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IB DP Maths: AI SL復(fù)習(xí)筆記4.7.4 The t-test

Category: IB課程, 教材筆記, 福利干貨 Date: 2022年7月19日 下午4:29

Two-Sample Tests

What is a?t-test?

  • A?t-test is used to?compare the means?of?two normally distributed?populations
  • In the exam the?population variance will always be unknown

What assumptions are needed for the?t-test?

  • The underlying distribution for?each variable?must be?normal
  • In the exam you will need to assume the variance for the two groups are equal
    • You will need to use the?pooled two-sample?t-test

What are the steps for a pooled two-sample?t-test?

  • STEP 1: Write the?hypotheses
    • H0?:?μx?=?μy
      • Where?μx?and?μy?are the?population means
      • Make sure you make it clear which mean corresponds to each population
      • In words this means the?two population means are equal
    • H1?:?μx?<?μy?or H1?:?μx?>?μy?or H1?:?μx?≠?μy
      • The alternative hypothesis will depend on what is being tested (see sections for one-tailed and two-tailed tests)
  • STEP 2: Enter the data into your GDC
    • Enter two lists of data – one for each sample
    • Choose the pooled option
    • Your GDC will then give you the?p-value
  • STEP 3: Decide whether there is?evidence to reject?the?null?hypothesis
    • Compare the?p-value?with the given?significance level
      • If?p-value?<?significance level then?reject H0
      • If?p-value?>?significance level then?accept H0
  • STEP 4: Write your?conclusion
    • If you?reject H0
      • There is sufficient evidence to suggest that the population mean of?X?is bigger than/smaller than/different to the population mean of?Y
      • This will depend on the alternative hypothesis
    • If you?accept H0
      • There is insufficient evidence to suggest that the population mean of?X?is bigger than/small than/different to the population mean of?Y
      • Therefore this suggests that the?population means are equal

One-tailed Tests

How do I perform a one-tailed?t-test?

  • A?one-tailed test?is used to test one of the two following cases:
    • The population mean of?X?is?bigger?than the population mean of?Y
      • The alternative hypothesis will be: H1?:?μx?>?μy
      • Look out for words such as increase, bigger, higher, etc
    • The population mean of?X?is?smaller?than the population mean of?Y
      • The alternative hypothesis will be: H1?:?μx?<?μy
      • Look out for words such as decrease, smaller, lower, etc
  • If you?reject the null?hypothesis then
    • This suggests that the population mean of?X?is?bigger?than the population mean of?Y
      • If the alternative hypothesis is H1?:?μx?>?μy
    • This suggests that the population mean of?X?is?smaller?than the population mean of?Y
      • If the alternative hypothesis is H1?:?μx?<?μy

Worked Example

The times (in minutes) for children and adults to complete a puzzle are recorded below.

Children 3.1 2.7 3.5 3.1 2.9 3.2 3.0 2.9
Adults 3.1 3.6 3.5 3.6 2.9 3.6 3.4 3.6 3.7 3.0

 

The creator of the puzzle claims children are generally faster at solving the puzzle than adults. A?t-test is to be performed at a 1% significance level.

a)Write down the null and alternative hypotheses.

4-7-4-ib-ai-sl-t-test-one-tail-a-we-solution

b)Find the p-value for this test.4-7-4-ib-ai-sl-t-test-one-tail-b-we-solution

c)State whether the creator’s claim is supported by the test. Give a reason for your answer.

4-7-4-ib-ai-sl-t-test-one-tail-c-we-solution

Two-tailed Tests

How do I perform a two-tailed?t-test?

  • A?two-tailed?test?is used to test the following case:
    • The population mean of?X?is?different?to the population mean of?Y
      • The alternative hypothesis will be: H1?:?μx≠?μy
      • Look out for words such as change, different, not the same, etc
  • If you?reject the null?hypothesis then
    • This suggests that the population mean of?X?is?different?to the population mean of?Y
    • You can not state which one is bigger as you were not testing for that
      • All you can conclude is that there is evidence that the means are not equal
      • To test whether a specific one is bigger you would need to use a one-tailed test

Worked Example

In a school all students must study either French or Spanish as well as maths. 18 students in a maths class complete a test and their scores are recorded along with which language they study.

Studies French 61 82 77 80 99 69 75 71 81
Studies Spanish 74 79 83 66 95 79 82 81 85

 

The maths teacher wants to investigate whether the scores are different between the students studying each language. A?t-test is to be performed at a 10% significance level.

a)Write down the null and alternative hypotheses.

4-7-4-ib-ai-sl-t-test-two-tail-a-we-solution

b)Find the?p-value for this test.

4-7-4-ib-ai-sl-t-test-two-tail-b-we-solution

c)Write down the conclusion to the test. Give a reason for your answer.

4-7-4-ib-ai-sl-t-test-two-tail-c-we-solution

 

 

 

 

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