In a hypothesis test, when you are testing for a decrease, which probability do you work out, and which when testing for an increase?

"At least as extreme" always means extreme in the direction the alternative hypothesis points . Testing for a decrease , work out the probability of a value less than or equal to the test statistic. Testing for an increase , work out the probability of a value greater than or equal to the test statistic. The test statistic itself is included either way.

Why can the actual significance level of a test differ from the level the test was set at?

The actual significance level differs when the test statistic can only take discrete values , because then no critical region has a probability exactly equal to the stated level. The critical value has to be the first value that falls inside the region, which usually makes the probability of landing in the critical region smaller than the stated significance level. That smaller probability is the actual significance level .

True or False? If a two-tailed test rejects the null hypothesis, the conclusion should say whether the population parameter has increased or decreased.

False. A two-tailed test asks only whether the parameter has changed , so the conclusion should report evidence of a change and stop there. Naming a direction claims more than the test was set up to detect. To conclude that a parameter has increased, the test has to be one-tailed from the start, with the direction chosen before the data is seen.

A LevelMathsEdexcelStatisticsFlashcardsHypothesis TestingIntroduction to Hypothesis Testing

Introduction to Hypothesis Testing (Edexcel A Level Maths: Statistics): Flashcards

Exam code: 9MA0

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FrontHypothesis Testing
Define null hypothesis.
FrontHypothesis Testing
How do you tell whether a hypothesis test should be one-tailed or two-tailed, and what is the alternative hypothesis in each case?
BackHypothesis Testing
The question's wording decides whether a test is one-tailed or two-tailed. Ask whether it is looking for a change in a stated direction, or for any change at all.
A one-tailed test looks for an increase or a decrease specifically, giving $H_{1} : θ > \dots$ or $H_{1} : θ < \dots$
A two-tailed test looks only for a change, giving $H_{1} : θ \neq \dots$
So, for example, "has the coin become biased towards heads" is one-tailed, but "is the coin fair or not" is two-tailed.

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Define null hypothesis.
The null hypothesis, written $H_{0}$ , is the statement that is assumed to be true for the duration of the test: the population parameter has its original value and nothing has changed.
It is always stated as an equation:
$H_{0} : θ = \dots$
So, for example, testing whether a coin is fair gives $H_{0} : p = \frac{1}{2}$ , where $p$ is the probability that the coin lands on heads.
How do you tell whether a hypothesis test should be one-tailed or two-tailed, and what is the alternative hypothesis in each case?
The question's wording decides whether a test is one-tailed or two-tailed. Ask whether it is looking for a change in a stated direction, or for any change at all.
- A one-tailed test looks for an increase or a decrease specifically, giving $H_{1} : θ > \dots$ or $H_{1} : θ < \dots$
- A two-tailed test looks only for a change, giving $H_{1} : θ \neq \dots$
So, for example, "has the coin become biased towards heads" is one-tailed, but "is the coin fair or not" is two-tailed.
Define test statistic.
The test statistic is the numerical value calculated from the sample, and it is sometimes called the observed value.
It is what the test is actually run on. Everything after it consists of asking how likely a value at least this extreme would be if $H_{0}$ were true, either by working out a probability or by checking whether the value lies in the critical region.
Define significance level.
The significance level is the threshold probability, fixed before the test is run, below which a result counts as too unlikely to have happened by chance.
It is also the probability of rejecting $H_{0}$ when $H_{0}$ is in fact true, so a smaller level makes the test harder to pass.
It is usually 1%, 5% or 10%, and fixing it in advance matters, because choosing it after seeing the data would let you pick whichever value gave the answer you wanted.
In a hypothesis test, when you are testing for a decrease, which probability do you work out, and which when testing for an increase?
"At least as extreme" always means extreme in the direction the alternative hypothesis points.
- Testing for a decrease, work out the probability of a value less than or equal to the test statistic.
- Testing for an increase, work out the probability of a value greater than or equal to the test statistic.
The test statistic itself is included either way.
Define p-value.
The p-value is the probability of getting a result at least as extreme as the test statistic, worked out on the assumption that the null hypothesis is true.
That assumption is the part most often forgotten. A p-value does not tell you the probability that $H_{0}$ is true; it tells you how surprising the data would be if it were.
Reject $H_{0}$ when the p-value is less than the significance level.
A hypothesis test is carried out at the $α %$ significance level. Complete the sizes of the critical region:
In a one-tailed test, the critical region holds $______$ in the tail being tested.
In a two-tailed test, it holds $______$ in each tail.
The completed sizes are:
In a one-tailed test, the critical region holds $α %$ in the tail being tested.
In a two-tailed test, it holds $\frac{α}{2} %$ in each tail.
So, for example, at the 5% level a one-tailed test puts all 5% in one tail, while a two-tailed test puts 2.5% in each.
Define critical region.
The critical region is the set of values of the test statistic that would lead to the null hypothesis being rejected.
Its boundary is the critical value: the least extreme value that still lies inside the region, and therefore the least extreme result that would cause you to reject $H_{0}$ .
The significance level is what fixes where that boundary falls.
Why can the actual significance level of a test differ from the level the test was set at?
The actual significance level differs when the test statistic can only take discrete values, because then no critical region has a probability exactly equal to the stated level.
The critical value has to be the first value that falls inside the region, which usually makes the probability of landing in the critical region smaller than the stated significance level. That smaller probability is the actual significance level.
True or False?
If the test statistic does not fall in the critical region, you should conclude that you accept the null hypothesis.
False.
You conclude that you do not reject $H_{0}$ , and that there is insufficient evidence to support the alternative hypothesis.
The difference is not pedantry: a test that fails to find evidence against $H_{0}$ has not shown $H_{0}$ to be true, and mark schemes can treat "accept $H_{0}$ " as a contradictory statement and withhold the mark for it.
Whatever wording you use, your comparison and your conclusion must agree with each other.
What two things must the conclusion of a hypothesis test do, whatever the outcome?
It must be written in the context of the question, using the situation's own terms rather than $H_{0}$ and $H_{1}$ , so that it answers what was actually asked.
It must not be definitive: say that there is sufficient, or insufficient, evidence to suggest something at the stated significance level.
A test is run on one sample, and a different sample could point the other way, so nothing is ever proved.
True or False?
If a two-tailed test rejects the null hypothesis, the conclusion should say whether the population parameter has increased or decreased.
False.
A two-tailed test asks only whether the parameter has changed, so the conclusion should report evidence of a change and stop there.
Naming a direction claims more than the test was set up to detect. To conclude that a parameter has increased, the test has to be one-tailed from the start, with the direction chosen before the data is seen.

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