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A LevelFurther Maths: Further Statistics 1EdexcelRevision NotesHypothesis TestingQuality of TestsSize & Power of Test

Size & Power of Test (Edexcel A Level Further Maths: Further Statistics 1)

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Mark

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Mark

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Size

What is the size of a test?

  • The size of a test is the probability of rejecting H0 when it was in fact true
    • P(in critical region | H0 is true) 
    • The situation being described is not a good outcome 
      • Something has been rejected when it was actually true!
    • A better test has a smaller size
      • You want to minimise this error happening
  • Size is related to the significance level, α%
    • A better test has a smaller significance level (e.g. 1%)
    • For continuous distributions (e.g. normal)
      • Size = significance level, α
      • You can often write this down with no calculation
    • For discrete distributions (e.g. binomial, Poisson, geometric)
      • Size = actual significance level (≤α)
      • As close to α% as a discrete variable can get, whilst still being critical

How does size relate to Type I errors?

  • The size is exactly the same as the probability of a Type I error
    • Both want to know the probability of rejecting H0 when it was in fact true

Worked example

A student wants to test, at a 10% significant level, whether a coin is biased towards heads by counting the number of heads in 20 flips of the coin.

Calculate the size of this test.

size-1size-2

Power

What is the power of a test?

  • The power of a test is the probability of rejecting H0 when it was false
    • P(in critical region | H0 is false) 
    • The situation being described has a good outcome 
      • The null hypothesis was false and it rightly got rejected
    • A better test has a higher power
      • You want to maximise this happening
  • In practice, you need to be given the actual population parameter to calculate the power
    • For example, H0 assumed p equals 1 half but actually p equals 1 third
      • This is more helpful than just saying p not equal to 1 half 
    • Power is P(in the critical region | actual population parameter)

How does power relate to Type II errors?

  • The power of a test is 1 - P(Type II error) 
    • Power is when H0 is false and it gets rejected
      • That's a good outcome
    • A Type II error is when H0 is false and it does not get rejected
      • That's a bad outcome
  • You ideally want the power of a test to be greater than 0.5 
    • That way it's less likely to produce a Type II error
      • And more likely to reach the correct conclusion

Worked example

Let X tilde Po open parentheses lambda close parentheses. A hypothesis test is conducted at the 5% significance level in which straight H subscript 0 colon space lambda equals 8 and straight H subscript 1 colon space lambda less than 8.

If is later discovered that lambda equals 6, find the power of the test.

power-of-test

Power Functions

What is a power function?

  • The power function is the power of a test written algebraically
    • In terms of p or lambda
    • For when you're not given the actual population parameter in the question
  • In reality, it's very unlikely you'll know the actual population parameter anyway
    • Otherwise you wouldn't be doing a hypothesis test on it!
    • Power functions don't need this information
  • The power function is P(in the critical region | population parameter is p)
    • Or, for Poisson 
      • P(in the critical region | population parameter is lambda)

How do I find power functions?

  • It's easier to show in an example
    • If the critical region is X less or equal than 2 for a binomial hypothesis test with n equals 50 
    • Then the power function is P(in the critical region | population parameter is p)
      • Let X tilde straight B open parentheses 50 comma p close parentheses
      • straight P open parentheses X less or equal than 2 space vertical line space p close parentheses equals open parentheses table row 50 row 0 end table close parentheses p to the power of 0 open parentheses 1 minus p close parentheses to the power of 50 plus open parentheses table row 50 row 1 end table close parentheses p to the power of 1 open parentheses 1 minus p close parentheses to the power of 49 plus open parentheses table row 50 row 2 end table close parentheses p squared open parentheses 1 minus p close parentheses to the power of 48
    • Simplify
      • open parentheses 1 minus p close parentheses to the power of 50 plus 50 p open parentheses 1 minus p close parentheses to the power of 49 plus 1225 p squared open parentheses 1 minus p close parentheses to the power of 48
    • Factorise and collect like terms
      • The power function is open parentheses 1 minus p close parentheses to the power of 48 open parentheses 1 plus 48 p plus 1176 p squared close parentheses

What can I do with power functions?

  • You can plot them against p (or lambda) 
    • You can then see where the power is biggest
  • You can input different values of p (or lambda) 
    • To compare two (or more) different hypothesis tests
      • The better test is the one with the higher power
    • To check if the power of a test is greater than 0.5
      • So that it's more likely to reach the correct conclusion
      • And less likely to produce a Type II error
      • Because Power = 1 - P(Type II error)

Worked example

Residents suspect that the number of accidents on a main road has decreased. They test, at a 5% significance level, the hypotheses straight H subscript 0 colon space lambda equals 9 and straight H subscript 1 colon space lambda less than 9.

a)table row blank row blank end table
Show that the power function is 1 over 6 straight e to the power of negative lambda end exponent open parentheses a plus b lambda plus c lambda squared plus d lambda cubed close parentheses, where a comma space b comma space c and d are integers to be found.

power-functions-1power-functions-2

b)
Find the largest integer value of lambda for which the probability of a Type II error is less than 20%.

power-functions-3

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    Mark

    Author: Mark

    Expertise: Maths

    Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.