True or False? A power function gives you the power of a test without needing the actual parameter value.

False. A power function gives the power for every possible parameter value, as an expression rather than as a number. You still have to substitute a value before it produces a power, but unlike a single calculation it lets you see the whole picture and compare tests across a range.

What can you do with a power function once you have it?

Substitute different parameter values to compare two tests, since the better test is the one with the higher power at the values that matter. You can also plot it, or find where the power rises above 0.5, which is the point beyond which the test is more likely to reach the right conclusion than the wrong one.

A LevelFurther MathsEdexcelFurther Statistics 1FlashcardsHypothesis TestingQuality of Tests

Quality of Tests (Edexcel A Level Further Maths: Further Statistics 1): Flashcards

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FrontType I & Type II Errors
What are the four possible outcomes of a hypothesis test?

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What are the four possible outcomes of a hypothesis test?
Two are correct: $H_{0}$ was true and was not rejected, or $H_{0}$ was false and was rejected.
Two are errors: $H_{0}$ was true but was rejected, and $H_{0}$ was false but was not rejected.
Define a Type I error.
A Type I error occurs when a test gives sufficient evidence to reject $H_{0}$ even though $H_{0}$ was actually true.
It is a false positive: in a court case it would be convicting someone who is innocent.
What is the probability of a Type I error?
The probability of a Type I error is the probability of landing in the critical region, given that $H_{0}$ is true.
For a continuous distribution that is exactly $α$ , while for a discrete one it is the actual significance level, normally a little below $α$ .
Define a Type II error.
A Type II error occurs when a test gives insufficient evidence to reject $H_{0}$ even though $H_{0}$ was actually false.
It is a false negative: in a court case it would be acquitting someone who is guilty.
Why do you need the actual population parameter to find the probability of a Type II error?
Because $H_{1}$ only says the parameter is different, which is not specific enough to give you a distribution to calculate with.
The probability is that of not landing in the critical region given the true parameter value, so that value has to be supplied.
True or False?
Lowering the significance level reduces the probability of both types of error.
False.
Lowering it shrinks the critical region, which does reduce the probability of a Type I error, but it makes rejecting $H_{0}$ harder and so increases the probability of a Type II error.
The only way to reduce both at once is to increase the size of the sample.
A test rejects $H_{0}$ when the count exceeds 77 out of 100. How do you find the probability of a Type I error?
Work out the probability of the count exceeding 77 using the value of $p$ that $H_{0}$ assumes, since a Type I error is only possible when $H_{0}$ is true.
Nothing about the alternative hypothesis is used, which is why this probability can often be written down with very little work.
True or False?
The size of a test and the probability of a Type I error are the same thing.
True.
Both are the probability of rejecting $H_{0}$ when it was in fact true, so they are two names for one quantity.
A better test has a smaller size, because that is the error you want the test to make as rarely as possible.
Complete the relationship between the power of a test and the probability of a Type II error:
$power =______- P (Type II error)$
The completed relationship is:
$power = 1 - P (Type II error)$
When $H_{0}$ is false there are only two possibilities, rejecting it or failing to, so the two probabilities must add up to 1.
Define the power of a hypothesis test.
The power of a test is the probability of rejecting $H_{0}$ when it was false.
That is a good outcome, so a better test has a higher power, and ideally it should be above 0.5.
A test has a size of 0.0432. What does that number mean?
That number is the probability, 4.32%, that the test will reject $H_{0}$ even though $H_{0}$ is true.
It sits below 5%, which is what you would expect from a discrete test carried out at the 5% level.
What is a power function?
A power function is the power of a test written algebraically, in terms of the unknown parameter $p$ or $λ$ rather than for one specific value.
It is what you use when the actual population parameter has not been given, which in practice is almost always the case.
True or False?
A power function gives you the power of a test without needing the actual parameter value.
False.
A power function gives the power for every possible parameter value, as an expression rather than as a number.
You still have to substitute a value before it produces a power, but unlike a single calculation it lets you see the whole picture and compare tests across a range.
The critical region is $X \leq 2$ for a test on $X ~ B (50, p)$ . How do you build the power function?
Write $P (X \leq 2)$ as a sum of binomial terms in $p$ , giving ${(1 - p)}^{50} + 50 p {(1 - p)}^{49} + 1225 p^{2} {(1 - p)}^{48}$ .
Then factorise out the lowest common power to simplify it, which gives ${(1 - p)}^{48} (1 + 48 p + 1176 p^{2})$ .
What can you do with a power function once you have it?
Substitute different parameter values to compare two tests, since the better test is the one with the higher power at the values that matter.
You can also plot it, or find where the power rises above 0.5, which is the point beyond which the test is more likely to reach the right conclusion than the wrong one.

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