Introduction to Hypothesis Testing (College Board AP® Statistics)

Introduction to hypothesis testing

What is a hypothesis test?

• A hypothesis test is a procedure for determining whether or not a population parameter has changed significantly from its previous (or previously assumed or accepted) value

  • e.g. you suspect the mean height of all the trees on an island has increased from its original value of 5.2 meters

What is the null hypothesis?

• A null hypothesis, , is the assumption that the population parameter has not changed

  • e.g. : The mean height of all trees on an island is still 5.2 meters ()

    • It is assumed to be correct, unless evidence proves otherwise

• Sometimes a null hypothesis can include an inequality with an equals

  • 

    • but these are still tested at the boundary,

What is the alternative hypothesis?

• An alternative hypothesis, , is how you think the population parameter has changed

  • e.g. The mean height of all trees on an island has increased from 5.2 meters ()

    • It is the situation for which evidence is being collected

What is a one-tailed test?

• A one-tailed test is when you suspect the population parameter has either increased or decreased

  • i.e. has changed in a particular direction

• The alternative hypothesis is a strict one-sided inequality

  • 

  • 

What is a two-tailed test?

• A two-tailed test is when you suspect the population parameter has changed

  • but you do not know if it has increased or decreased

• The alternative hypothesis is a not-equal-to statement

  • 

What is a test statistic?

• To try to prove your case, you take a recent sample from the population and calculate an unbiased estimate of the population parameter from that sample

  • This estimate is called a test statistic

  • e.g. you randomly sample 30 trees and calculate their mean height of 6.5 m

    • The test statistic is 6.5 m

• The type of test statistic will depend on what you are trying to prove

  • Use a sample mean to prove a change in population mean

  • Use a sample proportion to prove a change in population proportion

What is the standardized test statistic?

• It is not enough to just take a sample, work out a test statistic from it, then use that value to say the population parameter has changed

  • A test statistic will generally not be exactly equal to the population parameter it is estimating

  • You need to know whether the test statistic is extreme or not, compared to the population parameter assumed by

• Instead, you need a measure of how far the test statistic is from the population parameter

  • This is called the standardized test statistic, given by:

    • 

  • It shows how many standard errors you are from the population parameter

    • The standard error is an estimate of the population standard deviation from the data

What are conditions for a hypothesis test?

• Each type of hypothesis test has its own set of conditions that must be met

• A common condition is the independence condition which states that items in the sample (or experiment) must be independent

  • This is checked in two stages, first:

    • verify that data is collected by random sampling

    • or random assignment (in an experiment)

  • Second, if sampling without replacement:

    • verify that the sample size is less than or equal to 10% of the population size

    • Sometimes written

• Another common condition is normality of the population or sampling distribution

  • The distribution needs to be approximately symmetric

  • There should be no outliers

What is the p-value?

• To prove a test statistic is extreme when compared to the old population parameter

  • you need a probability that shows it is unlikely to have happened under the old population parameter

    • i.e. unlikely under the null hypothesis

• The p-value is the probability of obtaining a test statistic as extreme, or more extreme, than the one observed in the sample, assuming the null hypothesis is true

  • Note that this is a probability of an extreme region

    • not just of one extreme value

• The more extreme the test statistic is compared to the sample, the smaller the p-value

  • and the more evidence there is to suggest that the null hypothesis can be rejected

How do I calculate the p-value?

• The p-value is calculated by finding probabilities from the sampling distribution for that test statistic

  • Use the population parameter from the null hypothesis

    • as p-values are calculated assuming the null hypothesis is true

  • Work out the standardized test statistic

    • This allows you to use the appropriate probability tables

What is the significance level?

• The significance level, , is a probability boundary (threshold) at which you reject the null hypothesis

  • If the p-value is less than the significance level, you reject the null hypothesis

• It must be set before a hypothesis test is carried out

  • It will usually be 1%, 5% or 10%

    • written as , or

    • if it is not given in the question, use 5%

How do I conclude a hypothesis test?

• Conclusions to a hypothesis test need to contain two things:

  • a decision about the null hypothesis

  • an interpretation of this decision in the context of the question

• To make the decision, compare the p-value to the significance level

  • If then the null hypothesis should be rejected

  • If then the null hypothesis should not be rejected

• In a two-tailed test, double the p-value and compare this to

  • This is because there are two regions that are as extreme, or more extreme, than the one observed in the sample

How do I interpret a conclusion in context?

• One or two sentences are needed at the end that interpret the conclusion in context

• Start by saying whether the data provides sufficient evidence

  • or does not provide sufficient evidence

    • It must mention sufficient

    • It is not correct to say 'no / zero evidence' or 'this proves' or 'we accept that...'

• Then continue by writing out the relevant hypothesis in context

  • Copy out exact phrases from the question

    • Make sure the population parameter is mentioned in context

• For example, a good interpretation is:

  • 'The data provides sufficient evidence that the mean height of all trees on the island has increased'

  • but not:

    • 'this proves the heights of all trees have increased'

How do I lay out a hypothesis test in the exam?

• STEP 1

  Hypotheses:

  • 

  • or or

  • Define the population parameter in context

    • '...where is the ...'

  • State the significance level,

    • e.g.

• STEP 2
  Identify the appropriate inference procedure:

  • This is the type of test you want to perform

    • e.g. 'The appropriate inference procedure is a one-sided t-test for a population mean'

• STEP 3
  Verify any conditions relevant to that test

• STEP 4
  Find the p-value:

  • Calculate the standardized test statistic

    • 

  • Calculate the p-value

    • Mention any degrees of freedom where relevant

• STEP 5
  Conclusion:

  • Compare the p-value to

  • Provide a correct decision about

    • e.g. ' should be rejected'

  • Interpret the conclusion in context

    • e.g. the data provides sufficient evidence that the mean of ... etc