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CIE A Level Maths: Probability & Statistics 2复习笔记3.1.1 Hypothesis Testing

Category: A-level课程, 教材笔记, 福利干货 Date: 2022年8月18日下午4:30

Language of Hypothesis Testing

What is a hypothesis test?

A hypothesis test uses a sample of data in an experiment to test a statement made about the value of a population parameter
A hypothesis test is used when the value of the assumed population parameter is questioned
The hypothesis test will look at the which outcomes are unlikely to occur if assumed population parameter is true
The probability found will be compared against a given significance level to determine whether there is evidence to believe that the assumed population parameter is not true

What are the key terms used in statistical hypothesis testing?

Every hypothesis test must begin with a clear null hypothesis (what we believe to already be true) and alternative hypothesis (how we believe the data pattern or probability distribution might have changed)
A hypothesis is an assumption that is made about a particular population parameter
- A population parameter is a numerical characteristic which helps define a population
  - One example of a population parameter is the probability, p of an event occurring
  - Another example is the mean of a population
- The null hypothesis is denoted H₀ and sets out the assumed population parameter given that no change has happened
- The alternative hypothesis is denoted H₁ and sets out how we think the population parameter could have changed
- When a hypothesis test is carried out, the null hypothesis is assumed to be true and this assumption will either be accepted or rejected
A hypothesis test could be a one-tailed test or a two-tailed test
The null hypothesis will always be H₀ : θ = ...
The alternate hypothesis will depend on if it is a one-tailed or two-tailed test
- A one-tailed test would test to see if the population parameter, θ, has either increased or decreased
  - The alternative hypothesis, H₁ will be H₁ : θ > ... or H₁ : θ < ...
- A two-tailed test would test to see if the population parameter, θ , has changed
  - The alternative hypothesis, H₁ will be H₁ : θ ≠ ...
- It is important to read the wording of the question carefully to decide whether your hypothesis test should be one-tailed or two-tailed
To carry out a hypothesis test an experiment will be carried out on a sample of data, the result of this experiment will be the observed value
- A sample of data is a subset of data taken from the population
- The observed value is a numerical value calculated from the of data
A hypothesis test will always be carried out at an appropriate significance level
- The significance level sets the smallest probability that an event could have occurred by chance.
  - Any probability smaller than the significance level would suggest that the event is unlikely to have happened by chance
- The significance level must be set before the hypothesis test is carried out
- The significance level will usually be 1%, 5% or 10%, however it may vary

Worked Example

A hypothesis test is carried out at the 5% level of significance to test if a normal coin is fair or not.

(i)Describe what the population parameter could be for the hypothesis test.

(ii)State whether the hypothesis test should be a one-tailed test or a two-tailed test, give a reason for your answer.

(iii)Clearly defining your population parameter, state suitable null and alternative hypotheses for the test.

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Exam Tip

Make sure you read the question carefully to determine whether the test you are carrying out is for a one-tailed or a two-tailed test.

Critical Regions

How do we decide whether to reject or accept the null hypothesis?

The null hypothesis would be rejected if the observed value falls within the critical region
- The critical region is the range of values that the observed value could take which will lead to the null hypothesis being rejected
The critical value is the boundary of the critical region
- It is the least extreme value that would lead to the rejection of the null hypothesis
- The critical value is determined by the significance level
- In a two-tailed test the significance level is halved and both the upper and the lower tails are tested
- For discrete distributions the critical value is the first value that falls within the critical region and so the probability of the observed value falling within the critical region may be lower than the given significance level
  - This probability will be known as the actual significance level
  - The actual significance level is the probability of incorrectly rejecting the null hypothesis
- Finding the critical region will be different for a two-tailed test than it is for a one-tailed test

Do we always need to find the critical region?

In most cases the best method of conducting a hypothesis test is to find the critical region
- It allows you to see how far the observed value is from the critical value and make decisions about whether further testing is necessary
In some cases a hypothesis test can be carried out without finding the critical region
The null hypothesis would be rejected if the probability of a value being at least as extreme as the observed value, assuming that the null hypothesis is true, is less than the significance level
- If the test is looking for a decrease then extreme values are smaller than the observed value, so find the probability of less than or equal to the observed value
- If the test is looking for an increase then extreme values are bigger than the observed value, so find the probability of greater than or equal to the observed value
- This probability is called the "p-value"
In a two-tailed test it is common to half the significance level and compare this with the probability found in one of the tails

Worked Example

3-1-1-critical-regions-and-p-values-we-solution-

Conclusions of Hypothesis Testing

How is a hypothesis test carried out?

There are a number of ways that a hypothesis test can be carried out for different models, however the following steps should form the base for your test:
Step 1. Define the test statistic and population parameter
Step 2. Write the null and alternative hypotheses clearly
Step 3. Calculate the critical value(s) or the necessary probability for the test
Step 4. Compare the observed value with the critical value(s) or the probability with the significance level
Step 5. Decide whether there is enough evidence to reject H₀ or whether it has to be accepted
Step 6. Write a conclusion in context

How should a conclusion be written for a hypothesis test?

Your conclusion must be written in the context of the question
Use the wording in the question to help you write your conclusion
- If rejecting the null hypothesis your conclusion should state that there is sufficient evidence to suggest the alternative hypothesis is true at this level of significance
- If accepting the null hypothesis your conclusion should state that there is not enough evidence to suggest the alternative hypothesis is true at this level of significance
Your conclusion must not be definitive
- There is a chance that the test has led to an incorrect conclusion
- The outcome is dependent on the sample, a different sample might lead to a different outcome
The conclusion of a two-tailed test can state if there is evidence of a change
- You should not state whether this change is an increase or decrease

Worked Example

A teacher carried out a hypothesis test at the 10% significance level to test if her students perform better in exams after using a new revision technique. Under the null hypothesis she calculates the probability that a value will be at least as extreme as the observed value to be 0.09142. Write a conclusion for her hypothesis test.

5-1-1-conclusions-of-hypothesis-testing-we-solution