Hypothesis Testing (DP IB Maths: AI HL)

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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 population
    • The statement is either about a population parameter or the distribution of the population
  • The hypothesis test will look at the probability of observed outcomes happening under set conditions
  • The probability found will be compared against a given significance level to determine whether there is evidence to support the statement being made

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 or the distribution of the population
    • A population parameter is a numerical characteristic which helps define a population
      • Such as the mean value of the population
    • The null hypothesis is denoted straight H subscript 0 and sets out the assumed population parameter or distribution given that no change has happened
    • The alternative hypothesis is denoted straight H subscript 1 and sets out how we think the population parameter or distribution 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
      • When a null hypothesis is accepted or rejected a statistical inference is made
  • 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

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One-tailed Tests

What are one-tailed tests?

  • A one-tailed test is used for testing:
    • Whether a distribution can be used to model the population
    • Whether the population parameter has increased
    • Whether the population parameter has decreased
  • One-tailed tests can be used with:
    • Chi-squared test for independence
    • Chi-squared goodness of fit test
    • Test for proportion of a binomial distribution
    • Test for population mean of a Poisson distribution
    • Test for population mean of a normal distribution
    • Test to compare population means of two distributions

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Two-tailed Tests

What are two-tailed tests?

  • A two-tailed test is used for testing:
    • Whether the population parameter has changed
  • Two-tailed tests can be used with:
    • Test for population mean of a normal distribution
    • Test to compare population means of two distributions

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Conclusions of Hypothesis Testing

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

  • A sample of the population is taken and the test statistic is calculated using the observations from the sample
    • Your GDC can calculate the test statistic for you (if required)
  • To decide whether or not to reject the null hypothesis you first need either the p-value or the critical region
  • The p - value is the probability of a value being at least as extreme as the test statistic, assuming that the null hypothesis is true
    • Your GDC will give you the p-value (if required)
    • If the p-value is less than the significance level then the null hypothesis would be rejected
  • The critical region is the range of values of the test statistic which will lead to the null hypothesis being rejected
    • If the test statistic falls within the critical region then the null hypothesis would be 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

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 that the null hypothesis is unlikely to be true
    • If accepting the null hypothesis your conclusion should state that there is not enough evidence to suggest that the null hypothesis is unlikely to be true
  • 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
    • If you are testing the difference between the means of two populations then you can only conclude that the means are not equal
      • You can not say which population mean is bigger
      • You’d need to use a one-tailed test for this

Examiner Tip

  • Accepting the null hypothesis does not mean that you are saying it is true
    • You are simply saying there is not enough evidence to reject it

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Dan

Author: Dan

Expertise: Maths

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.