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IB DP Maths: AI HL復習筆記4.12.1 Hypothesis Testing for Mean (One Sample)

Category: IB課程, 教材筆記, 福利干貨 Date: 2022年7月21日下午6:05

One-Sample z-tests

What is a one-sample?z-test?

A?one-sample?z-test?is used to?test the mean (μ)?of a?normally distributed?population
- You use a?z-test when the?population variance (σ2) is known

What are the steps for performing a one-sample?z-test on my GDC?

STEP 1:?Write the hypotheses
- H_0?:?μ?=?μ₀
  - Clearly state that?μ?represents the?population mean
  - μ_{0 ?}is the?assumed population mean

- For a?one-tailed?test H_1?:?μ?<?μ_0?or H_1?:?μ?>?μ₀
- For a?two-tailed?test: H_1?:?μ?≠?μ₀
  - The alternative hypothesis will depend on what is being tested

STEP 2:?Enter the data?into your GDC and choose the?one-sample?z-test
- If you have the raw data
  - Enter the data as a list
  - Enter the value of?σ
- If you have summary statistics
- Your GDC will give you the?p-value
STEP 3:?Decide?whether there is?evidence to reject the null hypothesis
- If the?p-value < significance level then reject H₀
STEP 4:?Write your conclusion
- If you?reject H₀-?then there is?evidence?to suggest that...
  - The mean has decreased (for H_1?:?μ?<?μ₀)
  - The mean has increased (for H_1?:?μ?>?μ₀)
  - The mean has changed (for H_1?:?μ?≠?μ₀)
- If you?accept H-₀_?then there is?insufficient evidence?to reject the null hypothesis which suggests that...
  - The mean has not decreased (for H_1?:?μ?<?μ₀)
  - The mean has not increased (for H_1?:?μ?>?μ₀)
  - The mean has not changed (for H_1?:?μ?≠?μ₀)

How do I find the?p-value for a one-sample?z-test using a normal distribution?

How do I find the critical value and critical region for a one-sample z-test?

Exam Tip

Exam questions might specify a method for you to use so practise all methods (using GDC,?p-values, critical regions)
If the exam question does not specify a method then use whichever method you want
- Make it clear which method you are using
- You can always use a second method as a way of checking your answer

Worked Examplea)State the null and alternative hypotheses to test Kamala’s claim.b)Using a 5% level of significance, find the critical region for this test.c)Kamala calculates the mean of the 25 cats included in her sample to be 4.65 kg. Determine the conclusion of the test.

One-Sample t-tests

What is a one-sample?t-test?

A?one-sample?t-test?is used to?test the mean (μ)?of a?normally distributed?population
- You use a?t-test when the?population variance (σ2) is unknownt-distributions are similar to normal distributions
- - As the sample size gets larger the?t-distribution tends towards the standard normal distribution
You won’t be expected to find the critical value
- The?p-value can be found using the test function on your GDC

What are the steps for performing a one-sample?t-test on my GDC?

STEP 1:?Write the hypotheses
- H_0?:?μ?=?μ₀
  - Clearly state that?μ?represents the?population mean
  - μ_{0 ?}is the?assumed population mean

- For a?one-tailed?test H_1?:?μ?<?μ_0?or H_1?:?μ?>?μ₀
- For a?two-tailed?test: H_1?:?μ?≠?μ₀
  - The alternative hypothesis will depend on what is being tested

STEP 2:?Enter the data?into your GDC and choose the?one-sample?t-test
- If you have the raw data
  - Enter the data as a list
- If you have summary statistics
- Your GDC will give you the?p-value
STEP 3:?Decide?whether there is?evidence to reject the null hypothesis
- If the?p-value < significance level then reject H₀
STEP 4:?Write your conclusion
- If you?reject H₀-?then there is?evidence?to suggest that...
  - The mean has decreased (for H_1?:?μ?<?μ₀)
  - The mean has increased (for H_1?:?μ?>?μ₀)
  - The mean has changed (for H_1?:?μ?≠?μ₀)
- If you?accept H-₀then there is?insufficient evidence?to reject the null hypothesis which suggests that...
  - The mean has not decreased (for H_1?:?μ?<?μ₀)
  - The mean has not increased (for H_1?:?μ?>?μ₀)
  - The mean has not changed (for H_1?:?μ?≠?μ₀)

Worked Example

The IQ of a student at Calculus High can be modelled as a normal distribution with mean 126. The headteacher decides to play classical music during lunchtimes and suspects that this has caused a change in the average IQ of the students.

a)State the null and alternative hypotheses to test the headteacher’s suspicion.

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