less than 

    • It's value is the actual significance level

    • It represents the probability of rejecting  incorrectly (when was actually true)

• Some questions want a critical region that's as close to as possible, even if that means probabilities that exceed 

  • For example, if and  where 

    • Then  is the critical region that's as close to  as possible

    • The actual significance level is 0.0511