IB DP Maths: AI HL复习笔记4.12.1 Hypothesis Testing for Mean (One Sample)

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
  • H₀ : μ = μ₀
    • Clearly state that μ represents the population mean
    • μ₀  is the assumed population mean

• • For a one-tailed test H₁ : μ < μ₀ or H₁ : μ > μ₀
  • For a two-tailed test: H₁ : 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 H₁ : μ < μ₀)
    • The mean has increased (for H₁ : μ > μ₀)
    • The mean has changed (for H₁ : μ ≠ μ₀)
  • If you accept H­₀ then there is insufficient evidence to reject the null hypothesis which suggests that...
    • The mean has not decreased (for H₁ : μ < μ₀)
    • The mean has not increased (for H₁ : μ > μ₀)
    • The mean has not changed (for H₁ : 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

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
  • H₀ : μ = μ₀
    • Clearly state that μ represents the population mean
    • μ₀  is the assumed population mean

• • For a one-tailed test H₁ : μ < μ₀ or H₁ : μ > μ₀
  • For a two-tailed test: H₁ : 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 H₁ : μ < μ₀)
    • The mean has increased (for H₁ : μ > μ₀)
    • The mean has changed (for H₁ : μ ≠ μ₀)
  • If you accept H­₀then there is insufficient evidence to reject the null hypothesis which suggests that...
    • The mean has not decreased (for H₁ : μ < μ₀)
    • The mean has not increased (for H₁ : μ > μ₀)
    • The mean has not changed (for H₁ : μ ≠ μ₀)