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

Revision Note

Author

Mark

Expertise

Maths

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-1 size-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 $p = \frac{1}{2}$ but actually $p = \frac{1}{3}$
  - This is more helpful than just saying $p \neq \frac{1}{2}$
- 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 $X ~ Po (λ)$ . A hypothesis test is conducted at the 5% significance level in which $H_{0} : λ = 8$ and $H_{1} : λ < 8$ .

If is later discovered that $λ = 6$ , 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 $λ$
- 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 $λ$ )

How do I find power functions?

It's easier to show in an example
- If the critical region is $X \leq 2$ for a binomial hypothesis test with $n = 50$
- Then the power function is P(in the critical region | population parameter is $p$ )
  - Let $X ~ B (50, p)$
  - $P (X \leq 2 | p) = (\begin{matrix} 50 \\ 0 \end{matrix}) p^{0} {(1 - p)}^{50} + (\begin{matrix} 50 \\ 1 \end{matrix}) p^{1} {(1 - p)}^{49} + (\begin{matrix} 50 \\ 2 \end{matrix}) p^{2} {(1 - p)}^{48}$
- Simplify
  - ${(1 - p)}^{50} + 50 p {(1 - p)}^{49} + 1225 p^{2} {(1 - p)}^{48}$
- Factorise and collect like terms
  - The power function is ${(1 - p)}^{48} (1 + 48 p + 1176 p^{2})$

What can I do with power functions?

You can plot them against $p$ (or $λ$ )
- You can then see where the power is biggest
You can input different values of $p$ (or $λ$ )
- 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 $H_{0} : λ = 9$ and $H_{1} : λ < 9$ .

\begin{matrix}  \end{matrix}

Show that the power function is

\frac{1}{6} e^{- λ} (a + b λ + c λ^{2} + d λ^{3})

, where

a, b, c

and

d

are integers to be found.

power-functions-1 power-functions-2

Find the largest integer value of

λ

for which the probability of a Type II error is less than 20%.

power-functions-3

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

Revision Note

What is the size of a test?

How does size relate to Type I errors?

What is the power of a test?

How does power relate to Type II errors?

What is a power function?

How do I find power functions?

What can I do with power functions?

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Author: Mark